Abstract:
A method for deformable registration of 2 digital images includes providing a pair of digital images, including a fixed image and a moving image, extracting a set of edge images from each image of the pair of images, each edge set being extracted at a different resolution, selecting a pair of edge images with a lowest resolution, determining a mapping from edge points of the fixed image to edge points of moving image using a geodesic thin plate spline interpolation, applying the mapping to a next higher resolution edge point image of the moving image, selecting a pair of edge images at a next higher resolution, where a moving edge image is the moving edge image to which the mapping has been applied, repeating the steps at a next higher resolution for all edge images in the set of edge images, and applying the mapping to an entire moving image.

Description:
CROSS REFERENCE TO RELATED UNITED STATES APPLICATIONS 
     This application claims priority from “Geodesic Image Matching”, U.S. Provisional Application No. 60/968,994 of Khamene, et al., filed Aug. 30, 2007, the contents of which are herein incorporated by reference in their entireties. 
    
    
     TECHNICAL FIELD 
     This disclosure is directed to deformable image registration. 
     DISCUSSION OF THE RELATED ART 
     Image registration has been widely explored by numerous approaches. Rigid registration is now something people know well how to do. In essence, rigid registration involves finding the rotations, translations, and image scaling which make one image match another. This means that there are at most 16 parameters to work with. In the case of deformable registration, one wants to match one image with another that has been deformed, not just moved. In medical imaging, the displacement of the lungs is a good example, where one can not match a lung at full inhale with a lung at full exhale only by rotating, translating and scaling of the chest. Here one needs to actually move each pixel of the first image to match a pixel with the other image. This is a tedious task and the number of parameters is typically a multiple of the number of pixels in the image. Thus, there is a much bigger parameter space. One then would like to reduce this space by selecting only some of the pixels to be moved, hoping that the other pixels will follow the movements of the selected pixels. This is the interpolation framework. 
     SUMMARY OF THE INVENTION 
     Exemplary embodiments of the invention as described herein generally include methods and systems for a computationally efficient registration method using interpolation, where only some points on the edges are registered. The rest of the displacement is induced by the interpolation. One issue is that the displacement of the few points, the landmarks, can lead to tearing or folding of the image. Thus a one-to-one mapping may be enforced. A deformation according to an embodiment of the invention remains natural for all iterations and not just at the first and final stage. 
       FIG. 1  is a flowchart of a registration method according to an embodiment of the invention. An input to a registration according to an embodiment of the invention is the two images one wants to register, a moving image, and a fixed image, at steps  10   a  and  10   b . A process according to an embodiment of the invention is a multi-resolution process based on the detection of the edges at different scales of resolution. An exemplary edge detection technique is the Canny edge detection algorithm, applied to the input images at steps  11   a  and  11   b . The output of the edge filtering is n images of edges, for n resolutions. Starting with the lowest resolution edge image, some chosen points are selected from the edges of the fixed and moving images at steps  12   a  and  12   b . At step  13 , the two sets of points with unknown correspondences and different cardinals are matched by calculating a deformation mapping. According to an embodiment of the invention, the deformation mapping is calculated using geodesic thin plate splines, driven by a map of intensities built up with the knowledge of the edges and the intensities of the derivatives on the edges. For a first iteration of the multi resolution, the original filtered points are used, then the points warped using the mapping generated during the previous iteration are used for subsequent iterations. If, at step  14 , all the iterations have been used, the mapping is applied to the moving image at step  16  to recover the registered image, otherwise the mapping is applied to the next higher resolution moving edge image at step  15 . A method according to an embodiment of the invention does not generate the mapping until the end of the process, thus improving computational efficiency. 
     According to an aspect of the invention, there is provided a method for deformable registration of 2 digital images, the method including providing a pair of digital images, each image comprising a plurality of intensities associated with a 2-dimensional grid of points, said pair including a fixed image and a moving image, extracting a set of edge images from each image of said pair of images, each edge set being extracted at a different resolution, selecting a pair of edge images with a lowest resolution, determining a mapping from edge points of said fixed image to edge points of moving image using a geodesic thin plate spline interpolation, applying said mapping to a next higher resolution edge point image of said moving image, selecting a pair of edge images at a next higher resolution, wherein a moving edge image is the moving edge image to which said mapping has been applied, repeating said steps of determining a mapping, applying said mapping, and selecting a pair of edge images at a next higher resolution for all edge images in said set of edge images, and applying said mapping to an entire moving image. 
     According to a further aspect of the invention, the method includes reducing a number of points in each of said edge images, prior to determining said mapping. 
     According to a further aspect of the invention, reducing a number of points comprises following a curve defined by a set of edge points, keeping one point every α, wherein α is a predetermined distance interval, and keeping those points wherein an angle made by three points in β points is greater than θ, wherein β is a predetermined number of points and θ is a predetermined angle. 
     According to a further aspect of the invention, the method includes using a Gaussian pyramid to generate images at progressively lower resolutions to extract edge sets at different resolutions. 
     According to a further aspect of the invention, a geodesic thin plate spline interpolation comprises initializing moving points q i  from fixed points p i , wherein said fixed points are selected from edge points of said fixed image, wherein said moving points and said fixed points have 2-dimensions, initializing a set of N 2-dimensional curbs ζ i (t), iε[1,N], on said image domain parameterized by tε[0,T] associated with paths from said fixed points p i  to moving points q i  wherein 
                   ζ   i     ⁡     (   t   )       =       p   i     +       t   T     ⁢     (       q   i     -     p   i       )           ,         
updating said curbs for parameters tε[2,T−1] using a gradient descent of an energy associated with paths from said landmark points p i  to landmark points q i , calculating from said curbs for parameters tε[2,T−1] a set of coefficients a, b, α that define a diffeomorphic function g defined on said image domain wherein g(p i )≅q i , ∀iε[1,N], and
 
                 g   ⁡     (   x   )       =     ax   +   b   +       ∑     i   =   1     N     ⁢       α   i     ⁢       f     p   i       ⁡     (   x   )               ,         
wherein a is a 2×2 matrix, b is a 2D vector, α is a 2×N matrix, and ƒ p (x) are a set of basis functions, calculating g from said coefficients, updating said curbs for t=T from said diffeomorphic function g and a gradient descent of said constraint energy calculated for points transformed by g, and repeating said steps of updating said curbs for parameters tε[2,T−1], calculating said coefficients a, b, α, calculating g, and updating said curbs for t=T until g converges, where g is a deformation field that maps said fixed edge points to said moving edge points, and said curbs for t=T represent the image of the initial points as transformed by g.
 
     According to a further aspect of the invention, the q i  are initialized as 
                 q   i     =       p   i     -     γ   ⁢       ∂     C   F         ∂     p   i         ⁢     (       p   i     ,   …   ⁢           ,     p   N       )           ,     
     ⁢   wherein                   C   F     ⁡     (     P   I     )       =       ∑     x   ,   y       ⁢       (         S   F     ⁡     (     x   ,   y     )       -       S   M     ⁡     (     x   ,   y     )         )     2             
is said constraint energy, S F  and S M  are structural intensities for a fixed image and a moving image, respectively, wherein
 
                 S   I     =       ∑     i   ∈     P   I         ⁢       v   i   I     ⁢   K   *     δ   ⁡     (     P   i   I     )             ,         
where δ represents the Dirac distribution, K represents a Gaussian kernel, points P I ={P 1   I , . . . , P i   I , . . . , P n   I } are points defining the edges of image I, wherein I is either the moving or the fixed image, v I ={v 1   I , . . . , v i   I , . . . , v n   I } are their associated values, and γ is a predefined constant.
 
     According to a further aspect of the invention, points in the moving set are constrained to be in a set 
     
       
         
           
             
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     According to a further aspect of the invention, updating said curbs for parameters tε[2,T−1] using a gradient descent comprises calculating 
                 ζ   i     ⁡     (   t   )       ←         ζ   i     ⁡     (   t   )       -     β   ⁢       ∂   E       ∂       ζ   i     ⁡     (   t   )                         wherein                 E   ⁡     (     α   ,   ζ     )       =         ∑     i   ,     j   =   1       N     ⁢       ∑     t   =   0     T     ⁢       〈         α   i     ⁡     (   t   )       ,       α   j     ⁡     (   t   )         〉     ⁢     U   ⁡     (              ζ   i     ⁡     (   t   )       -       ζ   j     ⁡     (   t   )              )             +     λ   ⁢       ∑     i   =   1     N     ⁢       ∑     t   =   0       T   -   1       ⁢              Z   i     ⁡     (   t   )            2               ,     
     ⁢       U   ⁡     (   r   )       =     r   ⁢           ⁢   log   ⁢           ⁢   r       ,     
     ⁢         Z   i     ⁡     (   t   )       =         D   t     ⁡     (     ζ   i     )       -       a   ⁡     (   t   )       ⁢       ζ   i     ⁡     (   t   )         -     b   ⁡     (   t   )       -       ∑     k   =   1     N     ⁢       α   k     ⁢     U   ⁡     (              ζ   k     ⁡     (   t   )       -       ζ   i     ⁡     (   t   )              )               ,         
wherein
 
wherein for any time dependent function u, D t (u)=T×(u(t+1)−u(t)), tε[0,T−1], and β is a predefined constant.
 
     According to a further aspect of the invention, calculating said coefficients a, b, α comprises forming a 3×N matrix 
               P   =     [         1           ζ   1   x     ⁡     (     t   -   1     )               ζ   1   y     ⁡     (     t   -   1     )               1           ζ   2   x     ⁡     (     t   -   1     )               ζ   2   y     ⁡     (     t   -   1     )               …       …       …           1           ζ   N   x     ⁡     (     t   -   1     )               ζ   N   y     ⁡     (     t   -   1     )             ]       ;         
forming an N×N matrix K of U(r i,j ) as
 
               K   =     [         0         U   ⁡     (     r     1   ,   2       )           …         U   ⁡     (     r     1   ,   N       )                 U   ⁡     (     r     2   ,   1       )           0       …         U   ⁡     (     r     2   ,   N       )               …       …       …       …             U   ⁡     (     r     N   ,   1       )             U   ⁡     (     r     N   ,   2       )           …       0         ]       ,         
and matrix K λ =K+1/λK wherein λ is a smoothness parameter greater than 0, r i,j =|P i −p j | is a distance between the points i and j, and U(r)=r log(r), forming a 3×3 matrix of zeros O, forming a (N+3)×(N+3) matrix
 
               L   =     [           K   λ         P                   t     ⁢   P         O         ]       ,         
calculating L −1 Q wherein
 
               Q   =           t     ⁢     (               ζ   1   x     ⁡     (   t   )       ,   …   ⁢           ,       ζ   N   x     ⁡     (   t   )       ,   000                   ζ   1   y     ⁡     (   t   )       ,   …   ⁢           ,       ζ   N   y     ⁡     (   t   )       ,   000           )         ,         
wherein  t (α|a b)=L −1 Q is an (N+3)×2 matrix, and wherein | represents concatenation.
 
     According to a further aspect of the invention, calculating g comprises initializing g to an identity mapping, and evaluating 
               g   ⁡     (   x   )       ←         a   ⁡     (   t   )       ⁢     g   ⁡     (   x   )         +     b   ⁡     (   t   )       +       ∑     i   =   1     N     ⁢         α   i     ⁡     (   t   )       ⁢       f       ζ   i     ⁡     (   t   )         ⁡     (     g   ⁡     (   x   )       )                   
for all t=0, . . . , T.
 
     According to a further aspect of the invention, updating said curbs for t=T comprises calculating 
                 ζ   ⁡     (   T   )       ←       g   ⁡     (     ζ   ⁡     (   T   )       )       -     γ   ⁢       ∂     C   F         ∂     g   ⁡     (     ζ   ⁡     (   T   )       )           ⁢     (     g   ⁡     (     ζ   ⁡     (   T   )       )       )           ,     
     ⁢   wherein                   C   F     ⁡     (     P   I     )       =       ∑     x   ,   y       ⁢       (         S   F     ⁡     (     x   ,   y     )       -       S   M     ⁡     (     x   ,   y     )         )     2             
is said constraint energy, S F  and S M  are structural intensities for a fixed image and a moving image, respectively, wherein
 
                 S   I     =       ∑     i   ∈     P   I         ⁢       v   i   I     ⁢   K   *     δ   ⁡     (     P   i   I     )             ,         
where δ represents the Dirac distribution, K represents a Gaussian kernel, points P I ={P 1   I , . . . , P i   I , . . . , P n   I } are points defining the edges of image I, wherein I is either the moving or the fixed image, v I ={v 1   I , . . . , v i   I , . . . , v n   I } are their associated values, g(ζ(T)) represent the moving points, and γ is a predefined constant.
 
     According to another aspect of the invention, there is provided a program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for deformable registration of 2 digital images. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a flowchart of a registration method according to an embodiment of the invention. 
         FIG. 2  depicts an exemplary, non-limiting Gaussian pyramid according to an embodiment of the invention. 
         FIGS. 3(   a )-( b ) depict a case of matching one point to another, where one point is moving, according to an embodiment of the invention. 
         FIGS. 4(   a )-( b ) depict another case of matching one point to another, where one point is staying still, according to an embodiment of the invention. 
         FIGS. 5(   a )-( b ) depict a case of matching 2 points in a void with overlapping kernels, according to an embodiment of the invention. 
         FIGS. 6(   a )-( b ), which depicts the case of 2 points in a void with overlapping kernels, according to an embodiment of the invention. 
         FIGS. 7(   a )-( b ) depicts images used for testing a registration according to an embodiment of the invention. 
         FIGS. 8(   a )-( c ) depict results obtained on registering the cut “C” to the full “C”, according to an embodiment of the invention. 
         FIGS. 9(   a )-( c ) depict results obtained on registering the full “C” to the cut “C”, according to an embodiment of the invention. 
         FIG. 10  shows the steps of the completion of the cut “C” to the full “C”, according to an embodiment of the invention. 
         FIGS. 11(   a )-( c ) show the same type of experiment involving the registration of a bubbled square, according to an embodiment of the invention. 
         FIGS. 12(   a )-( c ) depict the results obtained for the registration of a house to which was applied a known deformation, according to an embodiment of the invention. 
         FIG. 13  is a block diagram of an exemplary computer system for implementing a method for image registration using geodesic image matching, according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Exemplary embodiments of the invention as described herein generally include systems and methods for image registration using geodesic image matching. Accordingly, while the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the invention to the particular forms disclosed, but on the contrary, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention. 
     As used herein, the term “image” refers to multi-dimensional data composed of discrete image elements (e.g., pixels for 2-D images and voxels for 3-D images). The image may be, for example, a medical image of a subject collected by computer tomography, magnetic resonance imaging, ultrasound, or any other medical imaging system known to one of skill in the art. The image may also be provided from non-medical contexts, such as, for example, remote sensing systems, electron microscopy, etc. Although an image can be thought of as a function from R 3  to R, the methods of the inventions are not limited to such images, and can be applied to images of any dimension, e.g., a 2-D picture or a 3-D volume. For a 2- or 3-dimensional image, the domain of the image is typically a 2- or 3-dimensional rectangular array, wherein each pixel or voxel can be addressed with reference to a set of 2 or 3 mutually orthogonal axes. The terms “digital” and “digitized” as used herein will refer to images or volumes, as appropriate, in a digital or digitized format acquired via a digital acquisition system or via conversion from an analog image. 
     Gaussian Pyramid 
     The process of Gaussian pyramid is simple. The idea behind it is to progressively blur an image and to work first on the blurriest image. The lower one is in the pyramid, the lower the frequencies to be registered. According to an embodiment of the invention, the image size is divided by 2 after each blur. An exemplary, non-limiting Gaussian pyramid is shown in  FIG. 2 . A Gaussian pyramid uses a Gaussian filter to blur the images.  FIG. 2  depicts an exemplary, non-limiting filter of size 5×5 pixels, with a standard deviation of 1.6 pixels. Five pixels at the highest resolution level g 0  map to one pixel at a next lower resolution level g 1 , and five pixels at resolution level g 0  map to 1 pixel at the next lower resolution level g 2 . 
     Canny Edge Detection 
     According to an embodiment of the invention, the gradient of the image is computed in x and y direction: ∇ x I(x, y) and ∇ y (x, y), and then the magnitude of the gradient and its angle with the horizontal direction are stored:
 
 M ( x,y )=|∇ x   I ( x,y )+ i∇   y   I ( x,y )|,  (1)
 
Θ( x,y )=arg  ument (∇ x   I ( x,y )+ i∇   y   I ( x,y )).  (2)
 
     According to an embodiment of the invention, {circumflex over (Θ)} is used, which is a discretization of Θ: {circumflex over (Θ)}ε{0, π/4, π/2, 3π/4}. 
     An exemplary, non-limiting Canny Edge Detector(M, {circumflex over (Θ)}) is as follows. 
                                                                                   1. G ← M           2. for each x, y in G                3.   do if M(x, y)&lt;M at neighbors in direction {circumflex over (Θ)} (x, y) and {circumflex over (Θ)}                (x, y)+π                4.   then G(x, y) ← 0           5.   else G(x, y) ← M(x, y)           6. return G                        
According to an embodiment of the invention, a Matlab version of a Canny Edge Detector can be used, which can eliminate spurious edges by hysteresis thresholding.
 
Combining Gaussian Pyramid and Canny Edge Detector in a Registration Process
 
     This disclosure will use herein below the convention of superscript F for fixed and M for moving, as it is desired to match a moving image with a fixed image. A superscript s is added to the variables to emphasize the scale. Thus, X M,s  represents the variable X for a moving image at scale s. 
     According to an embodiment of the invention, the edges of two different images are matched at different scales. If there is a set of points P M  defining the edges in the image I M , and a set P F  defining the edges in the image I F , then a registration process according to an embodiment of the invention can be summed up as follows. 
     Start at the lowest level (i.e. the smallest images).
         1. Match P M,s  with P F,s  and keep the displacement field F s .   2. Upscale the displacement field to F s+1 .   3. Apply the displacement field F s+1  to P M,s+1  and store it in P M,s+1 .   4. Increment s=s+1 and go to 1.       

     According to an embodiment of the invention, the displacement field is not actually upscaled because it is computed using thin plate spline coefficients, which are not dependent to the scale. 
     Geodesic Interpolating Splines 
     Reducing the number of points moving within the image, from all the pixels to only a set of landmarks, can reduce the computational cost of the registration. But it is desired to generate a dense mapping from the information on the displacement of this limited amount of points and to preserve the consistency of the displacement computed for the points in the image which are not in the set of landmarks. To preserve this consistency, the mapping should be one-to-one, and to prevent any folding or tearing of the mapping from happening, the mapping should be diffeomorphic. One approach for a numerically efficient way of making an interpolating spline diffeomorphic, described in Camion and Younes, “Geodesic interpolating splines”, in EMMCVPR, volume 2134 of Lecture Notes in Computer Science, pages 513-527, Springer, 2001, edited by Figueiredo, Zerubia, and Jain, the contents of which are herein incorporated by reference in their entirety, is as follows. 
     First, some notation. Here Ω is a bounded set in the plane, H is a Hilbert space of Ω,            ,           is an inner product on H, and (p 1 , . . . , p N ) and (q 1 , . . . , q N ) are two sets of matching landmarks in Ω.
     A goal according to an embodiment of the invention is to find a diffeomorphism gεΩ such that:
 
∀ iε[ 1 ,N], g ( p   i )≅ q   i ,  (3)
 
or more specifically, an approximate matching which fits best the diffeomorphism condition and the constraints on the landmarks matching.
 
Interpolating Splines
 
     The task of approximate matching using interpolating splines is finding a g such as: 
                   g   =         arg   ⁢           ⁢   min       h   ∈   H       ⁢       {            h        2     +     λ   ⁢       ∑     i   =   1     N     ⁢       (       〈       f     p   i       ,   h     〉     -     q   i       )     2           }     .               (   4   )               
For a fixed λ&gt;0, and where ƒ 1 , . . . , ƒ N εH is a set of functions to which the mapping should be as close as possible in the locations of the landmarks. That is to say, if there is a basis function ƒ p     i    centered on p i , the projection of h of this function should be as close as possible to q i .
 
     The parameter λ will be referred to as the smoothness, since reducing this parameter reduces the effect of the constraints, resulting in a smoother mapping. The solution should be searched in the space spanned by ƒ p     1   , . . . , ƒ p     N    so the unknown mapping can be expressed as a linear combination of ƒ i : 
                   g   =       ∑     i   =   1     N     ⁢       α   i     ⁢     f     p   i                   (   5   )               
If one defines the matrix S such as S i,j =           ƒ p     i   , ƒ p     j             , then the task becomes: find ω such as:
 
ω=arg min{ t   αSα+λ   t ( Sα−q )( Sα−q )},  (6)
 
where α and q are vectors with respective components α i  and q i .

     Since it is desired to find a function h as smooth as possible, one can evaluate this smoothness using an operator L. Define a Hilbert space H L  such that the inner product is: 
                       〈     f   ,   g     〉     L     =       ∫   Ω             ⁢       L   ⁡     (   f   )       ⁢     L   ⁡     (   g   )                   (   7   )               
and then define the norm on H L  as ∥ƒ∥ L   2 =           ƒ,ƒ           L .

     In this framework one needs to find ƒ x εH L , a basis function such that the projection of h on it is equivalent to evaluating h at the point where the basis function is given. That is to say:
 
∀ hεH   L   , h ( x )=             ƒ   x   ,h             .   (8)
 
This task then comprises minimizing

                    h        L     +     λ   ⁢       ∑     i   =   1     N     ⁢         (       h   ⁡     (     p   i     )       -     q   i       )     2     .               
The theoretical existence of functions ƒ x  can be proven. For example, one can use a Gaussian as a basis function:
 
     
       
         
           
             
               
                 
                   
                     
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     For the purpose of diffeomorphic matching, the use of the Thin Plate Splines as described by Bookstein, “Principal warps: Thin-plate splines and the decomposition of deformations”, IEEE Trans. Pattern Anal. Mach. Intell., 11(6):567-585, 1989, the contents of which are herein incorporated by reference in their entirety, provides a simple and computationally efficient way of interpolating. Nonetheless one could use the Gaussian basis and produce a diffeomorphic mapping with the same technique. 
     Thin Plate Splines 
     Using the Laplacian operator Δ in place of the L operator used before, the inner product can be defined as:
 
           ƒ,g         =∫ R     2     ΔƒΔg   (11)
 
The basis function can then be computed, using:

                   {               U   ⁡     (   r   )       =     r   ⁢           ⁢   log   ⁢           ⁢   r       ,                   f   x     ⁡     (   y   )       =       U   ⁡     (          x   -   y          )       .                     (   12   )               
There is now the equality up to the addition of an affine function:
 
                     ∀     h   ∈     H   Δ         ,     {             h   ⁡     (   x   )       =     〈       f   x     ,   h     〉                   h   ⁡     (   x   )       =       ∫     R   2               ⁢         f   x     ⁡     (   y   )       ⁢     Δ   2     ⁢     h   ⁡     (   y   )       ⁢     ⅆ   y                           (   13   )               
which provides g up to the addition of an affine function, that is to say:
 
                     g   ⁡     (   x   )       =     ax   +   b   +       ∑     i   =   1     N     ⁢       α   i     ⁢         f     p   i       ⁡     (   x   )       .                   (   14   )               
EQ. (6) can be used to find a, b and α as follows. Define:
         r i,j =|p i −p j |, the distance between the points i and j;       

               •α   =     (             α   1   x     ,   …   ⁢           ,     α   N   x                   α   1   y     ,   …   ⁢           ,     α   N   y             )       ;                   •   ⁢           ⁢   a     =         (           a   1   x           a   2   x               a   2   x           a   2   y           )     ⁢           ⁢   and   ⁢           ⁢   b     =     (           b   x               b   y           )         ;         
define the N×N matrix K of general term U(r i,j ):
 
                     K   =     [         0         U   ⁡     (     r     1   ,   2       )           …         U   ⁡     (     r     1   ,   N       )                 U   ⁡     (     r     2   ,   1       )           0       …         U   ⁡     (     r     2   ,   N       )               …       …       …       …             U   ⁡     (     r     N   ,   1       )             U   ⁡     (     r     N   ,   2       )           …       0         ]       ;           (   15   )               
define K λ =K+1/λK where λ is the parameter given in EQ. (6), and define the 3×N matrix P:
 
                     P   =     [         1         p   1   x           p   1   y             1         p   2   x           p   2   y             …       …       …           1         p   N   x           p   N   y           ]       ;           (   16   )               
one can then define the (N+3)×(N+3) matrix L:
 
                   L   =     [           K   λ         P                   t     ⁢   P         O         ]             (   17   )               
where O is a 3×3 matrix of zeros. Now if
 
               Q   =           t     ⁢     (             q   1   x     ,   …   ⁢           ,     q   N   x     ,   000                 q   1   y     ,   …   ⁢           ,     q   N   y     ,   000           )         ,         
then, it can be shown that
 
 t (α| ab )= L   −1   Q,   (18)
 
which is an (N+3)×2 matrix, where | represents concatenation, is a solution to EQ. (6). This defines Thin Plate Spline interpolation, according to an embodiment of the invention. The next section describes how to make this interpolation diffeomorphic.
 
Diffeomorphic Landmark Matching
 
     A method according to an embodiment of the invention combines the theory of diffeomorphism groups, generated as flows on Ω and thus solutions of an ordinary differential equation, with interpolating splines. The cost of a deformation g, can be defined as the sum of infinitely small costs, representing small deformations. Let d(t 1 ,), . . . , d(t T ,·) represent small displacements which incrementally lead to g. Then incrementally applying the deformation function Ψ(t k ,) leads to the desired deformation: 
                   {               h   0     =   id     ,                   h   k     ⁡     (           )       =         ψ   ⁡     (       t   k     ,     )       ∘       h     k   -   1       ⁡     (           )         .                     (   19   )               
The constrain h T =g is given in order to lead to the solution. Then one can write the energy Γ of this transformation as:
 
                   Γ   =       ∑     k   =   1     T     ⁢           ⁢              d   ⁡     (     t   k     )            2     .               (   20   )               
Now consider a continuous variable tε[0,1], which corresponds to having t T→∞ ˜t k /T, and:
 
v(t,) T→∞ ˜d(T·t)/T,
 
h(t,) T→∞ ˜h(T·t),  (21)
 
then EQ. (19) becomes:
 
                   {               ⁢           ∂   h       ∂   t       ⁢     (     t   ,     ⁢           )       =     v   ⁡     (     t   ,     h   (     t   ,     ⁢           )       )                         ⁢       h   (     0   ,     ⁢           )     =   id                     ⁢       h   (     T   ,     ⁢           )     =   g                     (   22   )               
and the energy becomes
 
Γ( v )=∫ 0   ∥v ( t )∥ 2   dt.   (23)
 
The knowledge of v thus allows one to determine g by integration of an ordinary differential equation.
 
     Now, add the spline requirements to the equation. Let ∀iε[1,N], ζ i (t) be curbs of Ω parameterized by t. These are in association with the flow paths v(t) leading from the landmark points p i  to the landmark points q i . That is to say: 
                   {                 ζ   i     ⁡     (   0   )       =     p   i       ,                   ζ   i     ⁡     (   1   )       =       q   i     .                     (   24   )               
According to an embodiment of the invention, the energy of this path can be minimized by:
 
                     E   ⁢     (     v   ,     ζ   1     ,   …   ⁢           ,     ζ   N       )       =         ∫   0   I     ⁢       ∫   Ω     ⁢              Lv   ⁢     (   t   )            2     ⁢           ⁢     ⅆ   x     ⁢     ⅆ   t           +       ∑     i   =   1     N     ⁢       ∫   0   I     ⁢                    ⅆ     ζ   i         ⅆ   t       ⁢     (   t   )       -     v   ⁡     (     t   ,       ζ   i     ⁡     (   t   )         )              2     .                   (   25   )               
There is also here defined a geodesic distance between two set of points I=(p 1 , . . . , p N ) and I′=(q 1 , . . . , q N ), which is:
 
 d ( I,I′ )= Inf {√{square root over ( E ( v ,ζ))}, vεH   L ,ζ(0)= I ,ζ(1)= I′}   (26)
 
Geodesic interpolating Thin Plate Splines
 
     Now, for each infinitely small displacement v(t,·) as defined in EQ. (21) is required to be a Thin Plate Spline mapping. Following EQ. (14), v is then: 
                     v   ⁡     (     t   ,   x     )       =         a   ⁡     (   t   )       ⁢   x     +     b   ⁡     (   t   )       +       ∑     i   =   1     N     ⁢           ⁢         α   i     ⁡     (   t   )       ⁢       f       ζ   i     ⁡     (   t   )         ⁡     (   x   )                     (   27   )               
One can combine the energy in EQ. (25) with the flow of EQ. (27), using U defined in EQ. (12) and discretizing for tε[0,T]:
 
                       E   ⁡     (     α   ,   ζ     )       =         ∑     i   ,     j   =   1       N     ⁢           ⁢       ∑     t   =   0     T     ⁢           ⁢       〈         α   i     ⁡     (   t   )       ,       α   j     ⁡     (   t   )         〉     ⁢     U   ⁡     (              ζ   i     ⁡     (   t   )       -       ζ   j     ⁡     (   t   )              )             +     λ   ⁢       ∑     i   =   1     N     ⁢           ⁢       ∑     t   =   0       T   -   1       ⁢           ⁢              Z   i     ⁡     (   t   )            2               ,     
     ⁢     with   ⁢     :               (   28   )                   Z   i     ⁡     (   t   )       =         D   t     ⁡     (     ζ   i     )       -       a   ⁡     (   t   )       ⁢       ζ   i     ⁡     (   t   )         -     b   ⁡     (   t   )       -       ∑     k   =   1     N     ⁢           ⁢       α   k     ⁢     U   ⁡     (              ζ   k     ⁡     (   t   )       -       ζ   i     ⁡     (   t   )              )                     (   29   )               
and for any time dependent function u,
 
∀ tε[ 0 ,T− 1 ], D   t ( u )= T ×( u ( t+ 1)− u ( t )).  (30)
 
     An exemplary, non-limiting two step algorithm according to an embodiment of the invention which converges for minimizing the energy of EQ. (28) is as follows. 
     G EODESIC  T HIN  P LATE  S PLINES  ((p 1 , . . . , p N ), (q 1 , . . . , q N ), β): 
                                                                           1.                 ∀     t   ∈     [     0   ,   T     ]         ,       α   ⁡     (   t   )       ←   0                         2.                 ∀     t   ∈     [     0   ,   T     ]         ,       b   ⁡     (   t   )       ←   0                         3.                 ∀     t   ∈     [     0   ,   T     ]         ,       a   ⁡     (   t   )       ←     (         1       0           0       1         )                           4.                 ∀     i   ∈     [     1   ,   N     ]         ,     ∀     t   ∈     [     0   ,   T     ]         ,         ζ   i     ⁡     (   t   )       =       p   i     +       t   T     ⁢     (       q   i     -     p   i       )                                    5.   repeat                    6.                 i   ∈     [     1   ,   N     ]       ,     ∀     t   ∈     [     2   ,     T   -   1       ]         ,         ζ   i     ⁡     (   t   )       ←         ζ   i     (   t   )     -     β   ⁢           ⁢       ∂   E       ∂       ζ   i     ⁡     (   t   )                                     7.       L is built using EQ. (17), where P is the matrix defined using               ζ(t − 1)               8.                 ∀     t   ∈     [     1   ,     T   -   1       ]         ,                       ⁢   t       ⁢     (       α   ⁡     (   t   )       ❘       a   ⁡     (   t   )       ⁢     b   ⁡     (   t   )           )       =       L     -   1       ⁢     ζ   ⁡     (   t   )                                    9.   until convergence       10.   return α, a, b, ζ                    
Here one just needs to write the derivative of E with respect to ζ:
 
                     ∀     i   ∈     [     1   ,   N     ]         ,         ∂   E       ∂       ζ   i     ⁡     (   t   )           =       2   ⁢       ∑     j   =   1     N     ⁢       〈         α   i     ⁡     (   t   )       ,       α   j     ⁡     (   t   )         〉     ⁢       ∇   1     ⁢     U   ⁡     (              ζ   j     ⁡     (   t   )       -       ζ   i     ⁡     (   t   )              )               -     2   ⁢   λ   ⁢       ∑     j   =   1     N     ⁢       (       〈         α   i     ⁡     (   t   )       ,       Z   j     ⁡     (   t   )         〉     +     〈         α   j     ⁡     (   t   )       ,       Z   i     ⁡     (   t   )         〉       )     ⁢       ∇   1     ⁢     U   ⁡     (              ζ   j     ⁡     (   t   )       -       ζ   i     ⁡     (   t   )              )               -     2   ⁢     λ   t     ⁢     a   ⁡     (   t   )       ⁢       Z   i     ⁡     (   t   )         -     2   ⁢   λ   ⁢           ⁢       TD     t   -   1       ⁡     (     Z   i     )             ,           (   31   )               
where ∇ 1 UεR denotes the gradient of U with respect to one of its variables (it does not matter which one since the mapping (a,b)           U(|a−b|) is symmetric). The points in the path define, for each time t, where the points have moved since the time t−δt. To recover the complete deformation, one sums the induced infinitely small deformations, which is equivalent to resolving the ODE of EQ. (22).
 
Geodesic Matching with Free Extremities

     In this section, it will be shown that a registration framework according to an embodiment of the invention can be viewed as a matching task with free extremities. 
     In the last section the minimization process was performed with fixed extremities that are the initial and target points. However, in a registration process according to an embodiment of the invention, it is desired to move the initial points toward a set of final points without any known correspondence, or even without knowing if the repartition of the points or their number is consistent with each other. Assuming one can define a constraint C Image) (I′) which describes how far from the target image the set of points is, one can then can use the framework given by Garcin and Younes, “Geodesic matching with free extremities”, J. Math. Imaging Vis., 25(3):329-340, 2006, the contents of which are herein incorporated by reference in their entirety, where not only the minimizing paths between two set of points are unknown but where the paths&#39; extremities themselves are unknown. 
     It has been shown that a constraint should be enforced on each extremity of the path to have a well posed problem. According to an embodiment of the invention, a soft constraint will be used on one of the extremities and a hard constraint on the other, to minimize the energy:
 
 E   marching ( I,I′ )= d ( I,I′ ) 2   +C   Image ( I′ ),  (32)
 
with the hard constraint:
 
∀iε[1,N], I i =P i ,  (33)
 
where d(I, I′) is defined in EQ. (26) and C Image (I′) is yet to be defined.
 
Kernel Matching
 
     Let P I  be a set of points defining the edges of an image I. The ‘Structural Intensity’ of an image I, S I , can be defined as the convolution by a Gaussian kernel of an image formed only by the points P I ={P 1   I , . . . , P i   I , . . . , P n   I } and their associated values v I ={v 1   I , . . . , v i   I , . . . , v n   I }. That is to say, if δ represents the Dirac distribution, and K represents the Gaussian kernel, one has: 
                     S   I     =       ∑     i   ∈     P   I         ⁢       v   i   I     ⁢   K   *       δ   ⁡     (     P   i   I     )       .                 (   34   )               
So, matching points has been replaced with matching ‘Structural Intensities’. A measure can be defined between two ‘Structural Intensities’, with superscript F for the fixed image and M for the moving image, as:
 
                     E   =       ∑     x   ,   y       ⁢       (         S   F     ⁡     (     x   ,   y     )       -       S   M     ⁡     (     x   ,   y     )         )     2         ,           (   35   )               
which with EQ. (34) expands and gives the constraint energy:
 
                       C   F     ⁡     (     P   M     )       =       ∑     x   ,   y       ⁢         (         S   F     ⁡     (     x   ,   y     )       -       [       ∑     i   ∈     P   M         ⁢       v   i   M     ⁢   K   *     δ   ⁡     (     P   i   M     )           ]     ⁢     (     x   ,   y     )         )     2     .               (   36   )               
Now, EQ. (36) can be derived with respect to the position of each point in P M  to carry out the gradient descent on each point&#39;s position:
 
                           ∂     C   F         ∂     P   j   M         =       -   2     ⁢       ∑     x   ,   y       ⁢           ⁢       (       S   F     -     [       ∑     i   ∈     P   M         ⁢       v   i   M     ⁢   K   *     δ   ⁡     (     P   i   M     )           ]       )     ×     v   j   M     ⁢     ∇   K     *     δ   ⁡     (     P   i   M     )               ⁢     
     ⁢     with   ⁢     :         ⁢                   (   37   )                 ∇   K     =       (             ∂   K       ∂   x                   ∂   K       ∂   y             )     =       (               -   x       σ   2       ⁢     exp   ⁡     (       -     (       x   2     +     y   2       )         2   ⁢     σ   2         )                       -   y       σ   2       ⁢     exp   ⁡     (       -     (       x   2     +     y   2       )         2   ⁢     σ   2         )               )     .               (   38   )               
Experiments
 
       FIGS. 3 ,  4 , and  5  present results of solving these equations without regularization with the Geodesic Splines.  FIGS. 3(   a )-( b ) depict a case of matching one point to another, where one point is moving.  FIGS. 4(   a )-( b ) depict another case of matching one point to another, where one point is staying still.  FIGS. 5(   a )-( b ) depict a case of matching 2 points in a void with overlapping kernels. In each of these cases, the (b) figure is a closer look at an outlined square(s)  31 ,  41 ,  51   a ,  51   b  of the (a) figure. The images  32 ,  42 ,  52  seen underneath are the fixed ‘structural intensities’. On top of the structural intensity, each outlined square  31 ,  41 ,  51   a ,  51   b  represents the kernel surrounding a point  33 ,  43 ,  53   a ,  53   b , and the arrow  34 ,  44 ,  54   a ,  54   b  starting from the center of each square and ending in a cross, shows the movement of this point in one iteration. 
     In  FIGS. 3 and 4 , it can be seen that when the kernel surrounding a point and the kernel of the fixed image are overlapping, as in  FIGS. 3(   a )-( b ), the point is moving. When there is no overlap, as in  FIGS. 4(   a )-( b ), there is no movement, and this is a consequence of the equations. 
     Another consequence of the equations is the fact that when, for example, two points are away from the information of the fixed image, but have overlapping kernels, as shown in  FIGS. 5(   a )-( b ), these two points tend to move away from one another to not overlap. If there are many points in a void, they will tend to spread quickly on the void, thus disrupting the contour. The points are moving whereas there is no information available from the fixed image. 
     Modified Kernel Matching 
     One can add a rule to the movement of the points: a point P i   M  is allowed to move only if: 
                       K   *     δ   ⁡     (     P   i   M     )         ⋂       ⋃   j     ⁢     (     K   *     δ   ⁡     (     P   j   F     )         )         ≠       {   ∅   }     .             (   39   )               
Defining {circumflex over (P)} as:
 
                     P   ^     =     {       k   ∈   N     ❘         K   *     δ   ⁡     (     P   k   M     )         ⋂       ⋃   j     ⁢     (     K   *     δ   ⁡     (     P   j   F     )         )         ≠     {   ∅   }         }             (   40   )               
implies the following energy:
 
                       C   F     ⁡     (     P   M     )       =       ∑     x   ,   y       ⁢         (         S   F     ⁡     (     x   ,   y     )       -       [       ∑     i   ∈     P     ^                   ⁢       v   i   M     ⁢   K   *     δ   ⁡     (     P   i   M     )           ]     ⁢     (     x   ,   y     )         )     2     .               (   41   )               
In other words, if a moving kernel has no information underneath, if it is not intersecting with the fixed image, then the point related to this kernel should not move, hence not be computed within the energy.  FIGS. 6(   a )-( b ), which depicts the case of 2 points  63   a ,  63   b  in a void with overlapping kernels  61   a,    61   b , show that this change has the desired effect, with  FIG. 6(   a ) depicting points staying still, and  FIG. 6(   b ) is a closer look at the outlined squares  61   a ,  61   b  of  FIG. 6(   a ).
 
     A Geodesic Thin Plate Spline algorithm according to an embodiment of the invention presented above can be modified to take into account the constraint of the image which leads the points. Exemplary, non-limiting pseudo-code for such an algorithm is as follows. 
     G EODESIC  T HIN  P LATE  S PLINES ((p 1 , . . . , P N ) Image F, γ, β): 
                                                                           1.                 ∀     t   ∈     [     0   ,   T     ]         ,       α   ⁡     (   t   )       ←   0                         2.                 ∀     t   ∈     [     0   ,   T     ]         ,       b   ⁡     (   t   )       ←   0                         3.                 ∀     t   ∈     [     0   ,   T     ]         ,       a   ⁡     (   t   )       ←     (         1       0           0       1         )                           4.                 ∀     i   ∈     [     1   ,   N     ]         ,       q   i     =       p   i     -     γ   ⁢           ⁢       ∂     C   F         ∂     p   i         ⁢     (       p   i     ,   …   ⁢           ,     p   N       )                               5.                 ∀     i   ∈     [     1   ,   N     ]         ,     ∀     t   ∈     [     0   ,   T     ]         ,         ζ   i     ⁡     (   t   )       =       p   i     +       t   T     ⁢     (       q   i     -     p   i       )                                    6.   repeat                    7.                 i   ∈     [     1   ,   N     ]       ,     ∀     t   ∈     [     2   ,     T   -   1       ]         ,         ζ   i     ⁡     (   t   )       ←         ζ   i     (   t   )     -     β   ⁢           ⁢       ∂   E       ∂       ζ   i     ⁡     (   t   )                                     8.       L is built using EQ. (17). where P is the matrix defined using               ζ(t − 1)               9.                 ∀     t   ∈     [     1   ,     T   -   1       ]         ,           ⁢             t     ⁢     (       α   ⁡     (   t   )       ❘       a   ⁡     (   t   )       ⁢     b   ⁡     (   t   )           )       =       L     -   1       ⁢     ζ   ⁡     (   t   )                               10.       h ← GEODESIC TPS GET WARP (ζ,α, a, b)               11.                 ζ   ⁡     (   T   )       ←       h   ⁡     (     ζ   ⁡     (   T   )       )       -     γ   ⁢           ⁢       ∂     C   F         ∂     h   ⁡     (     ζ   ⁡     (   T   )       )           ⁢     (     h   ⁡     (     ζ   ⁡     (   T   )       )       )                                  12.   until convergence       13.   return ζ(T), h                    
G EODESIC  TPS G ET  W ARP  (ζ, α, a, b):
 
                                                             1.   g ← id       2.   for t ← 0 to T                    3.             do   ⁢           ⁢     g   ⁡     (   x   )         ←         a   ⁡     (   t   )       ⁢     g   ⁡     (   x   )         +     b   ⁡     (   t   )       +       ∑     i   =   1     N     ⁢         α   i     ⁡     (   t   )       ⁢       f       ζ   i     ⁡     (   t   )         ⁡     (     g   ⁡     (   x   )       )                                      4.   return g                    
At convergence, ζ(T) represents the image of the initial points transformed by the deformation h.
 
Points Number Reduction
 
     Referring back to steps  12   a ,  12   b  of  FIG. 1 , even if the amount of data used is reduced by taking only the edges of the image as the only registration guide, it can be seen during the registration process that fewer points evenly distributed on the edges will give similar results. Points that are located on corners and on highly curbed edges have more effect on the registration than other points. Indeed the variations in the curve denote higher frequencies, so it seems consistent to have a higher frequency of points in these areas. From these observations filtering scheme may be set up to eliminate the less useful points. 
     The variations of the curve are directly linked to the second derivative of the parameterized curve, in that there is a density of points linked to the norm of the second derivative at each point. However, it is simpler and computationally more efficient to use a heuristic based on the angle of curvature when one follows a curve drawn by the points. 
     An exemplary, non-limiting algorithm for point number reduction according to an embodiment of the invention is as follows, presented here with a recursive function for the sake of comprehensiveness. This algorithm follows each of the curves in the edges to keep only one point every α, and keep the points where the angle made by three points in β points, is greater than θ. 
     P OINTS  F ILTERING  (AllPoints) 
                                                                     1. X ← AllPoints           2. while X≠∅                3.   do [sparseChain, X] = GetChain(X, nodePoint ← first)                4. remainingPoints ← {AllPoints \ sparseChain }           5. return remainingPoints, sparseChain                        
G ET C HAIN  (AllPoints, nodePoint):
 
                                                                                                                                                                                      1. sparseChain[1] ← AllPoints[nodeP oint]        2. sparseIndex←1        3. memory[1] ← AllPoints[nodeP oint]        4. AllPoints ← {AllPoints \ AllPoints[nodeP oint]}        5. i ← 1        6. repeat             7.   i ← i+1        8.   // vector of all the distances        9.   dist ← DISTANCE(AllPoints, memory[i])       10.   // vector of indices of all the points in the eight-neighborhood       11.   index 1  ← FIND(dist ≦ {square root over (2)})       12.   memory[i] ← AllPoints[index 1 [1]]       13.   AllPoints ← {AllPoints \ AllPoints[index 1 [1]]}       14.   if i&gt; α            15.   then if   ANGLE(memory[i- α ], memory[i- α /2],                memory[i])&gt; θ            16.   sparseChain[sparseIndex] ← memory[i- α ]]       17.   sparseIndex ← sparseIndex + 1            18.   if MODULO(i, β)            19.   then   sparseChain[sparseIndex] ← memory[i- β /2]            20.   sparseIndex ← sparseIndex + 1            21.   if LENGTH(index 1 ) ≠ 1       22.   then            23.   dist ← DISTANCE(AllPoints, memory[i-1])       24.   index 2  ← find(dist ≦ {square root over (2)})            25.   for j ← 1 to LENGTH(index 2 )            26.   do   [a, b, AllPoints] ← GETCHAIN(AllPoints,                index 2 [j])            27.   sparseChain ← {sparseChain | a}       28.   memory ← {memory | b}       29.   i ← SIZE(memory)       30.   sparseIndex ← SIZE(sparseChain)            31. until index 1  = ∅ ;       32. return sparseChain, memory, AllPoints                    
Results
 
       FIGS. 8-11  show results obtained with an algorithm according to an embodiment of the invention on the letter ‘C’. The original images are shown in  FIGS. 7(   a )-( b ). This test image involves the completion of the cut ‘C’, shown in  FIG. 7(   b ), which is a full dilatation as opposed to the full contraction of the reverse process. A full dilatation and full contraction are attempted, with the same parameters, to see if an algorithm according to an embodiment of the invention can perform both ways. 
       FIGS. 8(   a )-( c ) depict results obtained on registering the cut “C” to the full “C”.  FIGS. 9(   a )-( c ) depict results obtained on registering the full “C” to the cut “C”. Three types of images are shown: the moving points  81 ,  91  on the fixed edge  82 ,  92  in the (a) figure, the deformation field after registration in the (b) figure, and finally the difference image in the (c) figure. In the (c) figures, the surrounding gray areas  83 ,  93  are null. 
     An algorithm according to an embodiment of the invention can interpolate the ‘natural’ movement of the ‘C’.  FIG. 10  shows the steps of the completion of the cut “C” to the full “C”, showing the interpolation of the movement of the edges. 
       FIGS. 11(   a )-( c ) show the same type of experiment involving the registration of a bubbled square.  FIG. 11(   a ) depicts the original square, the moving image in this case,  FIG. 11(   b ) depicts the deformed image, which is taken as the fixed image, and  FIG. 11(   c ) depicts the registration result. 
       FIGS. 12(   a )-( c ) depict the results obtained for the registration of a house to which was applied a known deformation.  FIG. 12(   a ) is the original house, which is the moving image,  FIG. 12(   b ) depicts the deformed house, which is the fixed image, and  FIG. 12(   c ) depicts the registration result. 
     System Implementations 
     It is to be understood that embodiments of the present invention can be implemented in various forms of hardware, software, firmware, special purpose processes, or a combination thereof. In one embodiment, the present invention can be implemented in software as an application program tangible embodied on a computer readable program storage device. The application program can be uploaded to, and executed by, a machine comprising any suitable architecture. 
       FIG. 13  is a block diagram of an exemplary computer system for implementing a method for image registration using geodesic image matching, according to an embodiment of the invention. Referring now to  FIG. 13 , a computer system  131  for implementing the present invention can comprise, inter alia, a central processing unit (CPU)  132 , a memory  133  and an input/output (I/O) interface  134 . The computer system  131  is generally coupled through the I/O interface  134  to a display  135  and various input devices  136  such as a mouse and a keyboard. The support circuits can include circuits such as cache, power supplies, clock circuits, and a communication bus. The memory  133  can include random access memory (RAM), read only memory (ROM), disk drive, tape drive, etc., or a combinations thereof. The present invention can be implemented as a routine  137  that is stored in memory  133  and executed by the CPU  132  to process the signal from the signal source  138 . As such, the computer system  131  is a general purpose computer system that becomes a specific purpose computer system when executing the routine  137  of the present invention. 
     The computer system  131  also includes an operating system and micro instruction code. The various processes and functions described herein can either be part of the micro instruction code or part of the application program (or combination thereof) which is executed via the operating system. In addition, various other peripheral devices can be connected to the computer platform such as an additional data storage device and a printing device. 
     It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying figures can be implemented in software, the actual connections between the systems components (or the process steps) may differ depending upon the manner in which the present invention is programmed. Given the teachings of the present invention provided herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the present invention. 
     While the present invention has been described in detail with reference to a preferred embodiment, those skilled in the art will appreciate that various modifications and substitutions can be made thereto without departing from the spirit and scope of the invention as set forth in the appended claims.