Abstract:
The invention provides a reconstruction algorithm based on a theoretically exact analytic inversion formula that can be applied for cone beam data collected from the object that have been scanned by a moving source of radiation with two-dimensional array of detectors. 
     The said algorithm is applicable for arbitrary source trajectory that satisfies the completeness condition. The algorithm does not depend on the trajectory except for a precomputed weight function. 
     The algorithm does not contain the derivative of the cone beam data with respect to the position of the source. This guarantees its stability to noise and to numerical errors. 
     The number of elementary operations is optimally bounded with respect to the number of voxels in the object. 
     The said algorithm admits a high instruction level of parallelism for reducing the computing time.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    Not Applied 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                   
               
               
                 Related US Patent Documents 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 5,881,123 
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                 6,104,775 
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                 6,574,299 
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                 6,804,321 
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                 6,898,264 
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                 6,907,100 
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                 7,403,587 
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                 7,548,603 
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                 7,778,387 
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                 7,848,479 
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       OTHER RELATED REFERENCES 
       [0000]    
       
         Thy H. K.: An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983) N3, 546-552 
         Grangeat, P.: Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, Mathematical Methods in Tomography (Lecture Notes in Mathematics 1497) (1991) pp. 66-97 
         Tam, K. C., Samarasekera, S. and Sauer F.: Exact come beam CT with a spiral, Phys. Med. Bio. 43 (1998), 1015-1024. 
         F. Noo, F., Defrise, M. and Clack, R.: Direct reconstruction of cone-beam data acquired with a vertex path containing a circle, IEEE Trans. Medical Imaging, vol. 7, (1998) N6, 854-867 
         Katsevich, A.: A general scheme for constructing inversion algorithms for cone beam CT, Preprint, 2002, Dept. of Math. University of Central Florida, Orlando. (Int. J. Math. Math. Sci. 2003, 21, 1305-1321) 
         Katsevich, A.: Image reconstruction for the circle and line trajectory, Phys. Med. Biol. 49 (2004), 5059-5072 
         Katsevich, A.: An improved exact filtered backprojection algorithm for spiral computed tomography, Adv. in Appl. Math. 32 N4 (2004) 681-697 
         Pack, J. D., Noo, F. R. and Clackdoyle, R.: Cone-beam reconstruction using the backprojection of locally filtered projections, IEEE Trans. Medical Imaging. 24 (2005), 70-85 
         Pack, J. D. and Noo, F. R.: Cone-beam reconstruction using 1D filtering along the projection of M-lines, Inverse Problems 21 (2005) 1105-1120 
         Ye, Y., Zhao, S., Yu, H. and Wang, G.: A general exact reconstruction for cone-beam CT via backprojection-filtration, IEEE Trans. Medical Imaging 24 (2005) N9, 1190-1198 
         Zou, Y., Pan, X., Xia, D. and Wang, G.: PI-line-based image reconstruction in helical cone-beam computed tomography with a variable pitch, Medical Physics. 32 (2005), 2639-2648 
         Zhuang, T., Leng, S. Nett, B. E. and Chen, G.: Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data, Phys. Med. Biol. 51 (2006), 3189-3210 
         Katsevich, A.: 3PI algorithms for helical computer tomography, Adv. in Appl. Math. 36 (2006) N3, 213-250 
         Katsevich, A.: Zamyatin, A.: Analysis of a family of exact inversion formulas for cone beam computer tomography, Integral geometry and tomography. 59-73, Contemp. Math., 405, Amer. Math. Soc., Providence, R.I. (2006) 
         Chen, G. H., Zhuang, T. L., Leng, S. and Nett, B. E.: Shift-invariant and mathematically exact cone-beam FBP reconstruction using a factorized weighting function, IEEE Trans. Medical Imaging 51 (2006), 3189-3210 
         Zhuang, T. and Chen, G. H.: New families of exact fan-beam and cone-beam image reconstruction formulae via filtering the back projection image of differentiated projection data along singly measured lines, Inverse Problems. 22 (2006), 991-1006 
         Bontus, C., Koken, P., Kohler, Th. and Proksa, R.: Circular CT in combination with a helical segment, Phys. Med. Biol., 51 (2007), 107-120 
         Kapralov, M.; Katsevich, A.: A study of 1PI algorithms for a general class of curves, SIAM J. Imaging Sci. 1 (2008) N4, 418-459 
         Palamodov, V.: An algorithm for full 3D reconstruction with an arbitrary trajectory, Proceedings of the conference “Fully 3D Image Reconstruction in Radiology and Nuclear Medicine” (2013) Lake Tahoe, Calif. 
       
     
       STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVEL-OPMENT 
       [0021]    Not Applicable 
       THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT 
       [0022]    Not Applicable 
       REFERENCE TO A SEQUENCE LISTING, A TABLE, OR A COMPUTERPROGRAM, LISTING COMPACT DISC APPENDIX 
       [0023]    Not Applicable 
       BACKGROUND AND PRIOR ART 
       [0024]    Field of the invention. This invention relates to computer tomography, and in particular, to methods and systems for reconstructing three-dimensional images from the cone beam data obtained by a source of radiation moving along a piecewise smooth trajectory of arbitrary form. 
         [0025]    Here and further the data provided by the two-dimensional detectors will be re-ferred to as cone-beam (CB) data. Mathematically, the problem of image reconstruction in computer tomography (CT) consists of finding an unknown function from its integrals along rays with sources on a curve. The curve is called the source trajectory, and the collection of line integrals is called the cone beam data. 
         [0026]    A typical CT scanner consists of two major parts: a gantry and patient table. The gantry is a C-arm shaped device, where an x-ray source and x-ray detector are located opposite each other. The patient lies on the table, which is then inserted into the rotating gantry. By varying the motion of the table through the gantry one obtains different source trajectories. A saddle trajectory can be obtained by means of two rotating gantries. 
         [0027]    The simple circular trajectory is incomplete. For this reason a pure circular scan is complemented by an additional trajectory, such as a line, arc, helical segment, etc., which makes it complete. So the problem of developing a reconstruction algorithm for more general trajectories is of significant practical interest. Theoretically exact and efficient filtered back projection (FBP) algorithms have been developed for spiral trajectories, for a number of specific circle-plus trajectories. 
         [0028]    An exact algorithm is an algorithm based on an exact analytic formula. Under ideal circumstances, an exact algorithm provides a replication of an exact image. In real circumstances it produces images of good quality that depends on technical level of the equipment used and of density of the cone beam data. However, such algorithms are known to take many hours to provide an image reconstruction, and can take up great amounts of computer power. These algorithms can require keeping considerable amounts of cone beam projections in memory. 
         [0029]    The most of reconstruction algorithms can be divided in several groups: 
         [0030]    (I) The Feldkamp algorithm. 
         [0031]    (II) Slice-by-slice application of the inverse two-dimensional Radon transform. 
         [0032]    (III) Computing of the first derivative of the Radon transform by the method of Grangeat followed by inversion by means of Lorentz&#39;s formula. 
         [0033]    (IV) Grangeat&#39;s method followed by the Fourier transform. 
         [0034]    (V) FBP algorithms for reconstruction by means by Hilbert filtration on specially chosen planes of the derivative of the CB data along the trajectory. 
         [0035]    (VI) BFP algorithms for reconstruction along R-lines. 
         [0036]    We compare these methods in view of the following desirable properties: 
         [0037]    (1) It is exact. 
         [0038]    (2) It is applicable for arbitrary trajectory Y that satisfies Tuy&#39;s completeness condition. Tuy&#39;s condition is theoretically necessary: if it is violated a stable reconstruction of the volume of interest (VOI) is impossible. 
         [0039]    (3) It does not use the derivative of the ray data with respect to the position of the source in Y. Comparing with other algorithms, this property makes the reconstruction more stable with respect to deviation of the source position and to any other noise. 
         [0040]    (4) The number of elementary operations equals O(N 4 ) (which is the minimal order of the number of operations) where N3 is the number of voxels in VOI. 
         [0041]    (5) It admits a high instruction level of parallelism. 
         [0042]    None of the above methods possesses all the properties 1-5. Method (I) is not exact, method (II) lacks property (2) and slow, since the trajectory is of length O(N). Method (III) does not fulfil (4) and (5). Implementation of (IV) includes an interpolation from spherical grid to the Cartesian one in the frequency space. This makes an obstruction to (5). Katsevich&#39;s algorithm works effective for spiral curves and was adapted for some non spiral curves. Property (5) does not hold for these methods. Algorithms (VI) can be applied only to points in VOI that belong to a R-line that is to a chord of the trajectory Y. They do not fulfil (2) and (5). 
       BRIEF SUMMARY OF THE INVENTION 
       [0043]    The objective of the invention is to provide a scheme for creating a numerical algorithm for reconstruction of an attenuation function in a volume of interest that have been scanned with two-dimensional array of detectors and a source trajectory Y. The said algorithm is theoretically exact, it can be adapted to an arbitrary source trajectory Y that fulfils Tuy&#39;s condition. The algorithm does not assume computation of derivative along the source trajectory. The reconstruction is more stable with respect to variations of the position of the radiation source, of impulse energy and to any other noise comparing with methods that depends on the trajectory derivative. The algorithm is numerically stable since no other derivative is to be computed pointwise. 
         [0044]    The proposed algorithm has a simple structure, the form of the trajectory is encoded in the weight function that must be precomputed only once. The said algorithm can be easily reformatted for another trajectory simply by replacing the weight function. Since there is no trajectory derivative in the algorithm, the computer memory keeps just one CB projection at a time. The algorithm contains only O(N 4 ) elementary operations where N 3  is the number of voxels. The computing time for elementary operations in the algorithm can be essentially reduced by means of instruction level parallelism in multicore processors. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0045]      FIG. 1  depicts a possible embodiment. The source of penetrating radiation emits a cone-shaped beam of radiation that passes through the examination region as the rotating gantry rotates. A subject support holds a subject being examined within the examination volume while two rotating gantries are rotated. In this manner, the source of penetrating radiation follows a chosen trajectory relative to the subject. A two-dimensional array of radiation detectors is arranged to receive the radiation emitted from the source after it has traversed the examination volume. 
           [0046]    The invention can be used for any other C-arm assembly. The standard construction with one rotating gantry and a moving bed is also a possible embodiment. 
           [0047]      FIG. 2  shows a possible embodiment with the trajectory consisting of two circles in non-parallel planes and illustrates determining the weight function for this trajectory. 
           [0048]      FIG. 3  shows a possible embodiment with a trajectory Y made of two circles in parallel planes and a connecting segment. 
           [0049]      FIG. 4  depicts the icosahedron that is used for construction of the mesh in the sphere. 
           [0050]      FIG. 5  depicts a fragment of the icosahedron mesh. 
           [0051]      FIG. 6  illustrates integration in Step 11 of the algorithm. 
           [0052]      FIG. 7  illustrates integration in Steps 12 and 13 of the algorithm. 
           [0053]      FIG. 8  illustrates construction of the weight function for an arbitrary closed trajectory. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0054]    This is a flexible new algorithm for accurate cone beam reconstruction with source positions on a curve (or set of curves). The algorithm disclosed herein is based on exact analytical reconstruction formulas. The only condition is that each plane which touches the volume of interest meets the trajectory Y at a point with a non zero angle.
       Step 1 Let S 2  be the sphere of radius 1, in R 3  and N be a natural number. Choose an icosahedral mesh S N  of cells σ(k)⊂S 2 , k=1, . . . , 20N 2 .       
 
         [0056]    Details: The icosahedral mesh S N  in the unit sphere S 2  is the image of a triangular mesh I N  on the icosahedron I inscribed in S 2 . The edges of I are equal to α=1.0515. The icosahedron has 12 vertices, 30 edges and 20 faces which are regular triangles. Each face is divided in N 2  cells that are regular triangles with the edge α/N see  FIG. 5 . The total of cells is 20N 2 . Typically N≧10 −3 . 
         [0057]    
               
       
         
           
             Step 2 Choose coefficients c k  (z) for interpolation of an arbitrary function h defined on the mesh S N  to a point zεS 2 . 
           
         
       
     
         [0059]    Details: A simple option is to interpolate by h(z)=1/3Σh(ξ(k)) where the sum is taken over all vertices of a cell σ that contains z. 
         [0060]    
               
       
         
           
             Step 3 Let X be a domain of interest (VOI) in R 3  of volume V.
           Let X N  be the cubic mesh in X with edges equal V/N 3  and the nodes x(i)εX, i=1, . . . , N 3 .   
         
           
         
       
     
         [0063]    
               
       
         
           
             Step 4 Choose an analytic parametrization y=y(s)=(y 1 (s), y 2 (s), y 3 (s)), 0≦s≦1 of Y such that y i (s), i=1, 2, 3 has bounded first derivatives. 
           
         
       
     
         [0065]    
               
       
         
           
             Step 5 Let y(j)εY, j=1, . . . , N Y  be positions of the X-ray source for acquisition of CB data such that |y(j+1)−y(j)|≦1/2N for all required j.
           Compute y′(j)=∂y/∂s| y=y(j)  for all required j.   
         
           
         
       
     
         [0068]    
               
       
         
           
             Step 6 Compute z(i,j)=|x(i)−y(j)| −1 (x(i)−y(j)) and determine the cell σ(i,j)εS N  such that z(i,j)εσ(i,j). 
           
         
       
     
         [0070]    
               
       
         
           
             Step 7 Compute a differntiable function w(y,ξ) defined for yεY,ξεS 2  (weight function) that satisfies Σ j w(y(j),ξ(k))=1,
           where the sum is taken for all j such that l/N≦&lt;(y(j),ξ(k)&gt;≦(l+1)/N for all l=1, . . . , N Y −1 and all required k.   
         
           
         
       
     
         [0073]    Details: For a vector ξ=(ξ 1 ,ξ 2 ,ξ 3 ), we write &lt;y,ξ&gt;=y 1 ξ 1 +y 2 ξ 2 +y 3 ξ 3 . A weight function w must fulfil the equation 
         [0000]    
       
         
           
             
               
                 ∑ 
                 
                   
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         [0000]    for any ξεS 2  and pεR such that there is at least one point yεY such that &lt;y,ξ&gt;=p. We assume that the number of such points is finite. 
         [0074]    Moreover, w has to satisfy the condition w(y,ξ)=0 if &lt;y′,ξ&gt;=0, where y′ 1 ,y′ 2 ,y′ 3 ) and there are at least two such points y. 
         [0075]    The function 
         [0000]    
       
         
           
             
               
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         [0000]    is continuous and fulfils these conditions.  FIG. 8  illustrates this formula. 
         [0076]    
               
       
         
           
             Step 8 Compute ∂w(y,ξ(k))/∂s| y=y(j)  and ∇ ξ w(y(j),ξ)| ξ=ξ(k)  for all j and k, where ∇ ξ =(∂/∂ξ 1 ,∂/∂ξ 2 ,∂/∂ξ 3 ). 
           
         
       
     
         [0078]    
               
       
         
           
             Step 9 Compute Ω(y(j),ξ(m),ξ(k))=sgn         y′(j),ξ(k)         (∂w(y(j),ξ(k))/∂s−         y′(j),ξ(k))                   ξ(m),∇ ξ           w(y(j),ξ(k))) for all j, k and m such that          ξ(m),ξ(k)         ≦λ/N. 
           
         
       
     
         [0080]    Details: The number of elementary operations is O(N 4 ). 
         [0081]    
               
       
         
           
             Step 10 Load in the computer the CB data g={g(y(j),v j,q )} obtained for an attenuation function ƒ at the position y(j) of the source and the positions d j,q , q=1, . . . , Q of detectors in the coordinate system of the object.
           Set v j,q =|d j,q −y(j)| −1 (d j,q −y(j)).   
         
           
         
       
     
         [0084]    Details: CB data g(y(j),v j,q ) can be used immediately for Step 11 and deleted before the next scan with the source at y j+1  is implemented. Also computations at Steps 12 and 13 depend only on this beam data. 
         [0085]    
               
     
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         [0086]    Details: This computation combines numerical differentiation &lt;(k),∇ v &gt;g(y(j),v) together with the integration of the result along the circle {v;&lt;ξ(k),v&gt;=0}. This numerical method is more stable than the pointwise differentiation which is ill-conditioned. The para-meter λ which regulates the with of the strip in the sphere S 2  the must be optimally chosen.  FIG. 6  illustrates this operation. 
         [0087]    Number of elementary operations at this step is O(N 4 ). 
         [0088]    
               
     
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                          
                       
                       ≤ 
                       λ 
                     
                   
                 
                  
                 
                   
                     Ω 
                      
                     
                       ( 
                       
                         
                           y 
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             m 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     G 
                      
                     
                       ( 
                       
                         
                           y 
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     
                        
                       σ 
                        
                     
                     
                       ( 
                       
                         m 
                         , 
                         σ 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 for 
                  
                 
                     
                 
                  
                 all 
                  
                 
                     
                 
                  
                 required 
                  
                 
                     
                 
                  
                 j 
               
               , 
               m 
               , 
               
                 where 
                  
                 
                     
                 
                  
                 
                   p 
                    
                   
                     ( 
                     
                       m 
                       , 
                       σ 
                     
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 is 
                  
                 
                     
                 
                  
                 the 
                  
                 
                     
                 
                  
                 number 
                  
                 
                     
                 
                  
                 of 
                  
                 
                     
                 
                  
                 points 
               
             
              
             
                 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 ξ 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               ∈ 
               
                 
                   σ 
                    
                   
                       
                   
                    
                   such 
                    
                   
                       
                   
                    
                   that 
                    
                   
                       
                   
                    
                   
                      
                     
                       〈 
                       
                         
                           ξ 
                            
                           
                             ( 
                             m 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       〉 
                     
                      
                   
                 
                 ≤ 
                 
                   λ 
                   . 
                 
               
             
           
         
       
     
         [0089]    Details: Number of elementary operations is O(N 4 ). 
         [0090]    
               
     
         [0000]    
       
         
           
             Step 
              
             
                 
             
              
             13 
           
         
       
       
         
           
             
               Compute 
                
               
                   
               
                
               sum 
                
               
                   
               
                
               
                 V 
                  
                 
                   ( 
                   
                     
                       y 
                        
                       
                         ( 
                         j 
                         ) 
                       
                     
                     , 
                     
                       ξ 
                        
                       
                         ( 
                         m 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 ∑ 
                 σ 
               
                
               
                 
                   ∑ 
                   
                     k 
                     , 
                     
                       
                         ξ 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ∈ 
                       σ 
                     
                     , 
                     
                       
                          
                         
                           〈 
                           
                             
                               ξ 
                                
                               
                                 ( 
                                 m 
                                 ) 
                               
                             
                             , 
                             
                               ξ 
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           〉 
                         
                          
                       
                       ≤ 
                       λ 
                     
                   
                 
                  
                 
                   
 
                 
                  
                 
                   
                      
                     
                       〈 
                       
                         
                           
                             y 
                             ′ 
                           
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       〉 
                     
                      
                   
                    
                   
                     〈 
                     
                       
                         ξ 
                          
                         
                           ( 
                           m 
                           ) 
                         
                       
                       , 
                       
                         ∇ 
                         ξ 
                       
                     
                     〉 
                   
                    
                   
                     w 
                      
                     
                       ( 
                       
                         
                           y 
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     G 
                      
                     
                       ( 
                       
                         
                           y 
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                         , 
                         
                           ξ 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     
                        
                       σ 
                        
                     
                     
                       ( 
                       
                         m 
                         , 
                         σ 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               for 
                
               
                   
               
                
               all 
                
               
                   
               
                
               required 
                
               
                   
               
                
               j 
             
             , 
             
               m 
               . 
             
           
         
       
     
         [0091]    Details: Number of elementary operations is O(N 4 ). 
         [0092]    
               
       
         
           
             Step 14 Compute L(x(i),y(j))=1/3 Σ m  [|x(i)−y(j)| −1 U(y(j),ξ(m))−|x(i)−y(j)| −2 V(y(j), ξ(m))], where the sum is taken over vertices ξ(m)εσ(i,j), for all required i,j. 
           
         
       
     
         [0094]    Details: Number of operations is again O(N 4 ). 
         [0095]    
               
       
         
           
             Step 15 If the curve Y is closed curve, that is y(s=1)=y(s=0), compute F(x(i))=(8π 2 ) −1 Σ j (y(j)−y(j−1))L(x(i),y(j)) for i=1, . . . , N 3 .
           The function F is the reconstruction of ƒ.   
         
           
         
       
     
         [0098]    Details: If a simple curve Y is not closed one more term should be added to the quantity obtained in the next step. 
         [0099]    
               
       
         
           
             Step 16 If the trajectory Y is not closed, compute number (8π 2 ) −1 [K(x(i),y(1))−K(x(i),y(N))] where K(x(i),y(j))=1/3|x(i)−y(j)| −1 Σ m sgn         (y′(j),ξ(m)         w(y,ξ(m))G(y(j),ξ(m)) and the sum is taken over all m such that ξ(m)εσ(i,j), for all required i and j=1, N. 
           
         
       
     
         [0101]    The algorithm works also for an arbitrary curve Y that is union of finite number of simple curves Y 1 , . . . , Y m . The sum of results of Steps 15 and 16 for all the pieces. 
         [0102]    The sum G computed at Step 11 is an approximation for the integral 
         [0000]                ξ,∇ v           ∫ &lt;ξ,v&gt;=0   g ( y,v ) dφ 
 
         [0000]    taken along the circle {v;         ξ,v         =0}. 
         [0103]    The sum U obtained in Step 12 is an approximation for the integral 
         [0000]    
       
         
           
             
               ∫ 
               
                 
                   〈 
                   
                     
                       y 
                       - 
                       z 
                     
                     , 
                     ξ 
                   
                   〉 
                 
                 = 
                 0 
               
             
              
             
               sgn 
                
               
                 〈 
                 
                   
                     y 
                     ′ 
                   
                   , 
                   ξ 
                 
                 〉 
               
                
               
                 ( 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         s 
                       
                     
                      
                     
                       w 
                        
                       
                         ( 
                         
                           y 
                           , 
                           ξ 
                         
                         ) 
                       
                     
                   
                   - 
                   
                     
                       〈 
                       
                         
                           y 
                           ′ 
                         
                         , 
                         z 
                       
                       〉 
                     
                      
                     
                       w 
                        
                       
                         ( 
                         
                           y 
                           , 
                           ξ 
                         
                         ) 
                       
                     
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     〈 
                     
                       ξ 
                       , 
                       
                         ∇ 
                         v 
                       
                     
                     〉 
                   
                    
                   
                     
                       ∫ 
                       
                         
                           〈 
                           
                             ξ 
                             , 
                             v 
                           
                           〉 
                         
                         = 
                         0 
                       
                     
                      
                     
                       
                         g 
                          
                         
                           ( 
                           
                             y 
                             , 
                             v 
                           
                           ) 
                         
                       
                        
                       
                          
                         θ 
                       
                     
                   
                 
                 ) 
               
                
               
                  
                 ϕ 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 where 
                  
                 
                     
                 
                  
                 z 
               
               = 
               
                 
                   
                      
                     
                       y 
                       - 
                       x 
                     
                      
                   
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     ( 
                     
                       y 
                       - 
                       x 
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0104]    The sum V calculated at Step 13 is a approximation for the integral 
         [0000]      ∫ &lt;y-x,ξ&gt;=0   |             y′,ξ             |             z,∇   ξ             w ( y ,ξ)(         ξ,∇ v           ∫ &lt;ξ,v&gt;=0   g ( y,v ) d θ) dφ.  
 
         [0105]    The function L found at Step 14 is an approximation for the integral 
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   
                     〈 
                     
                       
                         y 
                         - 
                         z 
                       
                       , 
                       ξ 
                     
                     〉 
                   
                   = 
                   0 
                 
               
                
               
                 sgn 
                  
                 
                   〈 
                   
                     
                       y 
                       ′ 
                     
                     , 
                     ξ 
                   
                   〉 
                 
                  
                 
                   ρ 
                    
                   
                     ( 
                     
                       
                         w 
                          
                         
                           ( 
                           
                             y 
                             , 
                             ξ 
                           
                           ) 
                         
                       
                       
                          
                         
                           y 
                           - 
                           x 
                         
                          
                       
                     
                     ) 
                   
                 
                  
                 
                   〈 
                   
                     ξ 
                     , 
                     
                       ∇ 
                       v 
                     
                   
                   〉 
                 
                  
                 
                   
                     ∫ 
                     
                       
                         〈 
                         
                           ξ 
                           , 
                           v 
                         
                         〉 
                       
                       = 
                       0 
                     
                   
                    
                   
                     
                       g 
                        
                       
                         ( 
                         
                           y 
                           , 
                           v 
                         
                         ) 
                       
                     
                      
                     
                        
                       θ 
                     
                      
                     
                        
                       ϕ 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               ρ 
               ≃ 
               
                 
                   
                     ∂ 
                     
                         
                     
                   
                   
                     ∂ 
                     s 
                   
                 
                 - 
                 
                   
                     
                       〈 
                       
                         
                           y 
                           ′ 
                         
                         , 
                         ξ 
                       
                       〉 
                     
                     
                        
                       
                         y 
                         - 
                         x 
                       
                        
                     
                   
                    
                   
                     〈 
                     
                       z 
                       , 
                       
                         ∇ 
                         ξ 
                       
                     
                     〉 
                   
                 
               
             
           
         
       
     
         [0106]    Finally, the function F at Step 15 approximates the integral 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     8 
                      
                     
                       π 
                       2 
                     
                   
                   ) 
                 
                 
                   - 
                   1 
                 
               
                
               
                 
                   ∫ 
                   0 
                   1 
                 
                  
                 
                   
                      
                     s 
                   
                    
                   
                     
                       ∫ 
                       
                         
                           〈 
                           
                             
                               y 
                               - 
                               z 
                             
                             , 
                             ξ 
                           
                           〉 
                         
                         = 
                         0 
                       
                     
                      
                     
                       sgn 
                        
                       
                         〈 
                         
                           
                             y 
                             ′ 
                           
                           , 
                           ξ 
                         
                         〉 
                       
                        
                       
                         ρ 
                          
                         
                           ( 
                           
                             
                               w 
                                
                               
                                 ( 
                                 
                                   y 
                                   , 
                                   ξ 
                                 
                                 ) 
                               
                             
                             
                                
                               
                                 y 
                                 - 
                                 x 
                               
                                
                             
                           
                           ) 
                         
                       
                        
                       
                         〈 
                         
                           ξ 
                           , 
                           
                             ∇ 
                             v 
                           
                         
                         〉 
                       
                        
                       
                         
                           ∫ 
                           
                             
                               〈 
                               
                                 ξ 
                                 , 
                                 v 
                               
                               〉 
                             
                             = 
                             0 
                           
                         
                          
                         
                           
                             g 
                              
                             
                               ( 
                               
                                 y 
                                 , 
                                 v 
                               
                               ) 
                             
                           
                            
                           
                              
                             θ 
                           
                            
                           
                              
                             ϕ 
                           
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    which provides an exact reconstruction. 
         [0107]    Properties of the algorithm: 
         [0108]    Steps 1-3 do not depend on the source trajectory Y. 
         [0109]    Steps 1-9 do not depend on the cone beam data g. 
         [0110]    Steps 8-14 are to be done for each position y(j), j=1, . . . , N Y  independently. 
         [0111]    Steps 9 and 10 can be implemented in parallel for all j, and l in a multicore processor. 
         [0112]    Step 11 can be implemented in parallel for all j and l. 
         [0113]    Step 12 can be implemented in parallel for all j and m. 
         [0114]    Step 13 can be implemented in parallel for all j and m. 
         [0115]    Step 14 can be implemented in parallel for all j and i. 
         [0116]    The algorithm does not contain derivative of cone beam data along Y. Moreover, no derivative is computed pointwise at all. The only the numerical integral of the normal ξderivative over the circle          ξ,v         =0 is computed at Step 11. 
         [0117]    The total number of elementary operation is O(N 4 ). 
         [0118]    While the invention has been described, disclosed, illustrated and shown in various terms of certain embodiments or modifications which it has presumed in practice, the scope of the invention is not intended to be, nor should it be deemed to be, limited thereby and such other modifications or embodiments as may be suggested by the teachings herein are particularly reserved especially as they fall within the breadth and scope of the claims here appended. 
       SEQUENCE LISTING 
       [0119]    Not Applicable