Abstract:
A data consistency condition is derived for an array of attenuation values acquired with a fan-beam x-ray CT system. Using this data consistency condition, estimates of selected attenuation values can be calculated from the other attenuation values acquired during the scan. Such estimates reduce artifacts caused truncated data and by loss of data due to x-ray absorption.

Description:
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH  
       [0001]     This invention was made with government support under Grant No. 1R21 EB001683-01 awarded by the National Institute of Health. The United States Government has certain rights in this invention. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     The field of the present invention is computed tomography and, particularly, computer tomography (CT) scanners used to produce medical images from x-ray attenuation measurements.  
         [0003]     As shown in  FIG. 1 , a CT scanner used to produce images of the human anatomy includes a patient table  10  which can be positioned within the aperture  11  of a gantry  12 . A source of highly columinated x-rays  13  is mounted within the gantry  12  to one side of its aperture  11 , and one or more detectors  14  are mounted to the other side of the aperture. The x-ray source  13  and detectors  14  are revolved about the aperture  11  during a scan of the patient to obtain x-ray attenuation measurements from many different angles through a range of at least 180° of revolution.  
         [0004]     A complete scan of the patient is comprised of a set of x-ray attenuation measurements which are made at discrete angular orientations of the x-ray source  13  and detector  14 . Each such set of measurements is referred to in the art as a “view” and the results of each such set of measurements is a transmission profile. As shown in  FIG. 2A , the set of measurements in each view may be obtained by simultaneously translating the x-ray source  13  and detector  14  across the acquisition field of view, as indicated by arrows  15 . As the devices  13  and  14  are translated, a series of x-ray attenuation measurements are made through the patient and the resulting set of data provides a transmission profile at one angular orientation. The angular orientation of the devices  13  and  14  is then changed (for example, 1°) and another view is acquired. An alternative structure for acquiring each transmission profile is shown in  FIG. 2B . In this construction, the x-ray source  13  produces a fan-shaped beam which passes through the patient and impinges on an array of detectors  14 . Each detector  14  in this array produces a separate attenuation signal and the signals from all the detectors  14  are separately acquired to produce the transmission profile for the indicated angular orientation. As in the first structure, the x-ray source  13  and detector array  14  are then revolved to a different angular orientation and the next transmission profile is acquired.  
         [0005]     As the data is acquired for each transmission profile, the signals are filtered, corrected and digitized for storage in a computer memory. These steps are referred to in the art collectively as “preprocessing” and they are performed in real time as the data is being acquired. The acquired transmission profiles are then used to reconstruct an image which indicates the x-ray attenuation coefficient of each voxel in the reconstruction field of view. These attenuation coefficients are converted to integers called “CT numbers”, which are used to control the brightness of a corresponding pixel on a CRT display. An image which reveals the anatomical structures in a slice taken through the patient is thus produced.  
         [0006]     The reconstruction of an image from the stored transmission profiles requires considerable computation and cannot be accomplished in real time. The prevailing method for reconstructing images is referred to in the art as the filtered back projection technique.  
         [0007]     Referring to  FIG. 3 , the proper reconstruction of an image requires that the x-ray attenuation values in each view pass through all of the objects located in the aperture  11 . If the object is larger than the acquired field of view, it will attenuate the values in some transmission profiles as shown by the vertically oriented view in  FIG. 3 , which encompasses the supporting table  10 , and it will not attenuate the values in other transmission profiles as shown by the horizontally oriented view in  FIG. 3 . As a result, when all of the transmission profiles are back projected to determine the CT number of each voxel in the reconstructed field of view, the CT numbers will not be accurate. This inaccuracy caused by truncated projection data can be seen in the displayed image as background shading which can increase the brightness or darkness sufficiently to obscure anatomical details.  
         [0008]     A similar problem is presented when transmission profiles are affected by metal objects such as dental filings in the patient being scanned. In this situation x-rays passing through the metal object are strongly absorbed and the attenuation measurement is very noisy causing strong artifacts in the reconstructed image.  
         [0009]     The data truncation problem and the x-ray absorption problem each corrupt the acquired attenuation data set in a unique way. Referring to  FIG. 4 , as views of the attenuation data are acquired the attenuation values  32  in each view are stored on one row of a two dimensional data array  33 . As indicated by the dashed line  34 , each such row of attenuation data provides a transmission profile of the object to be imaged when viewed from a single view angle. One dimension of the data array  33  is determined by the number of views which are acquired during the scan and the other dimension is determined by the number of detector cell signals acquired in each view.  
         [0010]     Referring particularly to  FIG. 5A , the truncated data problem can be visualized as a set of contiguous views  36  in the acquired data array  33  that are corrupted because they include attenuation information from objects (e.g., supporting table, patient&#39;s shoulder or arms) outside the field of view of all the remaining acquired views. On the other hand, as shown in  FIG. 5B  the absorbed x-ray problem can be visualized as the corruption of one or more attenuation values in all, or nearly all the acquired views as indicated at  38 . In the first problem a select few of the acquired views are significantly affected and in the second problem all or nearly all the acquired views are affected in a more limited manner.  
       SUMMARY OF THE INVENTION  
       [0011]     The present invention is a method for correcting individual attenuation values in fan-beam projections that have been corrupted. More particularly, the present invention is a method which employs a novel fan-beam data consistency condition to estimate individual attenuation measurements in one acquired fan-beam projection view from attenuation measurements made at the other view angles. Corrupted data acquired during a scan is replaced with estimated values produced according to this method.  
         [0012]     A general object of the invention is to replace corrupted x-ray attenuation data acquired during a scan with attenuation data calculated from other, uncorrupted attenuation data acquired during the scan. The estimated and replaced attenuation data may be one or more entire attenuation profiles as occurs when correcting data truncation problems, or it may be selected attenuation values in one or more attenuation profiles as occurs when correcting x-ray absorption problems. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]      FIG. 1  is a perspective view of an x-ray CT system which employs the present invention;  
         [0014]      FIGS. 2A and 2B  are pictorial representations of a parallel beam and fan-beam scan respectively that may be performed with a CT system;  
         [0015]      FIG. 3  is a pictorial representation of a fan-beam acquisition situation which results in a data truncation problem that is solved using the present invention;  
         [0016]      FIG. 4  is a pictorial representation of an attenuation profile acquired by the system of  FIG. 1  and its storage in a data array;  
         [0017]      FIGS. 5A and 5B  are pictorial representations of the data array of  FIG. 4  illustrating data that may be corrupted by two problems encountered when scanning subjects with the system of  FIG. 1 ;  
         [0018]      FIG. 6  is a block schematic diagram of the CT imaging system of  FIG. 1 ;  
         [0019]      FIG. 7  is a graphic representation of an x-ray beam scan which shows the geometric parameters used to derive data consistency condition;  
         [0020]      FIG. 8  is a graphic representation used in the derivation of the data consistency condition;  
         [0021]      FIG. 9  is a graphic representation showing the relationship of vectors;  
         [0022]      FIG. 10  is a block diagram of a preferred embodiment of an x-ray CT system which employs the present invention; and  
         [0023]      FIG. 11  is a pictorial representation of the data structures that are produced when practicing the steps of the present invention. 
     
    
     GENERAL DESCRIPTION OF THE INVENTION  
       [0024]     It is well known that if all the projection data are summed in one view of non-truncated parallel-beam projections, the result is a view angle independent constant. Mathematically, this is a special case (zero-order moment) of the so-called Helgason-Ludwig consistency condition on two-dimensional Radon transforms. Physically, this condition states that the integral of the attenuation coefficients over the whole transmission profile should be a view angle independent constant. This data consistency condition (DCC) plays an important role in correcting the X-ray CT image artifacts when some projection data are missing in parallel beam scans. In practice, this may happen when a portion of a scanned object is positioned outside the scan field-of-view (FOV) defined by a CT scanner.  
         [0025]     A novel data consistency condition is derived here which enables estimation of attenuation values for fan-beam projections. It will be called a fan-beam data consistency condition (FDCC). The new FDCC explicitly gives an estimation of the projection data for a specific ray by filtering all the available fan-beam projections twice. To derive the FDCC, the following definition of a fan-beam projection g[{right arrow over (r)}, {right arrow over (y)}(t)] is used as the starting point  
               g   ⁡     [       r   →     ,       y   →     ⁡     (   t   )         ]       =       ∫   0   ∞     ⁢           ⁢       ⅆ     sf   ⁡     [         y   →     ⁡     (   t   )       +     s   ⁢     r   →         ]         .               (   1   )             
 
 The source trajectory vector {right arrow over (y)}(t) is parameterized by a parameter t, and {right arrow over (r)} is a vector starting from the source position to the image object as shown in  FIG. 7 . The vector {right arrow over (y)}(t) denotes a source position and the vector {right arrow over (r)} represents a vector from the x-ray source to the imaged object. In a laboratory coordinate system o−xy, the vector {right arrow over (y)}(t) is parameterized by a polar angle t, and the vector {right arrow over (r)} is parameterized by polar angle φ. The fan angle γ is also defined from the iso-ray. All the angles have been defined according to a counterclockwise convention. The image function f({right arrow over (x)}) is assumed to have a compact support Ω, i.e., it is non-zero only in a finite spatial region. Throughout this discussion, a vector will be decomposed into its magnitude and a unit vector, e.g. {right arrow over (r)}=r {circumflex over (r)}. Although a circular scanning geometry is shown in  FIG. 7 , the present invention may be employed in any geometry and a general vector notation is used herein to reflect this fact. 
 
         [0026]     Eq. (1) defines a homogeneous extension of the conventional fan-beam projections {overscore (g)}[{circumflex over (r)}, {right arrow over (y)}(t)]. That is  
               g   ⁡     [       r   →     ,       y     →          ⁡     (   t   )         ]       =         1   r     ⁢       g   _     ⁡     [       r   ^     ,       y   →     ⁡     (   t   )         ]         =       1   r     ⁢       ∫   0   ∞     ⁢           ⁢       ⅆ     sf   ⁡     [         y   →     ⁡     (   t   )       +     s   ⁢     r   ^         ]         .                   (   2   )             
 
 A Fourier transform G[{right arrow over (k)}, {right arrow over (y)}(t)] of the generalized fan-beam projection g[{right arrow over (r)}, {right arrow over (y)}(t)] with respect to variable {right arrow over (r)} is defined as  
               G   ⁡     [       k   →     ,       y   →     ⁡     (   t   )         ]       =       ∫     ℜ   2       ⁢           ⁢         ⅆ   2     ⁢     rg   ⁡     [       r   →     ,       y   →     ⁡     (   t   )         ]         ⁢       exp   ⁡     (       -   2     ⁢   π   ⁢           ⁢   i   ⁢       k   →     ·     r   →         )       .                 (   3   )             
 
 Note that this Fourier transform is local, since the Fourier transform is taken with respect to the vectors that emanate from a source position labeled by {right arrow over (y)}(t). 
 
         [0027]     By choosing a separate polar coordinate system for vectors {right arrow over (k)} and {right arrow over (r)} and using Eq. (2), the Fourier transform G[{right arrow over (k)}, {right arrow over (y)}(t)] can be factorized into the product of a divergent radial component 1/k and an angular component {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)]. That is  
               G   ⁡     [       k   →     ,       y   →     ⁡     (   t   )         ]       =       1   k     ⁢         G   _     ⁡     [       k   ^     ,       y   →     ⁡     (   t   )         ]       .               (   4   )             
 
 Here {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] is similarly defined by Eq. (3), but the vector {right arrow over (k)} is replaced by a unit vector {circumflex over (k)}. 
 
         [0028]     A connection between G[{right arrow over (k)}, {right arrow over (y)}(t)] and the Fourier transform {tilde over (f)}({right arrow over (k)}) of the object function f({right arrow over (x)}) can be established by inserting Eq. (1) and Eq. (2) into Eq. (3). The result is  
               G   ⁡     [       k   →     ,       y   →     ⁡     (   t   )         ]       =       1   k     ⁢       ∫   0   ∞     ⁢           ⁢       ⅆ   s     ⁢       f   ~     ⁡     (     s   ⁢     k   ^       )       ⁢       exp   ⁡     [     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   s   ⁢       k   ^     ·       y   →     ⁡     (   t   )           ]       .                   (   5   )             
 
 Compared to Eq. (4), the angular component {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] is given by  
                 G   _     ⁡     [       k   ^     ,       y   →     ⁡     (   t   )         ]       =       ∫   0   ∞     ⁢           ⁢       ⅆ   k     ⁢       f   ~     ⁡     (     k   ⁢     k   ^       )       ⁢     exp   ⁡     [     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   k   ⁢       k   ^     ·       y   →     ⁡     (   t   )           ]                   (   6   )             
 
 The presence of the integral in Eq. (6) indicates that the function {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] does not explicitly depend on the vector {right arrow over (y)}(t). Rather, the function {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] depends directly on the projection of vector {right arrow over (y)}(t) onto a given unit vector {circumflex over (k)}. Therefore, it is appropriate to introduce a new variable 
 
 p={circumflex over (k)}·{right arrow over (y)} ( t ),  (7) 
 
 and rebin the data {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] into G r ({circumflex over (k)}, p) by the following relation 
 
 {overscore (G)}[{circumflex over (k)}, {right arrow over (y)} ( t )]= G   r ( {circumflex over (k)}, p={circumflex over (k)}·{right arrow over (y)} ( t )).  (8) 
 
 Using Eqs. (7) and (8), Eq. (6) can be rewritten as  
                 G   r     ⁡     (       k   ^     ,   p     )       =       ∫   0   ∞     ⁢           ⁢       ⅆ   k     ⁢       f   ~     ⁡     (     k   ⁢     k   ^       )       ⁢     exp   ⁡     (     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   kp     )                   (   9   )             
 
         [0029]     From the definition given in Eq. (3), it is apparent that the function G r ({circumflex over (k)}, p) is a complex function, and thus has both an imaginary part and a real part. In general, the imaginary part and the real part of the function G r ({circumflex over (k)}, p) are not correlated with one another. However, Eq. (9) imposes a strong constraint relating the imaginary and real parts of the function G r ({circumflex over (k)}, p). To better understand this hidden constraint, the following fact is insightful. The variable k is the magnitude of the vector {right arrow over (k)}, and thus it is intrinsically a non-negative number. Using this property, Eq. (9) may be extended to the range of (−∞,+∞) by introducing a following function  
               Q   ⁡     (       k   ^     ,   k     )       =     {             f   ~     ⁡     (     k   ⁢     k   ^       )             k   ≥   0             0         k   &lt;   0                     (   10   )             
 
 In other words, function Q({circumflex over (k)}, k) is a causal function with respect to the variable k. Using the function Q({circumflex over (k)}, k), Eq. (9) can be written as:  
                 G   r     ⁡     (       k   ^     ,   p     )       =       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ     kQ   ⁡     (       k   ^     ,   k     )         ⁢     exp   ⁡     (     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   pk     )                   (   11   )             
 
 Therefore, a standard inverse Fourier transforms yields:  
               Q   ⁡     (       k   ^     ,   k     )       =       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ       pG   r     ⁡     (       k   ^     ,   p     )         ⁢     exp   ⁡     (       -   ⅈ     ⁢           ⁢   2   ⁢   π   ⁢           ⁢   pk     )                   (   12   )             
 
         [0030]     Note that function Q({circumflex over (k)},k) satisfies the causal structure dictated by Eq. (10). This fact implies that, for negative k, the integral in Eq. (12) should universally converge and that the value of the integral should be zero. Therefore, the integral must be done in the upper half of the complex p-plane. In addition, according to a known mathematical theorem, the causal structure in Eq. (10) and (12) requires the function G r ({circumflex over (k)}, p) to be analytical in the upper half of the complex p-plane. An intuitive argument is also beneficial in order to demonstrate that the function G r ({circumflex over (k)}, p) is analytical in the upper half of the complex p-plane. For negative k, the contour of integration for Eq. (12) should be closed by a large semicircle that encloses the upper half of the complex plane as shown in  FIG. 8 . By Cauchy&#39;s theorem, the integral will vanish if G r ({circumflex over (k)}, p) is analytic everywhere in the upper half plane. Thus, the intuitive argument also leads to the conclusion that the function G r ({circumflex over (k)}, p) is analytical in the upper half of the complex p-plane.  
         [0031]     The complex function G r ({circumflex over (k)}, p) may be separated into a real part and an imaginary part as 
 
 G   r ( {circumflex over (k)},p )= ReG   r ( {circumflex over (k)},p )+ iImG   r ( {circumflex over (k)},p ).  (13) 
 
 The causal structure implied in Eq. (10) and the concomitant analytical structure shown in  FIG. 8  require that the real part and imaginary part of the function G r ({circumflex over (k)}, p) are mutually linked in the following way:  
               Re   ⁢           ⁢       G   r     ⁡     (       k   ^     ,   p     )         =       1   π     ⁢   ⁢       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ     p   ′       ⁢       Im   ⁢           ⁢       G   r     ⁡     (       k   ^     ,     p   ′       )             p   ′     -   p                     (     14   ⁢   a     )                   Im   ⁢           ⁢       G   r     ⁡     (       k   ^     ,   p     )         =       -     1   π       ⁢   ℘   ⁢       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ     p   ′       ⁢       Re   ⁢           ⁢       G   r     ⁡     (       k   ^     ,     p   ′       )             p   ′     -   p               ,           (     14   ⁢   b     )             
 
 where the symbol           represents Cauchy principal value. In other words, the imaginary part and real part of the function G r ({circumflex over (k)}, p) are related to each other by a Hilbert transform. 
 
         [0032]     The imaginary part and real part of function {overscore (G)}[{circumflex over (k)}, {right arrow over (y)}(t)] have been explicitly calculated. The final results are  
               Re   ⁢           ⁢       G   _     ⁡     [       k   ^     ,       y   ⇀     ⁡     (   t   )         ]         =       1   2     ⁢       ∫   0     2   ⁢   π       ⁢           ⁢       ⅆ     φδ   ⁡     (       k   ^     ·     r   ^       )         ⁢       g   _     ⁡     [       r   ^     ,       y   ⇀     ⁡     (   t   )         ]                     (   15   )                 Im   ⁢           ⁢       G   _     ⁡     [       k   ^     ,       y   ⇀     ⁡     (   t   )         ]         =       -     1     2   ⁢   π         ⁢       ∫   0     2   ⁢   π       ⁢           ⁢       ⅆ   φ     ⁢           g   _     ⁡     [       r   ^     ,       y   ⇀     ⁡     (   t   )         ]           k   ^     ·     r   ^         .                   (   16   )             
 
         [0033]     Here the angular variable φ is the azimuthal angle of the unit vector {circumflex over (r)}, i.e., {circumflex over (r)}=(cos φ,sin φ). An important observation is that there is a Dirac δ-function in Eq. (15). As shown in  FIG. 9 , for a given unit vector {circumflex over (k)} and source position {right arrow over (y)}(t), the real part Re{overscore (G)}[{circumflex over (k)},{right arrow over (y)}(t)] is completely determined by a single ray along the direction {circumflex over (r)}={circumflex over (k)} ⊥ . Note that the clockwise convention has been chosen to define the unit vector {circumflex over (r)} from a given unit vector {circumflex over (k)}. Thus, the real part is given by:  
               Re   ⁢           ⁢       G   _     ⁡     [       k   ^     ,       y   ⇀     ⁡     (   t   )         ]         =         1   2     ⁢       g   _     ⁡     [         r   ^     =       k   ^     ⊥       ,       y   ⇀     ⁡     (   t   )         ]         =     Re   ⁢           ⁢       G   r     ⁡     (       k   ^     ,     p   =       k   ^     ·       y   ⇀     ⁡     (   t   )             )                   (   17   )             
 
 This equation can also be written as 
 
 {overscore (g)}[{circumflex over (r)},{right arrow over (y)} ( t )]=2 ReG   r ( {circumflex over (k)}={circumflex over (r)}   ⊥   ,p={circumflex over (r)}   ⊥   ·{right arrow over (y)} ( t ))  (18) 
 
 Using Eq. (14) and Eq. (16), the following consistency condition on the fan-beam projection data may be derived  
                 g   _     ⁡     [         r   ^     0     ,       y   ⇀     ⁡     (     t   0     )         ]       =       2   π     ⁢   ℘   ⁢       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ     p   ′       ⁢       Im   ⁢           ⁢       G   r     ⁡     (         r   ^     0   ⊥     ,     p   ′       )             p   ′     -         r   ^     0   ⊥     ·       y   ⇀     ⁡     (     t   0     )                           (   19   )                 Im   ⁢           ⁢       G   r     ⁡     (         r   ^     0   ⊥     ,       p   ′     =         r   ^     0   ⊥     ·       y   ⇀     ⁡     (   t   )             )         =       -     1     2   ⁢   π         ⁢       ∫   0     2   ⁢   π       ⁢           ⁢       ⅆ   φ     ⁢           g   _     ⁡     [       r   ^     ,       y   ⇀     ⁡     (   t   )         ]             r   ^     0   ⊥     ·     r   ^         .                   (   20   )             
 
         [0034]     In order to obtain one specific attenuation profile of projection data {overscore (g)}[{circumflex over (r)} 0 , {right arrow over (y)}(t 0 )] from Eq. (19), all the possible values of Im G r ({circumflex over (k)}, p) are required at the specific orientation {circumflex over (k)}={circumflex over (r)} 0   ⊥ . Therefore, it is important to have a scanning path that fulfills at least the short-scan requirement, viz. angular coverage of the source trajectory of 180°+fan angle. Thus, an individual projection at a specific view angle is linked to the projection data measured from all the different view angles via Eqs. (19) and (20). In other words, an individual attenuation profile can be estimated from all the available projection data. The novel fan-beam data consistency condition (FDCC) in Eq. (19) and Eq. (20) is the basis for the data correction method according to the present invention.  
       DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0035]     With initial reference to  FIG. 10 , a computed tomography (CT) imaging system  110  includes a gantry  112  representative of a “third generation” CT scanner. Gantry  112  has an x-ray source  113  that projects a fan-beam of x-rays  114  toward a detector array  116  on the opposite side of the gantry. The detector array  116  is formed by a number of detector elements  118  which together sense the projected x-rays that pass through a medical patient  115 . Each detector element  118  produces an electrical signal that represents the intensity of an impinging x-ray beam and hence the attenuation of the beam as it passes through the patient. During a scan to acquire x-ray projection data, the gantry  112  and the components mounted thereon rotate about a center of rotation  119  located within the patient  115 .  
         [0036]     The rotation of the gantry and the operation of the x-ray source  113  are governed by a control mechanism  120  of the CT system. The control mechanism  120  includes an x-ray controller  122  that provides power and timing signals to the x-ray source  113  and a gantry motor controller  123  that controls the rotational speed and position of the gantry  112 . A data acquisition system (DAS)  124  in the control mechanism  120  samples analog data from detector elements  18  and converts the data to digital signals for subsequent processing. An image reconstructor  125 , receives sampled and digitized x-ray data from the DAS  124  and performs high speed image reconstruction according to the method of the present invention. The reconstructed image is applied as an input to a computer  126  which stores the image in a mass storage device  129 .  
         [0037]     The computer  126  also receives commands and scanning parameters from an operator via console  130  that has a keyboard. An associated cathode ray tube display  132  allows the operator to observe the reconstructed image and other data from the computer  126 . The operator supplied commands and parameters are used by the computer  126  to provide control signals and information to the DAS  124 , the x-ray controller  122  and the gantry motor controller  123 . In addition, computer  126  operates a table motor controller  134  which controls a motorized table  136  to position the patient  115  in the gantry  112 .  
         [0038]     The fan-beam data consistency condition (FDCC) derived generally above is applied to this preferred geometry by restricting the motion of the x-ray source {right arrow over (y)}(t) to a circle centered at the origin “0” with a radius R. The scanning path is parameterized by a polar angle t shown in  FIG. 7 . Therefore, we have the following parameterization of the source trajectory 
 
 {right arrow over (y)} ( t )= R (cos  t ,sin  t ).  (21) 
 
 In addition, it is also useful to consider the following parameterizations for the unit vectors {circumflex over (r)}, {circumflex over (r)} 0 , and {circumflex over (r)} 0   ⊥  in the laboratory coordinate system: 
 
 {circumflex over (r)} =(cos φ, sin φ),  (22) 
 
 {circumflex over (r)}   0 =(cos φ 0 , sin φ 0 ),  (23) 
 
 {circumflex over (r)}   0   ⊥ =(−sin φ 0 ,cos φ 0 )  (24) 
 
 For convenience, the notation g m (γ,t) is used to describe the measured fan-beam projections with an equi-angular curved detector. The projection angle γ is in the range  
       [       -       γ   m     2       ,       γ   m     2       ]       
 
 where γ m  is the fan angle. By definition, 
 
 g   m (γ, t )= {overscore (g)}[{circumflex over (r)},{right arrow over (y)} ( t )]  (25) 
 
 with the following relation between φ and γ
 
φ=π+ t+γ   (26) 
 
 In practice, it is beneficial to introduce the following definitions: 
 
 ImG   r ( {circumflex over (r)}   0   ⊥   ,p ′)= F   p (φ 0   ,p ′)= F   t (φ 0   ,t )  (27) 
 
 p′={circumflex over (r)}   0   ⊥   ·{right arrow over (y)} ( t )= R  sin( t−φ   0 )  (28) 
 
φ 0 =π+t 0 +γ 0   (29) 
 
 In the second equality in Eq. (27), a data rebinning has been introduced via Eq. (28). 
 
         [0039]     Using these definitions, the FDCC of Eq. (19) and Eq. (20) for this geometry may be expressed as follows:  
                   g   m     ⁡     (       γ   0     ,     t   0       )       =       2   π     ⁢       ∫     -   ∞       +   ∞       ⁢           ⁢       ⅆ     p   ′       ⁢     1       p   0     -     p   ′         ⁢       F   p     ⁡     (       φ   0     ,     p   ′       )               ⁢     
     ⁢   and           (   30   )                     F   p     ⁡     (       φ   0     ,     p   ′       )       =         F   t     ⁡     (       φ   0     ,   t     )       =       1     2   ⁢   π       ⁢       ∫       –γ   m     /   2         +     γ   m       /   2       ⁢           ⁢       ⅆ   γ     ⁢     1     sin   ⁡     (       φ   0     -   t   -   γ     )         ⁢       g   m     ⁡     (     γ   ,   t     )                 ,           (   31   )             
 
 where the number p 0  in Eq. (30) is given by: 
 
 p   0   =R  sin γ 0 .  (32) 
 
         [0040]     Eqs. (30) and (31) explicitly relate the attenuation data in a single projection to the measured attenuation data in all of the other view angles. Given a desired projection attenuation value labeled by parameters γ 0  and t 0 , the numerical procedure to estimate this specific attenuation value may be summarized in the following three steps: 
 
 Step 1: For each of the other view angles, filter the measured data by a filtering kernel  
       1     sin   ⁢           ⁢   γ         
 
 to obtain F t (φ 0 ,t) as set forth in Eq. (31). 
 
 Step 2: Rebin the filtered data F t (φ 0 ,t) into F p (φ 0 ,p′) as set forth in Eq. (28). 
 
 Step 3: Filter the rebinned data F p (φ 0 ,p′) by a Hilbert kernel 1/p′ to obtain the estimated projection data g m (γ 0 , t 0 ) as set forth in Eq. (30). 
 
         [0041]     This process is implemented by a program executed by the computer  126  after the scan is completed and the acquired attenuation data g m (γ, t) is stored in data array  33 . As shown in  FIG. 11 , the above step 1 is performed on the entire data set  140  to produce data set F t (φ 0 , t) which is stored as array  142 . This data set is rebinned as described above in step 2 to form F p (φ 0 , p′) which is stored in data array  144 . The attenuation values g m (γ 0 , t 0 ) at any view angle to can then be estimated using the data in array  144  and Eq. (30) as described above in step 3. It can be appreciated that any acquired attenuation profile can be estimated in this manner in its entirety, or only a particular attenuation value therein may be estimated. Thus, in the truncated data problem illustrated in  FIG. 5A , the views  36  in the acquired data array  33  are replaced with estimated values, whereas in the absorbed x-ray problem illustrated in  FIG. 5B , the corrupted attenuation values  38  in the data array  33  are replaced with estimated values.  
         [0042]     It should be apparent that the method can be repeated using the corrected attenuation data in array  33  to further improve the results. Such an iterative process is normally not necessary when only a small amount of the acquired data is corrupted, but further iterations are required as the proportion of corrupted data increases.  
         [0043]     While the present invention is described with reference to fan-beam x-ray CT systems, it is also applicable to other imaging modalities such as radiation therapy systems and PET/CT systems. Projection data acquired with a fan, or divergent, beam may be estimated using the present invention where projection data is acquired from the same subject at a sufficient number of other projection angles.