Patent Publication Number: US-7906965-B2

Title: Methods and apparatuses for estimating the elliptical cone of uncertainty

Description:
This application claims the benefit of U.S. Provisional Application No. 60/996,169 filed on Nov. 5, 2007, of the common assignee, entitled “Methods and Apparatuses for Estimating the Elliptical Cone of Uncertainty,” the contents of which are incorporated by reference in their entirety. 
    
    
     BACKGROUND 
     The following relates to the magnetic resonance arts. It particularly relates to the imaging, tracking, and displaying of neural fibers and fiber bundles by diffusion tensor magnetic resonance imaging (DT-MRI), and will be described with particular reference thereto. However, it will also find application in conjunction with the tracking and graphical rendering of other types of fibrous structures as well as with other imaging modalities such as single photon emission computed tomography imaging (SPECT), computed tomography (CT), positron emission tomography (PET), and also in crystallography and geophysics. 
     Nerve tissue in human beings and other mammals includes neurons with elongated axonal portions arranged to form neural fibers or fiber bundles, along which electrochemical signals are transmitted. In the brain, for example, functional areas defined by very high neural densities are typically linked by structurally complex neural networks of axonal fiber bundles. The axonal fiber bundles and other fibrous material are substantially surrounded by other tissue. 
     Diagnosis of neural diseases, planning for brain surgery, and other neurologically related clinical activities as well as research studies on brain functioning may benefit from non-invasive imaging and tracking of the axonal fibers and fiber bundles. In particular, diffusion tensor magnetic resonance imaging (DT-MRI) has been shown to provide image contrast that correlates with axonal fiber bundles. In the DT-MRI technique, diffusion-sensitizing magnetic field gradients are applied in the excitation/imaging sequence so that the magnetic resonance images include contrast related to the diffusion of water or other fluid molecules. By applying the diffusion gradients in selected directions during the excitation/imaging sequence, diffusion weighted images are acquired from which diffusion tensor coefficients are obtained for each voxel location in image space. 
     The intensity of individual voxels are fitted to calculate six independent variables in a 3×3 diffusion tensor. The diffusion tensor is then diagonalized to obtain three eigenvalues and corresponding three eigenvectors. The eigenvectors represent the local direction of the brain fiber structure at the voxel at issue. Within an imaging voxel, the directional information of white matter tracts is usually obtained from the major eigenvector of the diffusion tensor. 
     Using the local directional information, a global anatomical connectivity of white matter tracts is constructed. Each time the tracking connects a pixel to the next pixel, judgment is made whether the fiber is continuous or terminated based on randomness of the fiber orientation of the adjacent pixels. The interplay between the local directional and global structural information is crucial in understanding changes in white matter tracts. However, image noise may produce errors in the calculated tensor, and, hence, in its eigenvalues and eigenvectors. There is an uncertainty associated with every estimate of fiber orientation. Accumulated uncertainties in fiber orientation may lead to erroneous reconstruction of pathways. It is difficult to trace reliably in the uncertain areas. 
     It is desirable to determine and display the tract dispersion, e.g., the eigenvectors and the associated uncertainties. For example, the unit eigenvector may be displayed with a cone of uncertainty around its tip. This conveys the notion that the direction of fiber is not known precisely. 
     However, the methods known in the art are directed to computation and visualization of a circular cone of uncertainty. These methods are not suitable for practical computation and visualization of an elliptical cone of uncertainty. 
     There is a need for the apparatuses and methods to overcome the above-referenced problems and others. 
     BRIEF DESCRIPTION 
     One embodiment includes a magnetic resonance imaging apparatus, including: a magnetic resonance imaging scanner to acquire diffusion-weighted imaging data of a subject; a reconstruction engine to reconstruct the acquired diffusion-weighted imaging data into diffusion-weighted image representations of the subject; a diffusion tensor engine to construct a diffusion tensor map of an area of interest of the subject; an eigenvalue/eigenvector ordering engine to obtain and order pairs of eigenvectors and eigenvalues for at least one voxel in the diffusion tensor map; a covariance matrix determining engine to construct a covariance matrix of a major eigenvector for at least one voxel for which pairs of eigenvectors and eigenvalues have been obtained and ordered; a first normalized measure determining engine to compute a first normalized measure of at least one voxel for which the covariance matrix has been constructed; a second normalized measure determining engine to compute a second normalized measure of at least one voxel for which the covariance matrix has been constructed; and a rendering engine to generate a human-viewable display of an image representation of uncertainty. 
     One embodiment includes a magnetic resonance imaging method, including: acquiring a three-dimensional diffusion tensor map of an area of interest of a subject; processing the diffusion tensor for at least one voxel to obtain pairs of eigenvectors and eigenvalues; constructing a covariance matrix of a major eigenvector for at least one voxel; computing first and second normalized uncertainty measures of at least one voxel; and generating a human-viewable display of an image representation of uncertainty. 
     One embodiment includes a magnetic resonance imaging apparatus, including: means for acquiring diffusion-weighted imaging data of a subject; means for reconstructing the acquired diffusion-weighted imaging data into diffusion-weighted image representations of the subject; means for constructing a diffusion tensor map of an area of interest of the subject; means for obtaining and ordering eigenvectors and eigenvalues for at least one voxel; means for constructing a covariance matrix of a major eigenvector for at least one voxel; means for computing a first normalized uncertainty measure; means for computing a second normalized uncertainty measure; and means for generating a human-viewable display of an image representation of uncertainty. 
     One embodiment provides a computer-readable medium comprising software, which software, when executed by a computer system, causes the computer to perform operations to analyze diffusion-weighted imaging representations reconstructed from acquired diffusion-weighted imaging data of a subject, the computer-readable medium includes: instructions to construct a diffusion tensor map of an area of interest of the subject; instructions to obtain and order eigenvectors and eigenvalues for at least one voxel; instructions to construct a covariance matrix of a major eigenvector for at least one voxel; instructions to compute a first normalized uncertainty measure of at least one voxel; instructions to compute a second normalized uncertainty measure of at least one voxel; and instructions to generate a human-viewable display of an image representation of uncertainty. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and other features of various embodiments of the invention will be apparent from the following, more particular description of such embodiments of the invention, as illustrated in the accompanying drawings, wherein like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. 
         FIG. 1A  diagrammatically illustrates an exemplary magnetic resonance imaging system according to an exemplary embodiment of the invention; 
         FIG. 1B  diagrammatically illustrates a portion of an exemplary magnetic resonance imaging system according to an exemplary embodiment of the invention; 
         FIG. 1C  diagrammatically illustrates the eigenvectors and eigenvalues of the diffusion tensor and their relationship with axonal fibers or fiber bundles; 
         FIG. 2  illustrates a Gnomonic projection according to an exemplary embodiment of the invention; 
         FIG. 3  illustrates an inverse Gnomonic projection according to an exemplary embodiment of the invention; 
         FIG. 4  illustrates an exemplary a cone of uncertainty constructed according to an exemplary embodiment of the invention; 
         FIG. 5A  illustrates an exemplary normalized areal measure map; 
         FIG. 5B  illustrates an exemplary normalized circumference map; 
         FIG. 5C  illustrates an exemplary eccentricity map of the ellipse of the cone of uncertainty; 
         FIG. 5D  illustrates an exemplary cones of uncertainty of an axial slice corresponding to the normalized areal map of  FIG. 5A . 
     
    
    
     DETAILED DESCRIPTION 
     Exemplary embodiments are discussed in detail below. While specific exemplary embodiments are discussed, it should be understood that this is done for illustration purposes only. In describing and illustrating the exemplary embodiments, specific terminology is employed for the sake of clarity. However, the invention is not intended to be limited to the specific terminology so selected. A person skilled in the relevant art will recognize that other components and configurations may be used without parting from the spirit and scope of the invention. It is to be understood that each specific element includes all technical equivalents that operate in a similar manner to accomplish a similar purpose. Each reference cited herein is incorporated by reference. The examples and embodiments described herein are non-limiting examples. 
     With reference to  FIG. 1A , a magnetic resonance imaging scanner or system  100  may include a housing  102  defining a generally cylindrical scanner bore  104  defining an examination region  105 , inside of which an associated imaging subject  106  is disposed. The subject  106  may be, for example, a human patient, an animal, a phantom, etc. In  FIG. 1A , the housing  102  is shown in cross-section to illustrate the inside of the housing  102 . Main magnetic field coils  110  may be disposed inside the housing  102 , and may produce a main B 0  magnetic field parallel to a central axis  112  of the scanner bore  104 . In  FIG. 1A , the direction of the main B 0  magnetic field is parallel to the z-axis of the reference x-y-z Cartesian coordinate system. Main magnetic field coils  110  are typically superconducting coils disposed inside cryoshrouding  114 , although resistive main magnets may also be used. The main magnetic field coils  110  may generate the main B 0  magnetic field, which may be substantially uniform in an imaging volume of the bore  104 . 
     The housing  102  also houses or supports magnetic field gradient coil(s)  116  for selectively producing known magnetic field gradients parallel to the central axis  112  of the bore  104 , along in-plane directions transverse to the central axis  112 , or along other selected directions. In one embodiment, the gradient coil(s)  116  are shielded with shielding coil(s) (not shown). The shielding coils are designed to cooperate with the gradient coil  116  to generate a magnetic field which has a substantially zero magnetic flux density outside an area defined by the outer radius of the shielding coil(s). 
     The magnetic resonance imaging scanner  100  may include a radio frequency coil arrangement or system  120  disposed inside the bore  104  to selectively excite and/or detect magnetic resonances. In one embodiment, the coil system  120  may include a whole body coil  122  and a local coil positioned in a proximity to the intended volume of examination, such as a head coil  124 . The coils  122 ,  124  may include a TEM coil, a hybrid TEM-birdcage coil, a birdcage resonator, an arrangement of loop resonators, or the like. The coils  122 ,  124  may include a plurality of resonators positioned around or in the intended volume of examination. The coils  122 ,  124  may be, for example, circularly cylindrical, but, of course, might have other geometries, such as an elliptic cross-section, semi-circular cross-section, semi-elliptical cross-section, and the like. 
     An MRI controller  140  may operate magnetic field gradient controller or controllers  142  coupled to the gradients coils  116  to superimpose selected magnetic field gradients on the main magnetic field in the examination region, and may also operate a radio frequency transmitter or transmitters  144  each coupled to an individual radio frequency coil segment to inject selected radio frequency excitation pulses at about the magnetic resonance frequency into the examination region for imaging. The radio frequency transmitter or transmitters  144  may be individually controlled and may have different phases and amplitudes. 
     Magnetic resonance is generated and spatially encoded in at least a portion of a region of interest of the imaging subject  106 . By applying selected magnetic field gradients via the gradient coils  116 , a selected k-space trajectory is traversed, such as a Cartesian trajectory, a plurality of radial trajectories, or a spiral trajectory. Alternatively, imaging data may be acquired as projections along selected magnetic field gradient directions. During imaging data acquisition, a radio frequency receiver or receivers  146  coupled to the head coil  124  may acquire magnetic resonance samples that are stored in a magnetic resonance data memory  150 . Of course, it is contemplated that the magnetic resonance signals may be excited and received by a single coil array such as the whole body coil  122  or the local coil  124 . 
     The MRI sequence may include a complex series of magnetic field gradient pulses and/or sweeps generated by the gradient coils  116  which along with selected RF pulses generated by the coils  122 ,  124  result on magnetic resonance echoes. For diffusion tensor magnetic resonance imaging (DT-MRI), data is acquired without diffusion weighting, and with diffusion weighting in N directions. Six diffusion directions may provide sufficient information to construct the diffusion tensor at each voxel. A reconstruction engine  152  may reconstruct the imaging data into an image representation, e.g., diffusion-weighted image representations. The reconstructed image generated by the reconstruction engine  152  may be stored in an image memory  154 . 
     A diffusion tensor engine  160  may calculate a plurality of diffusion coefficients values on a per voxel basis, as known in the art, to construct a diffusion tensor map  162 . For each voxel, a tensor, e.g., a symmetric positive definite 3×3 matrix, that describes a three dimensional shape of diffusion may be calculated. The diffusion tensor at each voxel may be processed to obtain eigenvectors and eigenvalues. 
     As described in a greater detail below, a covariance matrix determining engine  164  may construct a covariance matrix of, for example, the major eigenvector at each voxel. It is contemplated that covariance matrices of the medium and minor eigenvectors may be constructed as well. A cone of uncertainty (COU) determining engine  170  may construct or determine a cone of uncertainty (COU) based on the covariance matrix. A first normalized measure engine  172  may determine a first normalized measure of the elliptical COU. A second normalized measure engine  174  may determine a second normalized measure or a normalized circumference of the elliptical COU. 
     A rendering engine  176  may format the constructed cones of uncertainty, normalized areal maps, normalized circumference maps, or any other appropriate image representations to be displayed on a user interface  180 , stored in non-volatile memory, transmitted over a local intranet or the Internet, viewed, stored, manipulated, or so forth. 
     Major eigenvector maps represent the fiber orientation and may be visualized as vector-field or color coded maps giving a cartography of the tracts&#39; position and direction. The brightness may be weighted by a fractional anisotropy which is a scalar measure of a degree of anisotropy for a given voxel. To visualize the fiber direction map in the context of a conventional structural image, the fiber map may be registered to and superimposed on a structural image, for example, on a high resolution MR image, as known in the art. The fiber orientation information provided by the diffusion imaging maps may be used to differentiate among nerve fiber pathways to determine abnormalities. Further, the fiber orientation information provided by the diffusion imaging maps may be used to delineate anatomical structures based on a priori distinct fiber architecture of each anatomical structure. 
     The user interface  180  may also enable a radiologist, technician, or other operator of the magnetic resonance imaging scanner  100  to communicate with the magnetic resonance imaging controller  140  to select, modify, and execute magnetic resonance imaging sequences. 
     With continuing reference to  FIG. 1A  and further reference to  FIGS. 1B and 1C , an eigenvector/eigenvalue ordering engine  182  may order the diffusion tensor eigenvectors and eigenvalues for each voxel to obtain ordered eigenvectors and eigenvalues. For example, the eigenvector/eigenvalue ordering engine  182  may order eigenvalues from largest to smallest eigenvalue λ 1 , λ 2 , λ 3 . The eigenvector which corresponds to the largest eigenvalue λ 1  is called the major eigenvector, and aligns with the spatial direction having the highest diffusion coefficient. The medium and smallest eigenvalues λ 2 , λ 3  correspond to the medium and minor eigenvectors. The medium and minor eigenvectors are orthogonal to the major eigenvector and align with spatial directions having lower diffusion coefficients. The relative values of the eigenvalues λ 1 , λ 2 , λ 3  are indicative of the spatial orientation and magnitude of the diffusion tensor anisotropy. 
     As shown in  FIG. 1C , the eigenvectors and eigenvalues are geometrically representable by an ellipsoid  183  whose long axis aligns with the major eigenvector e 1 , i.e. with the direction of the highest apparent diffusion coefficient. The deviation of the ellipsoid  183  from a perfect sphere is representative of the anisotropy of the diffusion tensor. An anisotropic diffusion coefficient tensor may reflect the influence of neural fiber bundles  184  which tend to inhibit diffusion in directions partially or totally orthogonal to the fibers  184 , e.g. the directions of eigenvectors e 2 , e 3 . In contrast, diffusion parallel to the fibers  184 , i.e. channeled along the direction of the major eigenvector e 1 , is enhanced and larger than along the e 2 , e 3  directions. 
     Generally, the diffusion tensor, D, is a 3 by 3 symmetric positive definite matrix given by: 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             D 
                             xx 
                           
                         
                         
                           
                             D 
                             xy 
                           
                         
                         
                           
                             D 
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                   ( 
                   
                     1 
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     In the discussion below, the measured (noisy) diffusion-weighted signals are denoted by s i . The diffusion-weighted function evaluated at γ, is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         s 
                         ^ 
                       
                       i 
                     
                     ⁡ 
                     
                       ( 
                       γ 
                       ) 
                     
                   
                   = 
                   
                     exp 
                     [ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         7 
                       
                       ⁢ 
                       
                         
                           W 
                           ij 
                         
                         ⁢ 
                         
                           γ 
                           j 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     1 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     The diffusion-weighted function evaluated at γ(ξ) is: 
                         s   ^     i     ⁡     (     γ   ⁡     (   ξ   )       )       =     exp   [       ∑     j   =   1     7     ⁢       W   ij     ⁢       γ   j     ⁡     (   ξ   )           ]             (     1   ⁢   C     )               
where γ is the ordinary representation of the diffusion tensor together with the logarithm of the reference signal:
 
                     γ   =         [           γ   1           γ   2           γ   3           γ   4           γ   5           γ   6           γ   7           ]     T     ⁢     
     ⁢           =       [           ln   ⁡     (     s   0     )             D   xx           D   yy           D   zz           D   xy           D   yz           D   xz           ]     T         ,           (     1   ⁢   D     )               
and γ(ξ) is the ordinary representation of the diffusion tensor in terms of the Euler diffusion tensor representation, where the Euler representation of the diffusion tensor is:
 
                   ξ   =         [           ξ   1           ξ   2           ξ   3           ξ   4           ξ   5           ξ   6           ξ   7           ]     T     ⁢     
     ⁢           =       [           ln   ⁡     (     s   0     )             λ   1           λ   2           λ   3         θ       ϕ       ψ         ]     T               (     1   ⁢   E     )               
where s 0  is the parameter for the non-diffusion weighted signal,
     λ 1 , λ 2 , and λ 3  denote major, medium, and minor eigenvalues, respectively, and   θ, φ, and ψ denote Euler angles.   

     Suppose that {circumflex over (γ)} NLS  is the nonlinear least square estimate of the parameter vector γ, then the residual vector is a vector whose individual component is the difference between the observed and the expected or estimated signals:
 
 r ({circumflex over (γ)} NLS )=[ r   1 ({circumflex over (γ)} NLS ) . . .  r   n ({circumflex over (γ)} NLS )] T ,  (1F)
 
where r i ({circumflex over (γ)} NLS )=s i −ŝ i ({circumflex over (γ)} NLS ) for i=1, . . . ,n.
 
     The design matrix, W, is given by: 
     
       
         
           
               
             
               
                 
                   
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     The objective functions for the nonlinear least squares problem in the diffusion tensor imaging (DTI) in the ordinary diffusion tensor representation and Euler diffusion tensor representation may be expressed, respectively: 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         NLS 
                       
                       ⁡ 
                       
                         ( 
                         γ 
                         ) 
                       
                     
                     = 
                     
                       
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                                       7 
                                     
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                                         ij 
                                       
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                       ⁡ 
                       
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                             ) 
                           
                         
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                   , 
                 
               
               
                 
                   ( 
                   2 
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     The covariance matrix Σ γ  of the ordinary diffusion tensor representation may be defined as:
 
Σ γ =σ DW   2   [W   T ( Ŝ   2   −RŜ ) W]   −1 ,  (3)
 
     The covariance matrix of the Euler diffusion tensor representation may be derived from the covariance matrix of the ordinary diffusion tensor representation as described below. Generally, the covariance matrix Σ ξ  of the Euler diffusion tensor representation may be defined as:
 
Σ ξ =σ DW   2   [J   ξ   T (γ({circumflex over (ξ)})) W   T ( Ŝ   2   =RŜ ) WJ   ξ (γ({circumflex over (ξ)}))] −1   (4A)
 
where the matrix or vector transposition is denoted by superscript T,
     J ξ  denotes a Jacobian matrix,   S and Ŝ are diagonal matrices including diagonal elements,   R is a residual matrix, and   σ DW   2  is the estimated variance.   

     A first diagonal matrix S includes observed diffusion weighted signals: 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             s 
                             1 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           ⋱ 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
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                             n 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     4 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     A second diagonal matrix Ŝ includes the estimated diffusion weighted signals, respectively: 
     
       
         
           
             
               
                 
                   
                     S 
                     ^ 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               s 
                               ^ 
                             
                             1 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           ⋱ 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
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                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     4 
                     ⁢ 
                     C 
                   
                   ) 
                 
               
             
           
         
       
     
     The residual matrix R is equal to a difference between the first and second diagonal matrices:
 
 R=S−Ŝ   (4D)
 
     The element of the Jacobian matrix, J ξ (γ) may be defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             J 
                             ξ 
                           
                           ⁡ 
                           
                             ( 
                             γ 
                             ) 
                           
                         
                         ] 
                       
                       ij 
                     
                     ≡ 
                     
                       
                         ∂ 
                         
                           γ 
                           i 
                         
                       
                       
                         ∂ 
                         
                           ξ 
                           j 
                         
                       
                     
                   
                   ; 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     4 
                     ⁢ 
                     E 
                   
                   ) 
                 
               
             
           
         
       
     
     The estimated variance σ DW   2  may be derived from the nonlinear fit (Equation 1H):
 
σ DW   2 =2 f   NLS ({circumflex over (γ)} NLS )/( n −7).  4F)
 
     The core idea of error propagation is to transform one covariance matrix to another covariance matrix of interest through an appropriate mapping between the underlying representations. In DT-MRI, the most fundamental covariance matrix is Σ γ , from which other covariance matrices may be obtained. 
     For example, the Euler covariance matrix Σ ξ  may be constructed from the ordinary covariance matrix Σ γ  as follows:
 
Σ ξ   =J   γ (ξ({circumflex over (γ)}))Σ γ   J   γ   T (ξ({circumflex over (γ)})),  (5)
 
where J γ (ξ({circumflex over (γ)})),
 
                     [       J   ξ     ⁡     (     ξ   ⁡     (     γ   ^     )       )       ]     ij     ≡       ∂       ξ   i     ⁡     (   γ   )           ∂     γ   j           ⁢     |     γ   =     γ   ^           ,         
is the Jacobian matrix of ξ with respect to γ evaluated at {circumflex over (γ)}.
 
     The Jacobian matrix is a locally linear map from dγ to dξ at {circumflex over (γ)}:
 
 dξ=J   γ (ξ({circumflex over (γ)})) dγ   (6A)
 
     As shown, the Jacobian matrix between representations plays a critical role in transforming one covariance matrix to another and the Jacobian matrix is, in turn, dependent upon the mapping between representations. 
     Equations (5) and (4A) are equivalent in principle. For example, equation (4A) may be derived by substituting equation (3) into equation (5): 
     
       
         
           
             
               Σ 
               ξ 
             
             = 
             
               
                 
                   
                     J 
                     γ 
                   
                   ⁡ 
                   
                     ( 
                     
                       ξ 
                       ⁡ 
                       
                         ( 
                         
                           γ 
                           ^ 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   Σ 
                   γ 
                 
                 ⁢ 
                 
                   
                     J 
                     γ 
                     T 
                   
                   ⁡ 
                   
                     ( 
                     
                       ξ 
                       ⁡ 
                       
                         ( 
                         
                           γ 
                           ^ 
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 
 
               
               ⁢ 
               
                   
               
               = 
               
                 
                   
                     σ 
                     DW 
                     2 
                   
                   ⁢ 
                   
                     
                       
                         
                           J 
                           γ 
                         
                         ⁡ 
                         
                           ( 
                           
                             ξ 
                             ⁡ 
                             
                               ( 
                               
                                 γ 
                                 ^ 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       [ 
                       
                         
                           
                             W 
                             T 
                           
                           ( 
                           
                             
                               
                                 S 
                                 ^ 
                               
                               2 
                             
                             - 
                             
                               R 
                               ⁢ 
                               
                                 S 
                                 ^ 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         W 
                       
                       ] 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       J 
                       γ 
                       T 
                     
                     ⁡ 
                     
                       ( 
                       
                         ξ 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             ^ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                     
                 
                 = 
                 
                   
                     
                       
                         
                           
                             
                               σ 
                               DW 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   J 
                                   γ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ξ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         γ 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         [ 
                         
                           
                             
                               W 
                               T 
                             
                             ( 
                             
                               
                                 
                                   S 
                                   ^ 
                                 
                                 2 
                               
                               - 
                               
                                 R 
                                 ⁢ 
                                 
                                   S 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           W 
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             J 
                             γ 
                             T 
                           
                           ⁡ 
                           
                             ( 
                             
                               ξ 
                               ⁡ 
                               
                                 ( 
                                 
                                   γ 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   = 
                   
                     
                       
                         
                           σ 
                           DW 
                           2 
                         
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   J 
                                   γ 
                                   T 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ξ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         γ 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   W 
                                   T 
                                 
                                 ( 
                                 
                                   
                                     
                                       S 
                                       ^ 
                                     
                                     2 
                                   
                                   - 
                                   
                                     R 
                                     ⁢ 
                                     
                                       S 
                                       ^ 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               W 
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   J 
                                   γ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ξ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         γ 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         
                           σ 
                           DW 
                           2 
                         
                         [ 
                         
                           
                             
                               J 
                               ξ 
                               T 
                             
                             ( 
                             
                               γ 
                               ⁡ 
                               
                                 ( 
                                 
                                   ξ 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               W 
                               T 
                             
                             ( 
                             
                               
                                 
                                   S 
                                   ^ 
                                 
                                 2 
                               
                               - 
                               
                                 R 
                                 ⁢ 
                                 
                                   S 
                                   ^ 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               WJ 
                               ξ 
                             
                             ( 
                             
                               γ 
                               ⁡ 
                               
                                 ( 
                                 
                                   ξ 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
             
           
         
       
     
     In the above derivation, the following well-known identity is used:
 
 J   ξ (γ({circumflex over (ξ)}))· J   γ (ξ({circumflex over (γ)}))= J   ξ (ξ({circumflex over (ξ)}))= I   (6B)
 
     Therefore:
 
 [J   γ (ξ({circumflex over (γ)}))] −1   =J   ξ (γ({circumflex over (ξ)}))  (6C)
 
     In practice, it is easier to compute J ξ (γ({circumflex over (ξ)})) than J γ (ξ({circumflex over (γ)})). An explicit mapping between representations is usually needed to construct the Jacobian matrix. Occasionally, such a mapping may not exist or may be too complicated to shed light on the problem at hand. If this is the case then the first order matrix perturbation method may be helpful in finding the Jacobian matrix in the form of dξ=J γ (ξ({circumflex over (γ)}))dγ without requiring an explicit mapping ξ(γ). 
     The covariance matrix determining engine  164  may construct an ordinary covariance matrix  185  of the major eigenvector based on the nonlinear least squares method by using a reformulated first order matrix perturbation method. A notation, a(·,·) is used below to simplify the discussion. 
     Let 
                     Q   =     (           Q   11           Q   12           Q   13               Q   21           Q   22           Q   23               Q   31           Q   32           Q   33           )       ,           (     7   ⁢   A     )                   q   1     =       (           q     1   ⁢   x                 q     1   ⁢   y                 q     1   ⁢   z             )     =     (           Q   11               Q   21               Q   31           )         ,           (     7   ⁢   B     )                   q   2     =       (           q     2   ⁢   x                 q     2   ⁢   y                 q     2   ⁢   z             )     =     (           Q   12               Q   22               Q   32           )         ,     
     ⁢   and           (     7   ⁢   C     )                   q   3     =       (           q     3   ⁢   x                 q     3   ⁢   y                 q     3   ⁢   z             )     =     (           Q   13               Q   23               Q   33           )         ,           (     7   ⁢   D     )               
where q 1 , q 2  and q 3  are major, medium and minor eigenvectors, respectively.
 
     The quadratic form, q i   T ·D·q j , as a dot product between two vectors, may be presented by:
 
 q   i   T   ·D·q   j   =a ( q   i   ,q   j ) T ·β  (7E)
 
where
 
 a ( q   i   ,q   j ) T   ≡[q   ix   q   jx    q   iy   q   jy    q   iz   q   jz    q   ix   q   jy   +q   iy   q   jx    q   iy   q   jz   +q   iz   q   jy    q   ix   q   jz   +q   iz   q   jx ]  (8)
 
and
 
β≡[ D   xx    D   yy    D   zz    D   xy    D   yz    D   xz ].  (9)
 
     The first order matrix perturbation method begins, respectively, with the eigenvalue equation and the orthonormality condition:
 
 D·q   i =λ i   q   i ,  (10)
 
and
 
q i   T q j =δ ij ,  (11)
 
where δ ij  is the Kronecker delta function, which assumes unity if i=j or is equal to zero otherwise.
 
     Assuming a small variation on both sides of equation (10), e.g. δ(D·q i )=δ(λ i q i ):
 
δ D·q   i   +D·δq   i =δλ i   q   i +λ i   δq   i  for  i, j =1,2,3  (12)
 
     Assuming a small variation on both sides of equation (11), e.g. δ(q i   T q j )=δ(δ ij );
 
δ q   i   T   ·q   j   +q   i   T   ·δq   j =δ(δ ij )=0 for  i, j =1,2,3  (13)
 
     Equation (13) implies that:
 
δ q   i   T   ·q   j   =−q   i   T   ·δq   j  for  i≠j,   (14)
 
     If q i   T q i =1:
 
 q   i   T   ·δq   i =0 or δ q   i   T   ·q   i =0 for  i =1,2,3  (15)
 
     Taking the dot product between equation (12) with q j   T : 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             q 
                             j 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           D 
                           · 
                           
                             q 
                             i 
                           
                         
                       
                       + 
                       
                         
                           
                             q 
                             j 
                             T 
                           
                           · 
                           D 
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           i 
                         
                       
                     
                     = 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           λ 
                           i 
                         
                         ⁢ 
                         
                           
                             
                               q 
                               j 
                               T 
                             
                             · 
                             
                               q 
                               i 
                             
                           
                           
                             ︸ 
                             0 
                           
                         
                       
                       + 
                       
                         
                           λ 
                           i 
                         
                         ⁢ 
                         
                           
                             q 
                             j 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           i 
                         
                       
                     
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             q 
                             j 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           D 
                           · 
                           
                             q 
                             i 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   D 
                                   T 
                                 
                                 · 
                                 
                                   q 
                                   j 
                                 
                               
                               ) 
                             
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           i 
                         
                       
                     
                     = 
                     
                       
                         λ 
                         i 
                       
                       ⁢ 
                       
                         
                           q 
                           j 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           q 
                           i 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Since the diffusion tensor D is symmetric, equation (16) may be further reduced by taking into account equation (10): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             q 
                             j 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           D 
                           · 
                           
                             q 
                             i 
                           
                         
                       
                       + 
                       
                         
                           
                             λ 
                             j 
                           
                           · 
                           
                             q 
                             j 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           i 
                         
                       
                     
                     = 
                     
                       
                         λ 
                         i 
                       
                       ⁢ 
                       
                         
                           q 
                           j 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         i 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           q 
                           j 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         i 
                       
                     
                     = 
                     
                       
                         1 
                         
                           ( 
                           
                             
                               λ 
                               i 
                             
                             - 
                             
                               λ 
                               j 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           q 
                           j 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         D 
                         · 
                         
                           q 
                           i 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           q 
                           j 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         i 
                       
                     
                     = 
                     
                       
                         1 
                         
                           ( 
                           
                             
                               λ 
                               i 
                             
                             - 
                             
                               λ 
                               j 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   q 
                                   j 
                                 
                                 , 
                                 
                                   q 
                                   i 
                                 
                               
                               ) 
                             
                           
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       β 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Although equation (18) differs from equation (6A) defining the transformation relationship between γ and ζ via the Jacobian matrix, equation (18) may be reformulated to achieve the form of equation (6), e.g. δq i =J β (q i )δβ. 
     Assuming that the relationship between the eigenvalues is λ 1 &gt;λ 2 &gt;λ 3 , and solving for the dispersion of the major eigenvector q 1 , e.g., i=1, equation (18) results in equations (19)-(21). Equation (19) may be derived from equation (15):
 
 q   1   T   ·δq   1 =[0 0 0 0 0 0]·δβ  (19)
 
     Equations (20)-(21) may be derived from equations (17)-(18): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           q 
                           2 
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         1 
                       
                     
                     = 
                     
                       
                         1 
                         
                           ( 
                           
                             
                               λ 
                               1 
                             
                             - 
                             
                               λ 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           
                             a 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   q 
                                   2 
                                 
                                 , 
                                 
                                   q 
                                   1 
                                 
                               
                               ) 
                             
                           
                           T 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       β 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         q 
                         3 
                         T 
                       
                       · 
                       δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       q 
                       1 
                     
                   
                   = 
                   
                     
                       1 
                       
                         ( 
                         
                           
                             λ 
                             1 
                           
                           - 
                           
                             λ 
                             3 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         
                           a 
                           ⁡ 
                           
                             ( 
                             
                               
                                 q 
                                 3 
                               
                               , 
                               
                                 q 
                                 1 
                               
                             
                             ) 
                           
                         
                         T 
                       
                       · 
                       δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       β 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Equations (19-21) may be combined into a single matrix expression: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       q 
                                       1 
                                       T 
                                     
                                   
                                 
                                 
                                   
                                     
                                       q 
                                       2 
                                       T 
                                     
                                   
                                 
                                 
                                   
                                     
                                       q 
                                       3 
                                       T 
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               ︸ 
                               
                                 Q 
                                 T 
                               
                             
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           1 
                         
                       
                       = 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         1 
                                         
                                           ( 
                                           
                                             
                                               λ 
                                               1 
                                             
                                             - 
                                             
                                               λ 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           a 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 q 
                                                 2 
                                               
                                               , 
                                               
                                                 q 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         T 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         1 
                                         
                                           ( 
                                           
                                             
                                               λ 
                                               1 
                                             
                                             - 
                                             
                                               λ 
                                               3 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           a 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 q 
                                                 3 
                                               
                                               , 
                                               
                                                 q 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         T 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               ︸ 
                               
                                 S 
                                 1 
                               
                             
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     or 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             Q 
                             T 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           q 
                           1 
                         
                       
                       = 
                       
                         
                           
                             S 
                             1 
                           
                           · 
                           δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     or 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         1 
                       
                     
                     = 
                     
                       
                         Q 
                         · 
                         
                           S 
                           1 
                         
                         · 
                         δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     In the above derivation, the orthonormality condition is used, e.g. QQ T =I. The dispersions of the medium and minor eigenvectors may be similarly derived. 
     The above reformulation based on β may be extended to γ resulting in:
 
δ q   1   =Q·T   1 ·δγ  (23A)
 
where T 1  is a 3×7 matrix given by
 
     
       
         
           
             
               
                 
                   
                     T 
                     1 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                         
                       
                       
                         
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               
                                 ( 
                                 
                                   
                                     λ 
                                     1 
                                   
                                   - 
                                   
                                     λ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       q 
                                       2 
                                     
                                     , 
                                     
                                       q 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               T 
                             
                           
                         
                       
                       
                         
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               
                                 ( 
                                 
                                   
                                     λ 
                                     1 
                                   
                                   - 
                                   
                                     λ 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       q 
                                       3 
                                     
                                     , 
                                     
                                       q 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               T 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     23 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     In terms of the Jacobian matrix for β:
 
 Q·S   1   =J   β ( q   1 )  (24)
 
     In terms of the Jacobian matrix for γ:
 
 Q·T   1   =J   γ ( q   1 )  (25)
 
     Finally, the covariance matrix Σ q     1    of the major eigenvector q 1  obtained via the reformulated first order matrix perturbation method may be determined as:
 
Σ q     1   ({circumflex over (γ)})= J   γ ( q   1 )Σ γ ({circumflex over (γ)}) J   γ   T ( q   1 )  (26)
 
     Optionally or alternatively, the covariance matrix Σ q     1    derived in equation (26) may be derived based on the error propagation method, known in the art. The covariance matrix of the major eigenvector of the diffusion tensor using the Euler diffusion tensor representation is:
 
Σ q     1     =J   ξ ( q   1 ({circumflex over (ξ)}))Σ ξ   J   ξ   T ( q   1 ({circumflex over (ξ)})),  (27)
 
     Substituting equation (4A) into equation (27) results in: 
                           Σ     q   1       =       ⁢       σ   DW   2     ⁢           J   ξ     (       q   1     ⁡     (     ξ   ^     )       )     [         J   ξ   T     (     γ   ⁡     (     ξ   ^     )       )     ⁢       W   T     (         S   ^     2     -     R   ⁢           ⁢     S   ^         )     ⁢       WJ   ξ     (     γ   ⁡     (     ξ   ^     )       )       ]       -   1       ⁢       J   ξ   T     (       q   1     ⁡     (     ξ   ^     )       )                     Σ     q   1       =       ⁢           J   ξ     (       q   1     ⁡     (     ξ   ^     )       )     ⁡     [       J   ξ     (     γ   ⁡     (     ξ   ^     )       )     ]         -   1                       ⁢                 σ   DW   2     [         W   T     (         S   ^     2     -     R   ⁢           ⁢     S   ^         )     ⁢   W     ]       -   1         ︸     Σ   γ         [       J   ξ   T     (     γ   ⁡     (     ξ   ^     )       )     ]       -   1       ⁢       J   ξ   T     (       q   1     ⁡     (     ξ   ^     )       )                     Σ     q   1       =       ⁢         J   ξ     (       q   1     ⁡     (     ξ   ^     )       )     ⁢       J   γ     ⁡     (     ξ   ⁡     (     γ   ^     )       )       ⁢     Σ   γ     ⁢       J   γ   T     ⁡     (     ξ   ⁡     (     γ   ^     )       )       ⁢       J   ξ   T     (       q   1     ⁡     (     ξ   ^     )       )                     Σ     q   1       =       ⁢         J   γ     ⁡     (       q   1     ⁡     (     γ   ^     )       )       ⁢     Σ   γ     ⁢       J   γ   T     ⁡     (       q   1     ⁡     (     γ   ^     )       )                       (     28   ⁢   A     )               
where J ξ (q 1 ({circumflex over (ξ)}))J γ (ξ({circumflex over (γ)}))=J γ (q 1 ({circumflex over (γ)})), and
 
 [J   ξ (γ({circumflex over (ξ)}))] −1   =J   γ (ξ({circumflex over (γ)})).
 
     Jacobian matrix J γ (q 1 ({circumflex over (γ)})) may be constructed as described above. 
     In one embodiment, the cone of uncertainty determining engine  170  may construct the elliptical cone of uncertainty as follows. From equation (28A), the 1−α confidence region for the major eigenvector q 1  is an ellipsoid given by:
 
( q   1   −q   1 ({circumflex over (γ)})) T Σ q     1     + ({circumflex over (γ)})( q   1   −q   1 ({circumflex over (γ)}))≦2 (2 ,n− 7;α)  (28B)
 
where  (2,n−7;α) is the upper α quantile for an   distribution with 2 and n−7 degrees of freedom.
 
     The plus sign on Σ q     1     +  denotes the matrix pseudoinverse which is needed because the matrix rank of Σ q     1    is two rather than three. 
     Equivalently:
 
( q   1   −q   1 ({circumflex over (γ)})) T (2 (2 ,n− 7;α)Σ q     1   ({circumflex over (γ)})) + ( q   1   −q   1 ({circumflex over (γ)}))≦1  (28C)
 
     Because the eigenvectors of the diffusion tensor are orthogonal and the eigenvectors of Σ q     1   ({circumflex over (γ)}) are the singular vectors, equation (28C) may be simplified:
 
 q   1   T (2 (2 ,n− 7;α)Σ q     1   ({circumflex over (γ)})) +   q   1 ≦1  (28D)
 
     Let the eigenvalue decomposition of Σ 1     1   ({circumflex over (γ)}) be:
 
Σ q     1   ({circumflex over (γ)})=ω 1   c   1   c   1   T +ω 2   c   2   c   2   T +0 q   1 ({circumflex over (γ)}) q   1   T  ({circumflex over (γ)}),  (28E)
 
where the last term is due to Q·T 1 =J γ (q 1 ) and equation (26).
 
     The principal directions of the ellipse of the COU whose area corresponds to the 1−α joint confidence region for the major eigenvector q 1  may be found by selecting q 1  to be equal to one of:
 
√{square root over (2 (2 ,n −7;α)ω 1 )} c   1  or √{square root over (2 (2 ,n −7;α)ω 2 )} c   2   (28F)
 
     The term  (2,n−7;α) may be found by solving equation (28G): 
                     α   -       I       (     n   -   7     )     /     (     n   -   7   +     2   ⁢       )         ⁡     (         (     n   -   7     )     /   2     ,   1     )         =   0           (     28   ⁢   G     )               
where I x (a,b) is the incomplete Beta function.
 
     With continuing reference to  FIGS. 1A and 1B  and further reference to  FIG. 2 , the first normalized measure determining engine  170  may determine the normalized first or areal measure Γ. More specifically, a simple closed or Jordan curve  200  on the unit sphere  210  divides the unit sphere into first and second regions  220 ,  230 . For example, the first region  220  is greater than the second region  230 . The first normalized measure determining engine  172  may determine the normalized areal measure Γ as a ratio of an area of the second, smaller region  230  on the unit sphere, that is enclosed by the simple closed curve  200  whose Gnomonic or central projection is on the u, v-plane  240 , to an area of a hemisphere. 
     Let s be a point on an upper hemisphere whose Gnomonic projection on the u, v-plane  240  is the point p. Let p be denoted in a vector form by: 
     
       
         
           
             
               
                 
                   p 
                   = 
                   
                     ( 
                     
                       
                         
                           u 
                         
                       
                       
                         
                           v 
                         
                       
                       
                         
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     The point s in a vector form may be obtained by normalizing p to a unit length: 
     
       
         
           
             
               
                 
                   
                     s 
                     ≡ 
                     
                       ( 
                       
                         
                           
                             
                               X 
                               s 
                             
                           
                         
                         
                           
                             
                               Y 
                               s 
                             
                           
                         
                         
                           
                             
                               Z 
                               s 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       p 
                       
                         
                           
                             p 
                             T 
                           
                           · 
                           p 
                         
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               u 
                               
                                 
                                   
                                     u 
                                     2 
                                   
                                   + 
                                   
                                     v 
                                     2 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 
                                   
                                     u 
                                     2 
                                   
                                   + 
                                   
                                     v 
                                     2 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               
                                 
                                   
                                     u 
                                     2 
                                   
                                   + 
                                   
                                     v 
                                     2 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     30 
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, the components of s are functions of u and v. 
     Suppose that major and minor axes  242 ,  244  of the ellipse  246  coincide with the u and the v axes of the u, v-plane. Analogously to equation (28F), let a length of the major and the minor axes  242 ,  244  be a and b, respectively, where:
 
 a =√{square root over (2 (2 ,n −7;α)ω 1 )} and  (30B)
 
 b =√{square root over (2 (2 ,n −7;α)ω 2 )}  (30C)
 
     The normalized areal measure, Γ, which is a dimensionless quantity, has a range of values from zero to unity: 
                   Γ   =       1     2   ⁢   π       ⁢     ∫       ∫   R     ⁢           ⁢         EG   -     F   2         ⁢     ⅆ   u     ⁢     ⅆ   v                     (     31   ⁢   A     )               
where R is the smaller region  230 .
 
     A region R may be is defined by:
 
( u/a ) 2 +( v/b ) 2 ≦1  (31B)
 
     The coefficients of the first fundamental form are: 
     
       
         
           
             
               
                 
                   
                     E 
                     = 
                     
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 X 
                                 s 
                               
                             
                             
                               ∂ 
                               u 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 Y 
                                 s 
                               
                             
                             
                               ∂ 
                               u 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 Z 
                                 s 
                               
                             
                             
                               ∂ 
                               u 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     C 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     G 
                     = 
                     
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 X 
                                 s 
                               
                             
                             
                               ∂ 
                               v 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 Y 
                                 s 
                               
                             
                             
                               ∂ 
                               v 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               ∂ 
                               
                                 Z 
                                 s 
                               
                             
                             
                               ∂ 
                               v 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     D 
                   
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             X 
                             s 
                           
                         
                         
                           ∂ 
                           u 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             X 
                             s 
                           
                         
                         
                           ∂ 
                           v 
                         
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             Y 
                             s 
                           
                         
                         
                           ∂ 
                           u 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             Y 
                             s 
                           
                         
                         
                           ∂ 
                           v 
                         
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             Z 
                             s 
                           
                         
                         
                           ∂ 
                           u 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             
                               Z 
                               s 
                             
                           
                           
                             ∂ 
                             v 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     E 
                   
                   ) 
                 
               
             
           
         
       
     
     The surface integral of equation (31A) may be simplified to show explicit dependence on a and b. First, several preliminary expressions are evaluated: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           X 
                           s 
                         
                       
                       
                         ∂ 
                         u 
                       
                     
                     = 
                     
                       
                         1 
                         + 
                         
                           v 
                           2 
                         
                       
                       
                         
                           ( 
                           
                             
                               u 
                               2 
                             
                             + 
                             
                               v 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     F 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           X 
                           s 
                         
                       
                       
                         ∂ 
                         v 
                       
                     
                     = 
                     
                       
                         uv 
                         
                           
                             ( 
                             
                               
                                 u 
                                 2 
                               
                               + 
                               
                                 v 
                                 2 
                               
                               + 
                               1 
                             
                             ) 
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                       = 
                       
                         
                           ∂ 
                           
                             Y 
                             s 
                           
                         
                         
                           ∂ 
                           u 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     G 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           Y 
                           s 
                         
                       
                       
                         ∂ 
                         v 
                       
                     
                     = 
                     
                       
                         1 
                         + 
                         
                           u 
                           2 
                         
                       
                       
                         
                           ( 
                           
                             
                               u 
                               2 
                             
                             + 
                             
                               v 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     H 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           Z 
                           s 
                         
                       
                       
                         ∂ 
                         u 
                       
                     
                     = 
                     
                       - 
                       
                         u 
                         
                           
                             ( 
                             
                               
                                 u 
                                 2 
                               
                               + 
                               
                                 v 
                                 2 
                               
                               + 
                               1 
                             
                             ) 
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     I 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           Z 
                           s 
                         
                       
                       
                         ∂ 
                         v 
                       
                     
                     = 
                     
                       - 
                       
                         v 
                         
                           
                             ( 
                             
                               
                                 u 
                                 2 
                               
                               + 
                               
                                 v 
                                 2 
                               
                               + 
                               1 
                             
                             ) 
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     J 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   X 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 u 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   Y 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 u 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   Z 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 u 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       = 
                       
                         
                           1 
                           + 
                           
                             v 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 u 
                                 2 
                               
                               + 
                               
                                 v 
                                 2 
                               
                               + 
                               1 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     K 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     G 
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   X 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 v 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   Y 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 v 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 
                                   Z 
                                   s 
                                 
                               
                               
                                 ∂ 
                                 v 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       = 
                       
                         
                           1 
                           + 
                           
                             u 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 u 
                                 2 
                               
                               + 
                               
                                 v 
                                 2 
                               
                               + 
                               1 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     L 
                   
                   ) 
                 
               
             
             
               
                 
                   F 
                   = 
                   
                     
                       
                         
                           
                             ∂ 
                             
                               X 
                               s 
                             
                           
                           
                             ∂ 
                             u 
                           
                         
                         ⁢ 
                         
                           
                             ∂ 
                             
                               X 
                               s 
                             
                           
                           
                             ∂ 
                             v 
                           
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             Y 
                             s 
                           
                         
                         
                           ∂ 
                           u 
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             
                               Z 
                               s 
                             
                           
                           
                             ∂ 
                             u 
                           
                         
                         ⁢ 
                         
                           
                             ∂ 
                             
                               Z 
                               s 
                             
                           
                           
                             ∂ 
                             v 
                           
                         
                       
                     
                     = 
                     
                       - 
                       
                         
                           uv 
                           
                             
                               ( 
                               
                                 
                                   u 
                                   2 
                                 
                                 + 
                                 
                                   v 
                                   2 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     M 
                   
                   ) 
                 
               
             
           
         
       
     
     The integrand of equation (31A) is then given by: 
     
       
         
           
             
               
                 
                   
                     
                       EG 
                       - 
                       
                         F 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   v 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   u 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   u 
                                   2 
                                 
                                 + 
                                 
                                   v 
                                   2 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             4 
                           
                         
                         - 
                         
                           
                             
                               u 
                               2 
                             
                             ⁢ 
                             
                               v 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   u 
                                   2 
                                 
                                 + 
                                 
                                   v 
                                   2 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             4 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       1 
                       
                         
                           ( 
                           
                             
                               u 
                               2 
                             
                             + 
                             
                               v 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     N 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, 
                   Γ   =       1     2   ⁢   π       ⁢     ∫       ∫   R     ⁢           ⁢       1       (       u   2     +     v   2     +   1     )       3   /   2         ⁢           ⁢     ⅆ   u     ⁢     ⅆ   v                     (     31   ⁢   O     )               
where R is the region defined by (u/a) 2 +(v/b) 2 ≦1.
 
     By a change of variables, u=aξ x  and v=bξ x : 
                   Γ   =       ab     2   ⁢           ⁢   π       ⁢     ∫       ∫     R   1       ⁢       1       (     1   +       a   2     ⁢     ξ   x   2       +       b   2     ⁢     ξ   y   2         )       3   /   2         ⁢           ⁢     ⅆ     ξ   x       ⁢     ⅆ     ξ   y                       (     31   ⁢   P     )               
where R 1  is a circular region (or a disk) defined by ξ x   2 +ξ y   2 ≦1
 
     The limits of integration due to the circular region R 1  may be introduced into Γ and the integral is now given by: 
     
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     
                       ab 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           1 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             
                               
                                 1 
                                 - 
                                 
                                   ξ 
                                   y 
                                   2 
                                 
                               
                             
                           
                           
                             
                               1 
                               - 
                               
                                 ξ 
                                 y 
                                 2 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       a 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ξ 
                                       x 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       b 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               x 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     Q 
                   
                   ) 
                 
               
             
           
         
       
     
     In equation (31Q), the term a 2  may be factored and β 2  may be defined as: 
     
       
         
           
             
               
                 
                   
                     β 
                     2 
                   
                   ≡ 
                   
                     
                       1 
                       
                         a 
                         2 
                       
                     
                     + 
                     
                       
                         
                           b 
                           2 
                         
                         
                           a 
                           2 
                         
                       
                       ⁢ 
                       
                         ξ 
                         y 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     R 
                   
                   ) 
                 
               
             
           
         
       
     
     The inner integral of equation (31Q) becomes: 
     
       
         
           
             
               
                 
                   ∫ 
                   
                     
                       1 
                       
                         
                           ( 
                           
                             
                               β 
                               2 
                             
                             + 
                             
                               ξ 
                               2 
                             
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       ξ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     S 
                   
                   ) 
                 
               
             
           
         
       
     
     The integral of equation (31S) may be evaluated as described below. Define 
     
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       ξ 
                       ⁡ 
                       
                         ( 
                         
                           
                             β 
                             2 
                           
                           + 
                           
                             ξ 
                             2 
                           
                         
                         ) 
                       
                     
                     
                       
                         - 
                         
                           m 
                           2 
                         
                       
                       + 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     T 
                   
                   ) 
                 
               
             
           
         
       
     
     The derivative of T with respect to ξ is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ⅆ 
                             T 
                           
                           
                             ⅆ 
                             ξ 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 
                                   m 
                                   2 
                                 
                               
                               + 
                               1 
                             
                           
                           - 
                           
                             
                               
                                 ξ 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   m 
                                   - 
                                   2 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     ξ 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 
                                   m 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 
                                   m 
                                   2 
                                 
                               
                               + 
                               1 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       β 
                                       2 
                                     
                                     + 
                                     
                                       ξ 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   β 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 m 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     ξ 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 
                                   m 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 
                                   m 
                                   2 
                                 
                               
                               + 
                               1 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 m 
                                 - 
                                 2 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     ξ 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 + 
                                 1 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               β 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                             
                               - 
                               
                                 m 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 3 
                                 - 
                                 m 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     ξ 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 + 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               
                                 β 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   m 
                                   - 
                                   2 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       β 
                                       2 
                                     
                                     + 
                                     
                                       ξ 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     U 
                   
                   ) 
                 
               
             
           
         
       
     
     Rearranging β 2  (m−2) in equation (31U): 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           β 
                           2 
                         
                         + 
                         
                           ξ 
                           2 
                         
                       
                       ) 
                     
                     
                       - 
                       
                         m 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             3 
                             - 
                             m 
                           
                           ) 
                         
                         
                           
                             β 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               m 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         ∫ 
                         
                           
                             ( 
                             
                               
                                 β 
                                 2 
                               
                               + 
                               
                                 ξ 
                                 2 
                               
                             
                             ) 
                           
                           
                             
                               - 
                               
                                 m 
                                 2 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         
                           
                             β 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               m 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           ⅆ 
                           T 
                         
                         
                           ⅆ 
                           ξ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     V 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         
                           
                             ( 
                             
                               
                                 β 
                                 2 
                               
                               + 
                               
                                 ξ 
                                 2 
                               
                             
                             ) 
                           
                           
                             - 
                             
                               m 
                               2 
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           ξ 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               3 
                               - 
                               m 
                             
                             ) 
                           
                           
                             
                               β 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           ∫ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     ξ 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 + 
                                 1 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               ξ 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             
                               β 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
                             ξ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             
                               - 
                               
                                 m 
                                 2 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         
                           
                             ( 
                             
                               
                                 β 
                                 2 
                               
                               + 
                               
                                 ξ 
                                 2 
                               
                             
                             ) 
                           
                           
                             - 
                             
                               3 
                               2 
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           ξ 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           β 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             ξ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   β 
                                   2 
                                 
                                 + 
                                 
                                   ξ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             1 
                             2 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     W 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, for a faster approach, a trigonometric substitution of tan θ=ξ/β may be used. 
     The areal measure Γ: 
     
       
         
           
             
               
                 
                   
                     Γ 
                     = 
                       
                     ⁢ 
                     
                       
                         b 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             1 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   1 
                                   
                                     β 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ξ 
                                       x 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           β 
                                           2 
                                         
                                         + 
                                         
                                           ξ 
                                           x 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     
                                       1 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             
                               
                                 ξ 
                                 x 
                               
                               = 
                               
                                 - 
                                 
                                   
                                     1 
                                     - 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ξ 
                                 x 
                               
                               = 
                               
                                 + 
                                 
                                   
                                     1 
                                     - 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         b 
                         
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             a 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             1 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 1 
                                 
                                   β 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 
                                   1 
                                   - 
                                   
                                     ξ 
                                     y 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       β 
                                       2 
                                     
                                     + 
                                     1 
                                     - 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   
                                     1 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         ab 
                         π 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             1 
                           
                           1 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     1 
                                     - 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                 
                                 
                                   1 
                                   + 
                                   
                                     
                                       b 
                                       2 
                                     
                                     ⁢ 
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 1 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                       + 
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         
                                           a 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   
                                     1 
                                     / 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
               =         2   ⁢           ⁢   ab     π     ⁢       ∫   0   1     ⁢       [           1   -     ξ   y   2           1   +       b   2     ⁢     ξ   y   2           ⁢     1       (     1   +       b   2     ⁢     ξ   y   2       +     a   2     -       a   2     ⁢     ξ   y   2         )       1   /   2           ]     ⁢           ⁢     ⅆ     ξ   y               ,         
since the integrand is an even function,
 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ab 
                       
                       
                         π 
                         ⁢ 
                         
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               
                                 1 
                                 - 
                                 
                                   ξ 
                                   y 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   
                                     b 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ξ 
                                     y 
                                     2 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             ξ 
                                             y 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       
                                         1 
                                         / 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           
                                             
                                               
                                                 
                                                   a 
                                                   2 
                                                 
                                                 - 
                                                 
                                                   b 
                                                   2 
                                                 
                                               
                                               
                                                 1 
                                                 + 
                                                 
                                                   a 
                                                   2 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               ξ 
                                               y 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       
                                         1 
                                         / 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           
                             ξ 
                             y 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     X 
                   
                   ) 
                 
               
             
           
         
       
     
     The integral I of equation (31X) may be expressed in terms of the complete elliptic integrals of the first and third kinds. 
     Define: 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       1 
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           + 
                           
                             
                               b 
                               2 
                             
                             ⁢ 
                             
                               ξ 
                               y 
                               2 
                             
                           
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   ξ 
                                   y 
                                   2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     ξ 
                                     y 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     Y 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         d 
                         2 
                       
                       = 
                       
                         1 
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   ξ 
                                   y 
                                   2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     ξ 
                                     y 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                         
                       
                     
                     ; 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     then 
                     ⁢ 
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     Z 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       K 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           1 
                           
                             d 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           
                             ξ 
                             y 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     AA 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Π 
                     ⁡ 
                     
                       ( 
                       
                         
                           - 
                           
                             b 
                             2 
                           
                         
                         , 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       1 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         1 
                         
                           d 
                           1 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         
                           ξ 
                           y 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     BB 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, the integral I may be expressed in two formats as shown in equations (31CC) and (31DD): 
     
       
         
           
             
               
                 
                   I 
                   = 
                   
                     
                       ∫ 
                       0 
                       1 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               1 
                               - 
                               
                                 ξ 
                                 y 
                                 2 
                               
                             
                             
                               1 
                               + 
                               
                                 
                                   b 
                                   2 
                                 
                                 ⁢ 
                                 
                                   ξ 
                                   y 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             1 
                             
                               d 
                               2 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         
                           ξ 
                           y 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     CC 
                   
                   ) 
                 
               
             
             
               
                 
                   I 
                   = 
                   
                     
                       ∫ 
                       0 
                       1 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             1 
                             - 
                             
                               ξ 
                               y 
                               2 
                             
                           
                           
                             d 
                             1 
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ⅆ 
                         
                           ξ 
                           y 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     DD 
                   
                   ) 
                 
               
             
           
         
       
     
     For the integral I of equation (31CC): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           I 
                           = 
                             
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               1 
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       1 
                                       - 
                                       
                                         ξ 
                                         y 
                                         2 
                                       
                                       + 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                       - 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     1 
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 
                                   ξ 
                                   y 
                                 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               1 
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       1 
                                       + 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                       - 
                                       
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               b 
                                               2 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ξ 
                                           y 
                                           2 
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     1 
                                     
                                       d 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 
                                   ξ 
                                   y 
                                 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 ∫ 
                                 0 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   1 
                                   
                                     d 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   
                                     ξ 
                                     y 
                                   
                                 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   1 
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ξ 
                                       y 
                                       2 
                                     
                                     
                                       d 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     
                                       ξ 
                                       y 
                                     
                                   
                                 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             K 
                             ( 
                             
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   
                                     ξ 
                                     y 
                                     2 
                                   
                                   
                                     d 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   
                                     ξ 
                                     y 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     EE 
                   
                   ) 
                 
               
             
           
         
       
     
     For the integral I of equation (31DD): 
     
       
         
           
             
               
                 
                   I 
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               1 
                               - 
                               
                                 ξ 
                                 y 
                                 2 
                               
                             
                             
                               d 
                               1 
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           
                             ξ 
                             y 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             ∫ 
                             0 
                             1 
                           
                           ⁢ 
                           
                             
                               1 
                               
                                 d 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               
                                 ξ 
                                 y 
                               
                             
                           
                         
                         - 
                         
                           
                             ∫ 
                             0 
                             1 
                           
                           ⁢ 
                           
                             
                               
                                 ξ 
                                 y 
                                 2 
                               
                               
                                 d 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               
                                 ξ 
                                 y 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       = 
                       
                         
                           Π 
                           ( 
                           
                             
                               - 
                               
                                 b 
                                 2 
                               
                             
                             , 
                             
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           
                             ∫ 
                             0 
                             1 
                           
                           ⁢ 
                           
                             
                               
                                 ξ 
                                 y 
                                 2 
                               
                               
                                 d 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ⅆ 
                                 
                                   ξ 
                                   y 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     FF 
                   
                   ) 
                 
               
             
           
         
       
     
     Since the expressions (31EE) and (31FF) are equivalent, the expressions (31EE) and (31FF) may be equated: 
     
       
         
           
             
               
                 
                   
                     
                       
                         K 
                         ( 
                         
                           
                             
                               a 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                           
                         
                         ) 
                       
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               b 
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             1 
                           
                           ⁢ 
                           
                             
                               
                                 ξ 
                                 y 
                                 2 
                               
                               
                                 d 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               
                                 ξ 
                                 y 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         Π 
                         ( 
                         
                           
                             - 
                             
                               b 
                               2 
                             
                           
                           , 
                           
                             
                               
                                 a 
                                 2 
                               
                               - 
                               
                                 b 
                                 2 
                               
                             
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       - 
                       
                         
                           ∫ 
                           0 
                           1 
                         
                         ⁢ 
                         
                           
                             
                               ξ 
                               y 
                               2 
                             
                             
                               d 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   or 
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     GG 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           
                             ξ 
                             y 
                             2 
                           
                           
                             d 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           
                             ξ 
                             y 
                           
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           
                             b 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             Π 
                             ( 
                             
                               
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               , 
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           - 
                           
                             K 
                             ( 
                             
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     HH 
                   
                   ) 
                 
               
             
           
         
       
     
     The integral, I, may be expressed in terms of K and Π: 
     
       
         
           
             
               
                 
                   I 
                   = 
                   
                     
                       
                         Π 
                         ( 
                         
                           
                             - 
                             
                               b 
                               2 
                             
                           
                           , 
                           
                             
                               
                                 a 
                                 2 
                               
                               - 
                               
                                 b 
                                 2 
                               
                             
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       - 
                       
                         
                           ∫ 
                           0 
                           1 
                         
                         ⁢ 
                         
                           
                             
                               ξ 
                               y 
                               2 
                             
                             
                               d 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               ξ 
                               y 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         
                           Π 
                           ( 
                           
                             
                               - 
                               
                                 b 
                                 2 
                               
                             
                             , 
                             
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           
 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             1 
                             
                               b 
                               2 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Π 
                                 ( 
                                 
                                   
                                     - 
                                     
                                       b 
                                       2 
                                     
                                   
                                   , 
                                   
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 K 
                                 ( 
                                 
                                   
                                     
                                       a 
                                       2 
                                     
                                     - 
                                     
                                       b 
                                       2 
                                     
                                   
                                   
                                     1 
                                     + 
                                     
                                       a 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       = 
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 1 
                                 
                                   b 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             Π 
                             ( 
                             
                               
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               , 
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             1 
                             
                               b 
                               2 
                             
                           
                           ⁢ 
                           
                             K 
                             ( 
                             
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     31 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     II 
                   
                   ) 
                 
               
             
           
         
       
     
     The areal measure Γ may be found from equations (31X) and (31II) as: 
                     Γ   =         2   ⁢           ⁢   ab       π   ⁢       1   +     a   2             ⁢     (               (     1   +     1     b   2         )     ⁢     Π   (       -     b   2       ,         a   2     -     b   2         1   +     a   2           )       -                 1     b   2       ⁢     K   (         a   2     -     b   2         1   +     a   2         )             )         ⁢     
     ⁢   or   ⁢     
     ⁢       Γ   =         2   ⁢   a       π   ⁢           ⁢   b   ⁢       1   +     a   2             [         (     1   +     b   2       )     ⁢     Π   (       -     b   2       ,         a   2     -     b   2         1   +     a   2           )       -     K   (         a   2     -     b   2         1   +     a   2         )       ]       ,             (     32   ⁢           ⁢   A     )               
where K and Π are, respectively, the complete elliptic integrals of the first kind and the third kind.
 
     The elliptic integral K of the first kind may be defined as: 
     
       
         
           
             
               
                 
                   
                     K 
                     ⁡ 
                     
                       ( 
                       m 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         
                           π 
                           / 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             
                               1 
                               - 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           1 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     mv 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           v 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     32 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     The elliptic integral Π of the third kind may be defined as: 
     
       
         
           
             
               
                 
                   
                     Π 
                     ⁡ 
                     
                       ( 
                       
                         n 
                         , 
                         m 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         
                           π 
                           / 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   n 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 1 
                                 - 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       ⁢ 
                       
                         
                           1 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   nv 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   mv 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           v 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     32 
                     ⁢ 
                     C 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, where the ellipse on the u, v-plane is becomes a circle, e.g., r≡a=b, the normalized areal measure is: 
     
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     1 
                     - 
                     
                       
                         1 
                         
                           
                             1 
                             + 
                             
                               r 
                               2 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     A normalized areal measure map engine  177  may construct a normalized areal measure map based on the determined normalized areal measure of an area of interest. The constructed normalized areal measurement map may be used by the rendering engine  176  to generate a human-viewable display of an image of uncertainty which may be displayed on the display  180 . 
     With continuing reference to  FIGS. 1B and 2  and further reference to  FIG. 3 , the second normalized measure determining engine  174  may determine the normalized second or circumferential measure as a ratio of the circumference of the simple closed curve  200  on the unit sphere  210  to the circumference of a great circle  260  of the unit sphere  210 . As explained in detail below, the normalized circumference of a simple closed curve on the sphere may be expressed in terms of the complete elliptical integrals of the first and third kinds. More specifically, the normalized circumference of an inverse projected curve  330  on the unit sphere  210  may be determined as: 
     
       
         
           
             
               
                 
                   Λ 
                   = 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 ⅆ 
                                 
                                   
                                     X 
                                     s 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               
                                 ⅆ 
                                 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     ⅆ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         Y 
                                         s 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                   
                                     ⅆ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     ⅆ 
                                     
                                       
                                         Z 
                                         s 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                   
                                     ⅆ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     The 2π term in the denominator is the normalization factor of the circumference of the great circle of the unit sphere. The components of the vector, s, may be parameterized in terms of a, b, and θ: 
     
       
         
           
             
               
                 
                   
                     
                       s 
                       ⁡ 
                       
                         ( 
                         θ 
                         ) 
                       
                     
                     ≡ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 X 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 Y 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 Z 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       a 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       b 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               b 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       a 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       b 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             1 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       a 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       b 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (34A) may be further simplified and expressed in terms of the complete elliptic integrals of the first and third kinds. Provided here are several preliminary expressions: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           ⅆ 
                           
                             
                               X 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         
                           ⅆ 
                           θ 
                         
                       
                       ) 
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             a 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 b 
                                 2 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       ⁢ 
                       
                         
                           sin 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             a 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     C 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           ⅆ 
                           
                             
                               Y 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         
                           ⅆ 
                           θ 
                         
                       
                       ) 
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             b 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             a 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     D 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           ⅆ 
                           
                             
                               Z 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         
                           ⅆ 
                           θ 
                         
                       
                       ) 
                     
                     2 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               b 
                               2 
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           sin 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             a 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     E 
                   
                   ) 
                 
               
             
             
               
                 
                   Φ 
                   ≡ 
                   
                     
                       
                         ( 
                         
                           
                             ⅆ 
                             
                               
                                 X 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             ⅆ 
                             θ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             ⅆ 
                             
                               
                                 Y 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             ⅆ 
                             θ 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             ⅆ 
                             
                               
                                 Z 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             ⅆ 
                             θ 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     F 
                   
                   ) 
                 
               
             
             
               
                 
                   Φ 
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     a 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         b 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 cos 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             a 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     E 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (34E) may be expressed as: 
     
       
         
           
             
               
                 
                   Φ 
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   b 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       a 
                                       2 
                                     
                                     - 
                                     
                                       b 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       a 
                                       2 
                                     
                                     - 
                                     
                                       b 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         [ 
                         
                           1 
                           + 
                           
                             a 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 sin 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     F 
                   
                   ) 
                 
               
             
           
         
       
     
     The square root of Φ may be written as: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Φ 
                       
                       = 
                       
                         
                           
                             
                               
                                 b 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           [ 
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     or 
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     F 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                   
                   = 
                   
                     
                       
                         
                           
                             b 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 a 
                                 2 
                               
                               - 
                               
                                 b 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               sin 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                       
                       
                         
                           [ 
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             
                               
                                 b 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     = 
                     
                       
                         b 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                           
                             1 
                             / 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       2 
                                     
                                     - 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 
                                   
                                     b 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           b 
                                           2 
                                         
                                         - 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         b 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     G 
                   
                   ) 
                 
               
             
           
         
       
     
     The normalized circumference is then given by: 
     
       
         
           
             
               
                 
                   Λ 
                   = 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           Φ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     H 
                   
                   ) 
                 
               
             
           
         
       
     
     Since an arc length for each quadrant is the same, the normalized circumference, Λ, may be reduced to: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       Λ 
                       = 
                       
                         
                           2 
                           π 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               π 
                               / 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               Φ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               θ 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     or 
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     I 
                   
                   ) 
                 
               
             
             
               
                 
                   Λ 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         b 
                       
                       
                         
                           π 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                         
                         
                           1 
                           / 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           π 
                           / 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       2 
                                     
                                     - 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 
                                   
                                     b 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           b 
                                           2 
                                         
                                         - 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         b 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     J 
                   
                   ) 
                 
               
             
           
         
       
     
     The integral I 1  of equation (34J) may be expressed in two equivalent equations (34K) and (34N): 
     
       
         
           
             
               
                 
                   
                     I 
                     1 
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       
                         π 
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     sin 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         b 
                                         2 
                                       
                                       - 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   
                                     
                                       b 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         a 
                                         2 
                                       
                                       - 
                                       
                                         b 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
                               1 
                               - 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         b 
                                         2 
                                       
                                       - 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   
                                     
                                       b 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         θ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     K 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         
                           ∫ 
                           0 
                           
                             π 
                             / 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           b 
                                           2 
                                         
                                         - 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         b 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             θ 
                           
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   b 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                               
                                 π 
                                 / 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 a 
                                                 2 
                                               
                                               - 
                                               
                                                 b 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 a 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   ⁢ 
                                   
                                     
                                       1 
                                       - 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 b 
                                                 2 
                                               
                                               - 
                                               
                                                 a 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                           
                                             
                                               b 
                                               2 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   a 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 θ 
                               
                             
                           
                           
                             ︸ 
                             II 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     L 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         K 
                         ( 
                         
                           
                             ( 
                             
                               
                                 b 
                                 2 
                               
                               - 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                           
                             
                               b 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   b 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         II 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     M 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       2 
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         
                           π 
                           / 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             1 
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       b 
                                       2 
                                     
                                     - 
                                     
                                       a 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 
                                   
                                     b 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           b 
                                           2 
                                         
                                         - 
                                         
                                           a 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       
                                         b 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     N 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         
                           ∫ 
                           0 
                           
                             π 
                             / 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             a 
                                             2 
                                           
                                           - 
                                           
                                             b 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       
                                         ( 
                                         
                                           1 
                                           + 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         sin 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 
                                   1 
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             b 
                                             2 
                                           
                                           - 
                                           
                                             a 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       
                                         
                                           b 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               a 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         sin 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             θ 
                           
                         
                       
                       - 
                       
                         
                           
                             ( 
                             
                               
                                 b 
                                 2 
                               
                               - 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                           
                             
                               b 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                               
                                 π 
                                 / 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                                 
                                   
                                     [ 
                                     
                                       1 
                                       - 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 a 
                                                 2 
                                               
                                               - 
                                               
                                                 b 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                           
                                             ( 
                                             
                                               1 
                                               + 
                                               
                                                 a 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   ⁢ 
                                   
                                     
                                       1 
                                       - 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 b 
                                                 2 
                                               
                                               - 
                                               
                                                 a 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                           
                                             
                                               b 
                                               2 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 
                                                   a 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 θ 
                               
                             
                           
                           
                             ︸ 
                             II 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     O 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         Π 
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   2 
                                 
                               
                               ) 
                             
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                           
                           , 
                           
                             
                               ( 
                               
                                 
                                   b 
                                   2 
                                 
                                 - 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 b 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                       - 
                       
                         
                           
                             ( 
                             
                               
                                 b 
                                 2 
                               
                               - 
                               
                                 a 
                                 2 
                               
                             
                             ) 
                           
                           
                             
                               b 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   a 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         II 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     P 
                   
                   ) 
                 
               
             
           
         
       
     
     Since the two integral expressions, I 1  and I 2 , are equivalent, II may be solved by equating the two expressions so that II is given in terms of a, b, K and Π. Substituting the expression of II from equation (34P) into equation (34M), the normalized circumference is: 
     
       
         
           
             
               
                 
                   
                     Λ 
                     ⁡ 
                     
                       ( 
                       
                         a 
                         , 
                         b 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         b 
                         ⁢ 
                         
                           
                             1 
                             + 
                             
                               a 
                               2 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       b 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   Π 
                                   ( 
                                   
                                     
                                       
                                         
                                           a 
                                           2 
                                         
                                         - 
                                         
                                           b 
                                           2 
                                         
                                       
                                       
                                         1 
                                         + 
                                         
                                           a 
                                           2 
                                         
                                       
                                     
                                     , 
                                     
                                       
                                         
                                           b 
                                           2 
                                         
                                         - 
                                         
                                           a 
                                           2 
                                         
                                       
                                       
                                         
                                           ( 
                                           
                                             1 
                                             + 
                                             
                                               a 
                                               2 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           b 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               K 
                               ( 
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         a 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     Q 
                   
                   ) 
                 
               
             
           
         
       
     
     The normalized circumferential measure is a dimensionless quantity. In one embodiment, where the ellipse is a circle, e.g., r≡a=b, the normalized circumferential measure may be reduced to: 
     
       
         
           
             
               
                 
                   Λ 
                   = 
                   
                     r 
                     
                       
                         1 
                         + 
                         
                           r 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     34 
                     ⁢ 
                     R 
                   
                   ) 
                 
               
             
           
         
       
     
     A normalized circumference measure map engine  178  may construct a normalized circumference map based on the determined normalized circumferential measure of an area of interest. The constructed normalized circumference map may be used by the rendering engine  176  to generate a human-viewable display of an image of uncertainty which may be displayed on the display  180 . 
     With continuing reference to  FIGS. 1B , and  2 - 3  and further reference to  FIG. 4 , the COU determining engine  170  may construct a cone of uncertainty  410  based on the inverse Gnomonic projection of an ellipse  340  of the u, v-plane  240  into the unit sphere  210 , illustrated in  FIG. 3 . 
     In an example provided below, the technique of error propagation via diffusion tensor representations is denoted by “EP” while the reformulated perturbation technique described above is denoted by “RP”. 
     A synthetic diffusion tensor, as known in the art, may be given by:
 
γ=[ ln (1000)×10 +4  s/mm 2  9.475 6.694 4.829 1.123 −0.507 −1.63] T ×10 −4  mm 2 /s
 
     The eigenvalue-eigenvector pairs of the example tensor are:
 
{10.4×10 −4  mm 2 /s, q   1 =[0.9027 0.3139 −0.2940] T },
 
{6.30×10 −4   ,q   2 =[0.3197 −0.9470 −0.0295] T }, and
 
{4.30×10 −4   ,q   3 =[0.2877 0.0673 0.9553] T }.
 
     Further, the trace and fractional anisotropy (FA) are 0.0021 mm 2 /s and 0.4176, respectively. 
     The mean covariance matrices of the major eigenvector of the example tensor, based on EP and RP, are substantially equal to one another: 
     
       
         
           
             
               
                 Σ 
                 
                   q 
                   1 
                 
                   
                   
                 RP 
                   
               
               ≈ 
               
                 Σ 
                 
                   q 
                   1 
                 
                   
                   
                 EP 
                   
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       6.0911 
                     
                     
                       
                         - 
                         13.269 
                       
                     
                     
                       4.5350 
                     
                   
                   
                     
                       
                         - 
                         13.269 
                       
                     
                     
                       40.379 
                     
                     
                       2.3675 
                     
                   
                   
                     
                       4.5350 
                     
                     
                       2.3675 
                     
                     
                       16.450 
                     
                   
                 
                 ) 
               
               × 
               
                 
                   10 
                   
                     - 
                     5 
                   
                 
                 . 
               
             
           
         
       
     
     The computation above is carried out based on a signal-to-noise ratio (SNR) of 50 and on a design matrix, W, that was constructed from a 35 gradient direction set with four spherical shells having b values of 0, 500, 1000, and 1500 s/mm 2 . 
     The eigenvalue-eigenvector pairs of  EP Σ 1     1    are:
 
{ω 1 =4.4935×10 −4 ,[0.3202, −0.9469, −0.0277] T },
 
{ω 2 =1.7985×10 −4 ,[0.2871 0.0691 0.9553] T },
 
and
 
{ω 3 =4.387×10 −20 ≅0,[0.9027 0.3139 −0.2940] T }.
 
     The only difference between the eigenvalue-eigenvector pairs of  EP Σ q     1    and of  RP Σ q     1    is in the minor eigenvalue, 2.5243×10 −20  for  RP Σ q     1   , which is of no consequence in practice. 
     The angle of deviation of the two eigenvectors of  RP Σ q     1    may be compared to that of the circular cone of uncertainty. Since the elements in δD of equation (26) are taken to be the standard deviation of the diffusion tensor elements, let α be equal to approximately 0.3173, which is the area covering the two tails of the normal distribution at one standard deviation apart from the center, i.e., at −1 and +1. Therefore:
 
 (2,140−7;0.3173)=1.1578
 
     The angles of deviation for the major and the minor eigenvectors of  RP Σ q     1    are given by θ 1 =tan −1 (√{square root over (2 (2,n−7;α)ω 1 )}), and θ 2 =tan −1 (√{square root over (2 (2,n−7;α)ω 2 )}), respectively, where ω 1  and ω 2  are the first two largest eigenvalues of  RP Σ q     1   . In the context of the example discussed here, θ 1 =1.847° and θ 2 =1.169°. 
     With reference to  FIGS. 5A ,  5 B,  5 C and  5 D, using experimental simulated human head tensor data together with the experimental design defined in the above example, the normalized areal measure map  510 , which corresponds to the 0.95 joint confidence region (or 95% confidence region) at an SNR level of 15 may be obtained. Likewise, the normalized circumference map  520  may be obtained. The eccentricity map  530  of the ellipse of the cone of uncertainty is given by √{square root over (1−b 2 /a 2 )}. It is assumed that b≦a. The cones of uncertainty  540  of an axial slice corresponding to the normalized areal map  510  are illustrated in  FIG. 5D . 
     As described above, the perturbation method is reformulated to obtain the covariance of the major eigenvector of the diffusion tensor. The covariance matrix is shown to be equivalent to that derived from the error propagation method based on the Euler representation. When a mapping between representations of interest is not available then it is likely that the first order matrix perturbation method may be helpful in finding the Jacobian matrix so that the transformation from one covariance structure to another may be realized. Finally, two normalized measures of the cone of uncertainty and a technique of visualizing cone of uncertainty are described. 
     The proposed measures, which are directly linked to the uncertainty in the major eigenvector of the diffusion tensor, may be important for probing the integrity of the white matter tracts in the brain. In addition to that, the measures have a dependence on signal-to-noise ratio and may be used as a calibration gauge for assessing MRI system or DT-MRI acquisition performance. The described measures for quantifying uncertainty of the major eigenvector of the diffusion tensor are dimensionless and normalized to unity. The measures have direct geometric interpretations. In particular, the areal measure corresponds directly to the projection of the 1+α joint confidence region for q 1  in the u, v-plane onto the unit sphere. Besides the areal measure, the circumferential measure may be important in gaining insights into the asymmetric nature of tract dispersion. The dispersion information contained in Σ q     1    may be incorporated into the modeling the prior distribution of fiber coherence in probabilistic (or Bayesian) tractography. 
     By using the techniques described above, the COU may be constructed without overlapping cones in neighboring regions. 
     In an exemplary embodiment, the engines  152 ,  160 ,  164 ,  170 ,  172 ,  174 ,  176 ,  177 ,  178 , and  182  may be implemented with, for example, a computer having a processing unit, persistent memory, non-persistent memory, input device(s), output device(s), etc. The computer may include software stored in memory, which when executed by the computer, causes the computer to perform the functions of one or more of the engines. The computer may include one or more dedicated devices, for example, one or more field programmable gated array (FPGA) devices, to implement one or more of the engines. 
     In an exemplary embodiment, the engines  152 ,  160 ,  164 ,  170 ,  172 ,  174 ,  176 ,  177 ,  178 , and  182  may be implemented with one or more dedicated devices, for example, one or more field programmable gated array (FPGA) devices. 
     In an exemplary embodiment, the engines  152 ,  160 ,  164 ,  170 ,  172 ,  174 ,  176 ,  177 ,  178 , and  182  may be implemented with one or more software modules comprising one or more software instructions stored on one or more computer-readable mediums and executable by a processing unit having one or more processors to cause the processing unit to perform the functions of the engines. 
     In an exemplary embodiment, the engines  152 ,  160 ,  164 ,  170 ,  172 ,  174 ,  176 ,  177 ,  178 , and  182  may be implemented with a combination of one or more dedicated devices and/or one or more software modules stored on one or more computer-readable mediums. 
     Embodiments described above may be implemented on the same computer, computer system, or a processor. Embodiments described above may also be implemented on different computers, computer systems or processors. Further, embodiments described above may be implemented with one or more software instructions or code residing on tangible computer-readable medium or mediums of one or more computers or computer systems. 
     No element, act, or instruction used in the description of the exemplary embodiments should be construed as critical or essential to the exemplary embodiments unless explicitly described as such. 
     The invention is described in detail with respect to exemplary embodiments, and it will now be apparent from the foregoing to those skilled in the art that changes and modifications may be made without departing from the invention in its broader aspects, and the invention, therefore, as defined in the claims is intended to cover all such changes and modifications as fall within the true spirit of the invention.