Abstract:
Methods, systems and processes for providing efficient image reconstruction using local cone beam tomography which provide a reduced level of artifacts without suppressing the strength of the useful features; and in a dynamic case provide reconstruction of objects that are undergoing a change during the scan. An embodiment provides a method of reconstructing an image from cone beam data provided by at least one detector. The method includes collecting CB projection data of an object, storing the CB projection data in a memory; and reconstructing the image from the local CB projection data. In the reconstructing step, a combination of derivatives of the CB projection data that will result in suppressing the artifacts are found. The combination of derivatives includes collecting cone beam data that represents a collection of integrals that represent the object.

Description:
[0001]    This Patent Application claims the benefit of priority to U.S. Provisional Patent Application No. 60/754,236 filed on Dec. 28, 2005 and was funded in part by NSF grant DMS-0104033 and the Alexander von Humboldt Foundation. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to local tomography and, in particular, to methods, systems, apparatus and devices for cone beam local tomography. 
       BACKGROUND AND PRIOR ART 
       [0003]    In computed tomography (CT) the goal is to reconstruct the distribution of the x-ray attenuation coefficient f inside the object being scanned. Local tomography (LT) computes not f, but Bf, where B is some operator that enhances singularities of f. In two dimensions (2D), B is an elliptic pseudo-differential operator (PDO) of order one as described by A. Katsevich, “Local Tomography for the Generalized Random Transform”, SIAM Journal on Applied Mathematics, Vol. 57, no. 4 (1997) pp. 1128-1162, Kuchment et al., “On local Tomography”, Inverse Problem 11 (1995), pp. 571-589, A. Ramm and A. Katsevich, “The Random Transform and Local Tomography”, CRC Press, Boca Raton, Fla., 1996 and A. G. Ramm, “Necessary and Sufficient Conditions for a PDO to be a local tomography operator”, Comptes Rend Acad. Sci., Paris 332 (1996) pp. 613-618. In the cone beam setting (three dimensions) a LT function is denoted by g Λ . The corresponding operator B: f→g Λ  is much more complicated than in 2D. It preserves the so-called visible (or, useful) singularities and creates non-local artifacts. Unfortunately, the strength of these artifacts is the same as that of the useful singularities of g Λ  as described in A. Katsevich “Cone Beam Local Tomography”, SIAM Journal on Applied Mathematics (1999), pp. 2224-2246 and D. Finch et al, “Microlocal analysis of the X-ray transform with sources on a curve”, Inside out: Inverse Problems and Applications, Cambridge Univ. Press, (2003) pp. 193-218. 
       SUMMARY OF THE INVENTION 
       [0004]    A primary objective of the invention is to provide a new method, system, apparatus and device for improved cone beam local tomography. 
         [0005]    A secondary objective of the invention is to provide new methods, systems, apparatus and devices for cone beam local tomography that produces artifacts that are one order smoother in the scale of Sobolev spaces. 
         [0006]    A third objective of the invention is to provide new methods, systems, apparatus and devices for cone beam local tomography using the flexibility of LT, its relative stability with respect to inconsistencies in the data and the ability to accurately reconstruct edges inside objects, which provides important information complementing well-established inversion techniques. 
         [0007]    A fourth objective of the invention is to provide methods, systems, apparatus, and devices for reconstructing an image from local cone beam projection data with suppressed artifacts. 
         [0008]    A fifth objective of the invention is to provide methods, systems, apparatus, and devices for reconstructing an image from local cone beam projection data with suppressed artifacts without suppressing the useful features of the image. 
         [0009]    A sixth objective of the invention is to provide methods, systems, apparatus, and devices for determining a shift between a location of the reconstruction point shown on the reconstructed image and an actual location of the point to determine a location error of the moving object to correct the location of the useful features of the actual image. 
         [0010]    A first embodiment provides a method of reconstructing an image from cone beam data provided by at least one detector. First, cone beam projection data of an object is collected and stored in memory. The image is reconstructed from the local cone beam projection data, wherein artifacts of the image are suppressed without suppressing the strength of the useful features. During the reconstructing step, a direction of a derivative of the cone beam projection data that will result in suppressing the artifacts is found, the direction being a tangent to the curve, which represents the scan trajectory. Based on the direction, the cone beam projection data is differentiated along the direction of the tangent of the curve. The derivative results are processed, then back projection is applied to the processed data to extract the useful features of the image with suppressed artifacts. 
         [0011]    In an embodiment, a shift between a location of the reconstruction point shown on the reconstructed image and an actual location of the point in the object is determined to correct for the location of the useful features of the actual image. 
         [0012]    Further objects and advantages of this invention will be apparent from the following detailed description of preferred embodiments which are illustrated schematically in the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0013]      FIG. 1  shows an overview of the basic process steps according to the invention. 
           [0014]      FIG. 2  is flow diagram for the back-projection substeps corresponding to step  40  of  FIG. 1 . 
           [0015]      FIG. 3   a  shows the new LT function g in the case of the clock phantom. 
           [0016]      FIG. 3   b  shows old LT function g Λ  in the case of the clock phantom. 
           [0017]      FIG. 4   a  shows the results of reconstructing the three-disk phantom according to the present invention. 
           [0018]      FIG. 4   b  shows the results of reconstructing the three-disk phantom according to the prior art. 
           [0019]      FIG. 5   a  shows the results for a phantom consisting of two identical balls having radius  40  centered at (−80,0,0) and (80,0,0), respectively, according to the present invention. 
           [0020]      FIG. 5   b  shows the results for a phantom consisting of two identical balls having radius  40  centered at (−80,0,0) and (80,0,0), respectively, according to the prior art. 
           [0021]      FIG. 6   a  shows reconstruction of an ellipsoid in the dynamic case, new LT function g according to the present invention wherein the ellipsoid moves either up or down. 
           [0022]      FIG. 6   b  shows reconstruction of an ellipsoid in the dynamic case, new LT function g according to the present invention wherein the ellipsoid moves either left or right 
           [0023]      FIG. 7   a  shows reconstruction of an ellipsoid in the dynamic case, new LT function g according to the present invention wherein the ellipsoid expands and contracts. 
           [0024]      FIG. 7   b  shows reconstruction of an ellipsoid in the dynamic case, the prior art LT function g wherein the ellipsoid expands and contracts as well as moves left-right. 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0025]    Before explaining the disclosed embodiments of the present invention in detail it is to be understood that the invention is not limited in its application to the details of the particular arrangements shown since the invention is capable of other embodiments. Also, the terminology used herein is for the purpose of description and not of limitation. 
       Local Tomography Function of the Present Invention: 
       [0026]    C is a smooth curve defined by I         s→y(s)⊂R 3 . Here 
         [0027]    s is a real parameter; 
         [0028]    I is a compact interval; 
         [0029]    y(s) denote the parametric equations of the curve; and 
         [0030]    R 3  is the three-dimensional Euclidean space. 
         [0000]    Fix an open set V, whose closure  V ⊂R 3 \C. Let D f (s,α) denote the cone beam transform of f: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    where fεC 0   ∞ (V). Here
       α is a unit vector;   S 2  is the unit sphere in R 3 ; and   f is the function representing the distribution of the x-ray attenuation coefficient inside the object being scanned
 
Fix φεC 0   ∞ (I×V) and define:
       
 
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         [0000]    where q is real parameter and 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    g(x) defined by equation (2) above is one example of a new local tomography function of the present invention. Introduce the following notation 
         [0000]      Π( x ,ξ):={ yεR   3 :ξ·( y−x )=0}, 
         [0000]        WF   v ( f ):={( x ,ξ)ε WF ( f ):Π( x ,ξ) intersects  C  transversely},  (4) 
         [0000]        L   + ( s,z ):={ xεR   3   :x=y ( s )+ t ( z−y ( s )), t&gt; 0}. 
       Here 
       [0000]    
       
         
           
             ξ is a non-zero vector in R 3 ; 
             z is a point in R 3 ; 
             WF(f) is the wave-front of f; and 
             WF v (f) denotes the visible singularities of f. The set WF v (f) and the term ‘visible singularity’ were introduced in E. T. Quinto, “Singularies of the X-ray transform and limited data tomography in R 2  and R 3 ”, SIAM Journal of Mathematical Analysis, vol. 24 (1993), pp. 1215-1225.
 
Analogously to A. Katsevich, “Cone beam local tomography”, SIAM Journal on Applied Mathematics, vol. 59 (1999), pp. 2224-2246, the following results can be established: the operator B defined by equation (2) extends to a map E′(V)→E′(V), and
 
           
         
       
     
         [0000]      WF(g)⊂WF v (f)∪E(f,C), 
         [0000]        E ( f,C ):={( x ,ξ)ε T*V\ 0 :ξ⊥{dot over (y)} ( s ),ξ⊥( x−y ( s )), xεL   + ( s,z ),( z ,ξ)ε WF ( f ),( s,x )ε suppφ}.   (5) 
       Here E′(V) is the space of distributions with compact support inside V, and V is an open set not intersecting C. 
       [0038]    An equivalent definition of the set E(f,C) is as follows. If (z,ξ)εWF(f) and the plane Π(z,ξ) is tangent to C at some point y(s), then (x,ξ)εE(f,C) for all xεL + (s,z), (s,x)εsuppφ. 
         [0039]    ξ (0) ≠0 and x (0) εR 3 \C are selected such that the plane Π(x (0) ,ξ (0) ) intersects C transversely. Then there exists a sufficiently small conic neighborhood U×Ω⊂T*V\0 of (x (0) ,ξ (0) ) such that B(x,ξ) is a classical amplitude from the class S 1 (U×Ω). Moreover, if Π(x (0) ,ξ (0) ) does not intersect C, then B(x,ξ) is from the class S −∞ (U×Ω) for some neighborhood U×Ω         (x (0) ,ξ (0) ). 
         [0040]    The principal symbol of B is computed as follows. Denote 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    and pick δ, 0&lt;δ&lt;m. Let w(t) be a function with the properties 
         [0000]        w ( t )ε C   0   ∞ ([ m−δ,M +δ]), w ( t )=1 ,tε[m,M].   (7) 
       Representing f in terms of its Fourier transform and using equation (7) we get from equations (1) and (2) 
       [0041]    
       
         
           
             
               
                 
                   
                     
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         [0000]    Pick (x,ξ)εT*V\0 such that Π(x,ξ) intersects C, and the intersections are transversal. Asymptotics of B(x,σξ) as σ→∞ are found. Here σ is a real parameter. The stationary points of the phase in equation (9) are determined by solving 
         [0000]      (ξ· {dot over (y)} ( s ))( t− 1)=0,ξ·( x−y ( s ))=0  (10) 
         [0000]    for s and t. Let s j =s j (x,ξ) be the points of intersections of the plane Π(x,ξ) and curve C. As is easily seen, the critical points are (s=s j ,t=1). They are non-degenerate, because by assumption ξ·{dot over (y)}(s j )≠0. Now the stationary phase method immediately yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
       Analysis of Artifacts: 
       [0042]    Pick any (z (0) ,ξ (0) )εWF(f) such that the plane Π(z (0) ,ξ (0) ) is tangent to C at some y(s 0 ). Fix x (0) εL + (s 0 ,z (0) ), x (0) ∉{z (0) ,y(s 0 )} and (s 0 ,x (0) )εsuppφ. The problem is to find the behavior of g, defined by equation (2), in a neighborhood of x (0) . This behavior, of course, depends on the nature of the singularity of f at z (0) . Assuming that f is a conormal distribution associated to a smooth hypersurface S, which has nonzero Gaussian curvature at z (0) : fεI comp   m (V,N*S) for some m. Here N*S is the conormal bundle of S. Let U×Ω be a sufficiently small conic neighborhood of (z (0) ,ξ (0) ). In view of the above examples, it is assumed without loss of generality that {tilde over (f)}(ξ)=A(ξ)e iH(ξ) +A 1 (ξ) (cf. L. Hormander, The Analysis of Linear Partial Differential Operators, Volume IV, Springer Verlag, New York, 1985, Proposition 25.1.3). Here AεS m−3/4 (R 3 ) and A(ξ)≡0 on R 3 \Ω, A 1 εS −∞ (R 3 ), H(ξ)εC 0   ∞ (R 3 \0) is real-valued and homogeneous of degree one, and N*S=(H′(ξ),ξ) on U×Ω. 
         [0043]    The coordinate system is introduced with the origin at z (0) , such that ξ (0)  is along the x 3 -axis, and y(s 0 )=(c,0,0), where c=|y(s 0 )−z (0) |. Then 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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                           ( 
                           
                             ξ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             H 
                             
                               j 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                            
                           
                             ( 
                             
                               ξ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                             ) 
                           
                         
                         = 
                         0 
                       
                     
                     , 
                     
                       j 
                       = 
                       1 
                     
                     , 
                     2 
                     , 
                     
                       3 
                       ; 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                        
                       
                         
                           
                             
                               
                                 H 
                                 11 
                               
                                
                               
                                 ( 
                                 
                                   ξ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 H 
                                 12 
                               
                                
                               
                                 ( 
                                 
                                   ξ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 H 
                                 21 
                               
                                
                               
                                 ( 
                                 
                                   ξ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 H 
                                 22 
                               
                                
                               
                                 ( 
                                 
                                   ξ 
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                        
                     
                     ≠ 
                     0. 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The inequality {dot over (y)} 2 (s 0 )≠0 is equivalent to the assumption that the line tangent to C at y(s 0 ) does not contain z (0) . This assumption is not very restrictive, because almost all scanning protocols satisfy it. By assumption, x (0) ∉{z (0) ,y(s 0 )}, so b∉{0,c}. Denote φ 1 (s,x):=−φ(s,x)|x−y(s)|/(2π) 3 . Substituting the expression for {tilde over (f)} into equation (8), using equation (9) produces 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         g 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             R 
                             2 
                           
                         
                          
                         
                           
                             
                               ϕ 
                               1 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                            
                           
                             
                               ∫ 
                               Ω 
                             
                              
                             
                               
                                 
                                   A 
                                    
                                   
                                     ( 
                                     ξ 
                                     ) 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                
                                               
                                                 x 
                                                 - 
                                                 
                                                   y 
                                                    
                                                   
                                                     ( 
                                                     s 
                                                     ) 
                                                   
                                                 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               ξ 
                                               · 
                                               
                                                 
                                                   y 
                                                   . 
                                                 
                                                  
                                                 
                                                   ( 
                                                   s 
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       O 
                                        
                                       
                                         ( 
                                         ξ 
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               × 
                               
                                  
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         H 
                                          
                                         
                                           ( 
                                           ξ 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         ξ 
                                         · 
                                         
                                           z 
                                            
                                           
                                             ( 
                                             
                                               s 
                                               , 
                                               t 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                  
                                 ξ 
                               
                                
                               
                                  
                                 t 
                               
                                
                               
                                  
                                 s 
                               
                             
                           
                         
                       
                     
                     ∈ 
                     
                       
                         C 
                         0 
                         ∞ 
                       
                        
                       
                         ( 
                         V 
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         z 
                          
                         
                           ( 
                           
                             s 
                             , 
                             t 
                           
                           ) 
                         
                       
                        
                       
                         : 
                       
                     
                     = 
                     
                       
                         y 
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                       + 
                       
                         t 
                          
                         
                           ( 
                           
                             x 
                             - 
                             
                               y 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the integral with respect to ξ is understood as an oscillatory one. For simplicity of notation, the dependence of z(s,t) on x is omitted. Let I denote the integral with respect to ξ in equation (13). Changing variables ξ→({circumflex over (ξ)},λ), where ξ=λ({circumflex over (ξ)},1), 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ξ 
                           ^ 
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 ξ 
                                 ^ 
                               
                               1 
                             
                             , 
                             
                               
                                 ξ 
                                 ^ 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       , 
                       
                         
                           
                             ξ 
                             ^ 
                           
                           j 
                         
                         = 
                         
                           
                             ξ 
                             j 
                           
                           / 
                           λ 
                         
                       
                       , 
                       
                         j 
                         = 
                         1 
                       
                       , 
                       
                         2 
                          
                         
                             
                         
                          
                         produces 
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         I 
                         = 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               λ 
                               4 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   Ω 
                                   ^ 
                                 
                               
                                
                               
                                 
                                   
                                     A 
                                      
                                     
                                       ( 
                                       
                                         λ 
                                          
                                         
                                           ( 
                                           
                                             
                                               ξ 
                                               ^ 
                                             
                                             , 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           w 
                                            
                                           
                                             ( 
                                             
                                               t 
                                                
                                               
                                                  
                                                 
                                                   x 
                                                   - 
                                                   
                                                     y 
                                                      
                                                     
                                                       ( 
                                                       s 
                                                       ) 
                                                     
                                                   
                                                 
                                                  
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             ( 
                                             
                                               
                                                 t 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       ξ 
                                                       ^ 
                                                     
                                                     , 
                                                     1 
                                                   
                                                   ) 
                                                 
                                               
                                               · 
                                               
                                                 
                                                   y 
                                                   . 
                                                 
                                                  
                                                 
                                                   ( 
                                                   s 
                                                   ) 
                                                 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                       
                                       + 
                                       
                                         O 
                                          
                                         
                                           ( 
                                           
                                             λ 
                                             
                                               - 
                                               1 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 × 
                                 
                                    
                                   
                                     
                                       λ 
                                        
                                       
                                         ( 
                                         
                                           
                                             H 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   ^ 
                                                 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           - 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   ξ 
                                                   ^ 
                                                 
                                                 , 
                                                 1 
                                               
                                               ) 
                                             
                                             · 
                                             
                                               z 
                                                
                                               
                                                 ( 
                                                 
                                                   s 
                                                   , 
                                                   t 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     ξ 
                                     ^ 
                                   
                                 
                                  
                                 
                                    
                                   λ 
                                 
                               
                             
                           
                         
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where {circumflex over (Ω)} is a small neighborhood of the origin in R 2 . The asymptotics of the integral with respect to {circumflex over (ξ)} in equation (14) is obtained by the stationary phase method: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       κ 
                       λ 
                     
                      
                     
                       
                         A 
                          
                         
                           ( 
                           
                             λ 
                              
                             
                               ( 
                               
                                 
                                   
                                     ξ 
                                     ^ 
                                   
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       t 
                                     
                                     ) 
                                   
                                 
                                 , 
                                 1 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               w 
                                
                               
                                 ( 
                                 
                                   t 
                                    
                                   
                                      
                                     
                                       x 
                                       - 
                                       
                                         y 
                                          
                                         
                                           ( 
                                           s 
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 { 
                                 
                                   
                                     t 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             ξ 
                                             ^ 
                                           
                                            
                                           
                                             ( 
                                             
                                               s 
                                               , 
                                               t 
                                             
                                             ) 
                                           
                                         
                                         , 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     
                                       y 
                                       . 
                                     
                                      
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                 
                                 } 
                               
                               2 
                             
                           
                           + 
                           
                             O 
                              
                             
                               ( 
                               
                                 λ 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     × 
                     
                        
                       
                         λ 
                          
                         
                           ( 
                           
                             
                               
                                 x 
                                 3 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 z 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     λ 
                     -&gt; 
                     ∞ 
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           x 
                           3 
                           
                             ( 
                             0 
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             s 
                             , 
                             t 
                           
                           ) 
                         
                       
                        
                       
                         : 
                       
                     
                     = 
                     
                       
                         H 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             
                               ξ 
                               ^ 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                           , 
                           1 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and {circumflex over (ξ)}(s,t) is obtained by solving 
         [0000]        H   j ({circumflex over (ξ)},1)= z   j ( s,t ), j= 1,2.  (17) 
         [0000]    In equation (15) and everywhere below κ denotes various non-zero constants. In equation (15) the critical point on {circumflex over (Ω)} determined by equation (17) is non-degenerate and the homogeneity of H(ξ): 
         [0000]        H=ξ·H′={circumflex over (ξ)}·H   {circumflex over (ξ)}   +H   3   ={circumflex over (ξ)}·{circumflex over (z)}+H   3 ,  (18) 
         [0000]      so 
         [0000]        x   3   (0) ( s,t )= H ({circumflex over (ξ)}( s,t ),1)−{circumflex over (ξ)}( s,t )· {circumflex over (z)} ( s,t ),  (19) 
         [0000]    where {circumflex over (z)}=(z 1 ,z 2 ). Substituting equations (15) and (14) into equation (13) produces: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         g 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       - 
                       
                         κ 
                          
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                            
                           
                             
                               λ 
                               3 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   R 
                                   2 
                                 
                               
                                
                               
                                 
                                   
                                     ϕ 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   A 
                                    
                                   
                                     ( 
                                     
                                       λ 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               ξ 
                                               ^ 
                                             
                                              
                                             
                                               ( 
                                               
                                                 s 
                                                 , 
                                                 t 
                                               
                                               ) 
                                             
                                           
                                           , 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 × 
                                 
                                   [ 
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                
                                               
                                                 x 
                                                 - 
                                                 
                                                   y 
                                                    
                                                   
                                                     ( 
                                                     s 
                                                     ) 
                                                   
                                                 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           { 
                                           
                                             
                                               t 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       ξ 
                                                       ^ 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         s 
                                                         , 
                                                         t 
                                                       
                                                       ) 
                                                     
                                                   
                                                   , 
                                                   1 
                                                 
                                                 ) 
                                               
                                             
                                             · 
                                             
                                               
                                                 y 
                                                 . 
                                               
                                                
                                               
                                                 ( 
                                                 s 
                                                 ) 
                                               
                                             
                                           
                                           } 
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       O 
                                        
                                       
                                         ( 
                                         
                                           λ 
                                           
                                             - 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                 × 
                                 
                                    
                                   
                                     λ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       a 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           , 
                                           t 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                  
                                 
                                    
                                   t 
                                 
                                  
                                 
                                    
                                   s 
                                 
                                  
                                 
                                    
                                   λ 
                                 
                               
                             
                           
                         
                       
                     
                     ∈ 
                     
                       
                         C 
                         0 
                         ∞ 
                       
                        
                       
                         ( 
                         V 
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         a 
                          
                         
                           ( 
                           
                             s 
                             , 
                             t 
                           
                           ) 
                         
                       
                        
                       
                         : 
                       
                     
                     = 
                     
                       
                         
                           x 
                           3 
                           
                             ( 
                             0 
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             s 
                             , 
                             t 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               
                                 y 
                                 3 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                             + 
                             
                               t 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     3 
                                   
                                   - 
                                   
                                     
                                       y 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
       From equations (16) and (17), 
       [0044]        x   3   (0) ( s,t )= h ( ŷ ( s )+ t ( {circumflex over (x)}−ŷ ( s ))),  (21) 
         [0000]    where z 3 =h(z 1 ,z 2 ) is the local equation of S in a neighborhood of z (0) . From equation (12), 
         [0000]        h   1 ( {circumflex over (z)}   (0) )= h   2 ( {circumflex over (z)}   (0) )=0.  (22) 
       Assume first x=x (0)  in (20). From equation (20) minus equation (22), a=0 and 
       [0045]        a   s =( h   1   {dot over (y)}   1   +h   2   {dot over (y)}   2   −{dot over (y)}   3 )(1 −t )=0, 
         [0000]        a   t   =h   1 ( x   1   (0)   −y   1 )+ h   2 ( x   2   (0)   −y   2 )+ y   3 =0  (23) 
         [0000]    when s=s 0 ,t=t 0 :=−c/(b−c). By assumption b≠c, so t 0  is defined. At the critical point 
         [0000]      (ξ( s   0   ,t   0 ),1)· {dot over (y)} ( s   0 )=0.  (24) 
       From equation (12), 
       [0046]    
       
         
           
             
               
                 
                   
                     
                       
                         a 
                         ss 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               b 
                               
                                 b 
                                 - 
                                 c 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 11 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                                 2 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 h 
                                 12 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 h 
                                 22 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           
                             y 
                             ¨ 
                           
                           3 
                         
                          
                         
                           b 
                           
                             b 
                             - 
                             c 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         a 
                         st 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       b 
                        
                       
                         ( 
                         
                           
                             
                               h 
                               11 
                             
                              
                             
                               
                                 y 
                                 . 
                               
                               1 
                             
                           
                           + 
                           
                             
                               h 
                               12 
                             
                              
                             
                               
                                 y 
                                 . 
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         a 
                         tt 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           h 
                           11 
                         
                          
                         
                           ( 
                           
                             b 
                             - 
                             c 
                           
                           ) 
                         
                       
                       2 
                     
                   
                   , 
                   
                     
 
                   
                    
                   so 
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     det 
                      
                     
                       ( 
                       
                         
                           a 
                           ″ 
                         
                          
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                             , 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       b 
                        
                       
                         [ 
                         
                           
                             b 
                              
                             
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       h 
                                       11 
                                     
                                      
                                     
                                       h 
                                       22 
                                     
                                   
                                   - 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           h 
                                           12 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 y 
                                 ¨ 
                               
                               3 
                             
                              
                             
                               
                                 h 
                                 11 
                               
                                
                               
                                 ( 
                                 
                                   b 
                                   - 
                                   c 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
       Hence, in addition to b=0 (i.e., x (0) =z (0) ), there is at most one more point on the ray L + (s 0 ,z (0) ), where det(a″) can be zero. 
       [0047]    Considering now x=(b,0,ρ) in (20) and arguing similarly to A. Katsevich, “Cone Beam Local Tomography”, SIAM Journal on Applied Mathematics (1999), pp. 2224-2246, gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         s 
                         0 
                       
                        
                       
                         ( 
                         ρ 
                         ) 
                       
                     
                     = 
                     
                       
                         s 
                         0 
                       
                       + 
                       
                         O 
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       
                         t 
                         0 
                       
                        
                       
                         ( 
                         ρ 
                         ) 
                       
                     
                     = 
                     
                       
                         t 
                         0 
                       
                       + 
                       
                         O 
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       a 
                        
                       
                         ( 
                         
                           
                             
                               s 
                               0 
                             
                              
                             
                               ( 
                               ρ 
                               ) 
                             
                           
                           , 
                           
                             
                               t 
                               0 
                             
                              
                             
                               ( 
                               ρ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           c 
                           
                             b 
                             - 
                             c 
                           
                         
                          
                         ρ 
                       
                       + 
                       
                         O 
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where all O(ρ) are smooth functions of b and p. Consequently, 
         [0000]      ({circumflex over (ξ)}( s   0 (ρ), t   0 (ρ)),1)· {dot over (y)} ( s   0 (ρ))= O (ρ).  (28) 
       Using equations (27) and (28), we get under the assumption det(a″)≠0: 
       [0048]    
       
         
           
             
               
                 
                   
                     
                       g 
                        
                       
                         ( 
                         
                           b 
                           , 
                           0 
                           , 
                           ρ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           Φ 
                            
                           
                             ( 
                             
                               
                                 λ 
                                 ; 
                                 b 
                               
                               , 
                               ρ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             λ 
                              
                             
                               c 
                               
                                 b 
                                 - 
                                 c 
                               
                             
                              
                             
                               ρ 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     O 
                                      
                                     
                                       ( 
                                       ρ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                         
                            
                           λ 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       Φ 
                        
                       
                         ( 
                         
                           
                             λ 
                             ; 
                             b 
                           
                           , 
                           ρ 
                         
                         ) 
                       
                     
                     = 
                     
                       O 
                        
                       
                         ( 
                         
                           λ 
                           
                             m 
                             + 
                             1 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     λ 
                     -&gt; 
                     ∞ 
                   
                   , 
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Φ admits an asymptotic expansion in decreasing powers of λ as λ→∞. The coefficients of the expansion are C ∞  functions of b and ρ and the expansion can be differentiated with respect to b, ρ and λ. Similarly, O(ρ) in equation (29) is a smooth function of b and ρ. 
         [0049]    The result analogous to equation (29) for the LT function proposed in A. K. Louis and P. Maass, “Contour reconstruction in 3-D X-ray CT”, IEEE Transactions on Medical Imaging, vol. 12 (1993), pp. 764-769 (see A. Katsevich, “Cone beam local tomography”, SIAM J. on Applied Mathematics, vol. 59 (1999) pp. 2224-2246) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         g 
                         Λ 
                       
                        
                       
                         ( 
                         
                           b 
                           , 
                           0 
                           , 
                           ρ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           Φ 
                            
                           
                             ( 
                             
                               
                                 λ 
                                 ; 
                                 b 
                               
                               , 
                               ρ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             λ 
                              
                             
                               c 
                               
                                 b 
                                 - 
                                 c 
                               
                             
                              
                             
                               ρ 
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     O 
                                      
                                     
                                       ( 
                                       ρ 
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                         
                            
                           λ 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       Φ 
                        
                       
                         ( 
                         
                           
                             λ 
                             ; 
                             b 
                           
                           , 
                           ρ 
                         
                         ) 
                       
                     
                     = 
                     
                       O 
                        
                       
                         ( 
                         
                           λ 
                           
                             m 
                             + 
                             2 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     λ 
                     -&gt; 
                     
                       ∞ 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (30) follows immediately from equations (6.10), (6.11) in A. Katsevich, “Cone beam local tomography”, SIAM J. on Applied Mathematics, vol. 59 (1999) pp. 2224-2246, if one recalls that the Fourier transform of a function in R 3 , which is discontinuous across a smooth surface S with nonzero Gaussian curvature, is of order O(|ξ| −2 ) as |ξ|→∞. 
         [0050]    Comparing equations (29) and (30) it is shown that the function in equation (2) has non-local artifacts, which are an order of magnitude smaller. Because of the direction along which the derivative is computed in equation (2), the leading term in the expansion of Φ of order O(λ m+2 ) disappears and the artifact is reduced. 
       Dynamic Local Tomography: 
       [0051]    Consider now the case when f changes with time. The parameter s in the definition of the curve C is regarded as time, it is assumed that f is of the form f(s,x), and the cone beam data are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         D 
                         f 
                       
                        
                       
                         ( 
                         
                           s 
                           , 
                           α 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           f 
                            
                           
                             ( 
                             
                               s 
                               , 
                               
                                 
                                   y 
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           t 
                         
                       
                     
                   
                   , 
                   
                     s 
                     ∈ 
                     I 
                   
                   , 
                   
                     α 
                     ∈ 
                     
                       
                         S 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
       In the dynamic case the LT function will still be defined by equation (2). The only difference is that the data are now given by equation (31) instead of equation (1). 
       [0052]    Even though in practice f(s,z) does not vanish when s∉I, to simplify the notation assume fεC 0   ∞ (I×V). Clearly, this does not result in any loss of generality, because an interval I′⊃I sufficiently close to I is selected, C is extended appropriately, and f is multiplied by a cut-off χεC 0   ∞ (I′), such that χ=1 on I. Since φεC 0   ∞ (I×V), replacing f by χf will not affect g. The same argument applies when f is a distribution. 
         [0053]    First, the operator B defined by equations (31) and (2) extends to a map E′(I×V)→E′(V). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       For 
                        
                       
                           
                       
                        
                       ψ 
                     
                     ∈ 
                     
                       
                         
                           C 
                           ∞ 
                         
                          
                         
                           ( 
                           V 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           Bf 
                           , 
                           ψ 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         V 
                       
                        
                       
                         
                           ψ 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                           
                             ∫ 
                             I 
                           
                            
                           
                             
                               ϕ 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                             × 
                             
                               
                                 
                                   ∂ 
                                   2 
                                 
                                 
                                   ∂ 
                                   
                                     q 
                                     2 
                                   
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                      
                                     
                                       x 
                                       - 
                                       
                                         y 
                                          
                                         
                                           ( 
                                           q 
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                    
                                   
                                     
                                       ∫ 
                                       0 
                                       ∞ 
                                     
                                      
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                
                                               
                                                 x 
                                                 - 
                                                 
                                                   y 
                                                    
                                                   
                                                     ( 
                                                     q 
                                                     ) 
                                                   
                                                 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         f 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             , 
                                             
                                               
                                                 y 
                                                  
                                                 
                                                   ( 
                                                   s 
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 t 
                                                  
                                                 
                                                   ( 
                                                   
                                                     x 
                                                     - 
                                                     
                                                       y 
                                                        
                                                       
                                                         ( 
                                                         q 
                                                         ) 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         t 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                      
                     
                        
                       
                         q 
                         = 
                         s 
                       
                     
                      
                     
                       
                          
                         s 
                       
                        
                       
                         
                            
                           x 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
       Changing variables x→z=y(s)+t(x−y(q)), after simple transformations 
       [0054]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       Bf 
                       , 
                       ψ 
                     
                     ) 
                   
                   = 
                   
                     
                       ∫ 
                       I 
                     
                      
                     
                       
                         ∫ 
                         V 
                       
                        
                       
                         
                           [ 
                           
                             
                               
                                  
                                 
                                   z 
                                   - 
                                   
                                     y 
                                      
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                 
                                  
                               
                                
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     t 
                                      
                                     
                                        
                                       
                                         z 
                                         - 
                                         
                                           y 
                                            
                                           
                                             ( 
                                             s 
                                             ) 
                                           
                                         
                                       
                                        
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∂ 
                                   2 
                                 
                                 
                                   ∂ 
                                   
                                     q 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ∫ 
                                     0 
                                     ∞ 
                                   
                                    
                                   
                                     
                                       ϕ 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           , 
                                           
                                             x 
                                              
                                             
                                               ( 
                                               · 
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ψ 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           , 
                                           
                                             x 
                                              
                                             
                                               ( 
                                               · 
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       t 
                                       
                                         - 
                                         4 
                                       
                                     
                                      
                                     
                                        
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 q 
                                 = 
                                 s 
                               
                             
                           
                           ] 
                         
                         × 
                         
                           f 
                            
                           
                             ( 
                             
                               s 
                               , 
                               z 
                             
                             ) 
                           
                         
                          
                         
                            
                           z 
                         
                          
                         
                           
                              
                             s 
                           
                           . 
                           
                             
 
                           
                            
                           where 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                        
                       
                         ( 
                         · 
                         ) 
                       
                     
                      
                     
                       : 
                     
                     = 
                     
                       x 
                        
                       
                         ( 
                         
                           z 
                           , 
                           s 
                           , 
                           q 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         t 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           z 
                           - 
                           
                             y 
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         y 
                          
                         
                           ( 
                           q 
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0055]    Since φεC 0   ∞ (I×V), t is bounded away from zero in equation (33), the expression in brackets is a C ∞ (I×V) function, and the desired assertion follows immediately by duality. 
         [0056]    Next the relation between the wave fronts of f and Bf is investigated without considering the most general situation, but concentrating on the practically relevant case when f is a conormal distribution, which depends smoothly on s. An arbitrary (s 0 ,z (0) ,η (0) )εI×(T*V\0) is selected. Because of linearity, the following assumptions are made about f without loosing any generality.
       1. supp f belongs to a small neighborhood of (s 0 ,z (0) );   2. For a sufficiently small open cone Ω         η (0)  the Fourier transform of f(s,z) with respect to z satisfies:       
 
         [0000]        {tilde over (f)} ( s ,η)= A ( s ,η) e   iH(s,η)   +A   1 ( s ,η).  (35)       3. Here
           A(s,η) is smooth, identically vanishes on R 3 , Ω, and (δ/δs) k A(s,η)εS m (R 3 ) uniformly in s for all k≧0;   H(s,η) is smooth (away from ξ=0), real-valued, and homogeneous of degree one in 77;   A 1 (s,η)εC ∞  and (δ/δs) k A 1 (s,η)=o(|η| −N ),|η|→∞, uniformly in s for all k≧0 and N≧0.   
                 
         [0063]    When f is as described above and WF(g) is a subset of all (x,ξ)εT*V\0 which satisfy 
         [0000]        y ( s )+ t ( x−y ( s ))= H   η ( s ,η) 
         [0000]      η·{dot over (y)}( s )(1 −t )= H   s ( s ,η)  (36) 
         [0000]      η·( x−y ( s ))=0 
         [0000]      ξ= tη   
         [0000]    for some (s,x)εsupp φ and ηεΩ. 
         [0064]    Clearly the derivative δ 2 /δq 2  in equation (2) can be omitted when finding WF(g). From equation (35) the integral is studied 
         [0000]    
       
         
           
             
               
                 
                   
                     ∫ 
                     
                       R 
                       3 
                     
                   
                    
                   
                     
                       ∫ 
                       Ω 
                     
                      
                     
                       
                         ∫ 
                         I 
                       
                        
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                          
                         
                           
                             
                               ϕ 
                               1 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 t 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                            
                           
                             A 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 η 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                                
                                
                               
                                 [ 
                                 
                                   
                                     H 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         , 
                                         η 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     η 
                                     · 
                                     
                                       ( 
                                       
                                         
                                           y 
                                            
                                           
                                             ( 
                                             s 
                                             ) 
                                           
                                         
                                         + 
                                         
                                           t 
                                            
                                           
                                             ( 
                                             
                                               x 
                                               - 
                                               
                                                 y 
                                                  
                                                 
                                                   ( 
                                                   s 
                                                   ) 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     ξ 
                                     · 
                                     x 
                                   
                                 
                                 ] 
                               
                             
                           
                            
                           
                              
                             t 
                           
                            
                           
                              
                             s 
                           
                            
                           
                              
                             η 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    for some smooth φ 1  with compact support in the domain t&gt;0. Differentiating the expression in brackets in equation (37) produces 
         [0000]      [•] η   =H   η ( s ,η)−( y ( s )+ t ( x−y ( s ))) 
         [0000]      [•] s   =H   s ( s ,η)−η· {dot over (y)} ( s )(1 −t )  (38) 
         [0000]      [•] t =−η·( x−y ( s )) 
         [0000]      [•] x   =ξ−tη.    
         [0065]    In an embodiment, a shift between the location of a reconstruction point shown on the reconstructed image and the actual location of the point of the useful feature. The step of determining the error of the moving object is especially useful when reconstructing an image of a moving object to correct for the location of the useful features of the actual image. Suppose first that η·{dot over (y)}(s)≠0 in equation (36). Then to (z,η)εWF(f(s,•)) such that y(s)εΠ(z,η)∩C there corresponds (x,ξ)εWF(g), where 
         [0000]    
       
         
           
             
               
                 
                   
                     x 
                     = 
                     
                       z 
                       + 
                       
                         
                           ( 
                           
                             z 
                             - 
                             
                               y 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               
                                 - 
                                 1 
                               
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     ξ 
                     = 
                     
                       t 
                        
                       
                           
                       
                        
                       η 
                     
                   
                   , 
                   
                     t 
                     = 
                     
                       1 
                       - 
                       
                         
                           
                             
                               H 
                               s 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 η 
                               
                               ) 
                             
                           
                           
                             η 
                             · 
                             
                               
                                 y 
                                 . 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
       If η·{dot over (y)}(s)=0 and H s (s,η)=0, then to (z,η)εWF(f(s,•)) such that y(s)εΠ(z,η)∩C there corresponds (x,ξ)εWF(g), where 
       [0066]        x=y ( s )+( z−y ( s ))/ t,    
         [0000]      ξ= tη,t&gt; 0.  (40) 
         [0067]    Thus, similarly to the static case, WF(g) consists of useful singularities described by equation (39), and of non-local artifacts, which are described by equation (40). However, as opposed to the static case, the useful singularities of g in general no longer coincide with WF(f). For example, the size of the shift between singsupp g and singsupp f depends on the quantity H s (s,η)/(η·{dot over (y)}(s)). Knowing this quantity can lead to the increased accuracy of locating singsupp f from singsupp g. If the motion of the object is approximately known, one can locate the singularities of g (represented as (x,ξ)εWF(g)), then estimate the quantity H s (s,η)/(η·{dot over (y)}(s)) based upon a priory information and equation (36), and then use equation (39) to find singsupp f (represented by z in that equation). 
         [0068]    The conditions leading to the appearance of the non-local artifact are somewhat different compared with the static case as well. The ray L + (s,z)⊂g only when the additional condition H s (s,η)=0 holds. Note that when H s (s,η)≡0, equation (36) transforms to equation (5). 
         [0069]    Next consider the quantitative relationship between the singularities of f and g. First, lemma 3 is needed. 
         [0070]    Suppose Π(z (0) ,η (0) )∩C=y(s 0 ) and η (0) ·{dot over (y)}(s 0 )≠0, where z (0) =H η (s 0 ,η (0) ). Find (x (0) ,ξ (0) ) by solving equation (36). Let Ω be a sufficiently small open cone containing ξ (0) . Solving the system equation (36) for s,t,x, and η in terms of ξεΩ determines C ∞ (Ω) functions s(ξ),t(ξ),x(ξ), and η(ξ). s,t,x are homogeneous of order 0, and η is homogeneous of order 1. 
         [0071]    Choose the coordinate system in which the x 3 -coordinate is along η (0) . The homogeneity of H implies δ 2 /(δη j δη 3 )H(s,η)| η=η     (0)   =0, j=1,2,3. Using equation (38), the matrix of the second derivatives evaluated at (s 0 ,x (0) ,η (0) ,ξ (0) ,t), where t is determined from the second equation in equation (36), becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                      
                     
                       : 
                     
                   
                   = 
                   
                     
                        
                       
                         
                           
                             
                               H 
                               11 
                               ″ 
                             
                           
                           
                             
                               H 
                               12 
                               ″ 
                             
                           
                           
                             0 
                           
                           
                             
                               
                                 H 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 ″ 
                               
                               - 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 Δ 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               t 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               H 
                               21 
                               ″ 
                             
                           
                           
                             
                               H 
                               22 
                               ″ 
                             
                           
                           
                             0 
                           
                           
                             
                               
                                 H 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ″ 
                               
                               - 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 Δ 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               - 
                               t 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 H 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                                 ″ 
                               
                               - 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   3 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               t 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     t 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   3 
                                 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     t 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 
                                   η 
                                   3 
                                 
                                  
                                 
                                   
                                     
                                       y 
                                       ¨ 
                                     
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       t 
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               
                                 η 
                                 3 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 Δ 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               
                                 Δ 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               
                                 η 
                                 3 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               
                                 η 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               t 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               t 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               t 
                             
                           
                           
                             0 
                           
                           
                             
                               - 
                               
                                 η 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                        
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
       Here Δ j =x j −y j (s), H″ jk =δ 2 /(δη j δη k )H(s,η), H″ sj =δ 2 /(δsδη j )H(s,η), j,k=1,2,3. In equation (41), Δ 3 =0. After simple transformations, 
       [0072]        D=t   4 η 3   2 ( {dot over (y)}   3   −H   s3 ) 2 .  (42) 
       Because of the homogeneity of H, H s =ξ·H sξ . Using equations (36) and (42) produces 
       [0073]        D=t   4 (η· {dot over (y)}−η·H   sη ) 2   =t   4 (η· {dot over (y)}−H   s ) 2   =t   6 (η· {dot over (y)} ) 2 &gt;0.  (43) 
       This proves that the functions s(ξ),t(ξ),x(ξ), and η(ξ) are C ∞  near ξ (0) . The statement about homogeneity follows from equation (36) and the homogeneity of H. Therefore, the functions are C ∞  inside the cone Ω. 
       [0074]    Using G(ξ):=ξ·x(ξ), the Lagrangian submanifold Λ of a conic neighborhood of (x (0) ,ξ (0) ) is defined as: Λ={(G′(ξ),ξ):ξεΩ}. 
         [0075]    Again using f as previously described, suppose Π(z (0) ,η (0) )∩C=y(s 0 ) and η (0) ·{dot over (y)}(s 0 )≠0. (x (0) ,ξ (0) ) is found by solving equation (36). Then equation (2) defines a distribution gεI m+7/4 (V,Λ), and for |ξ|&gt;1: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             
                               G 
                                
                               
                                 ( 
                                 ξ 
                                 ) 
                               
                             
                           
                         
                          
                         
                           
                             g 
                             _ 
                           
                            
                           
                             ( 
                             ξ 
                             ) 
                           
                         
                       
                       + 
                       
                         2 
                          
                         π 
                          
                         
                           
                             ϕ 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                              
                             
                                
                               
                                 x 
                                 - 
                                 
                                   y 
                                    
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                                
                             
                           
                           t 
                         
                          
                         
                            
                           
                             η 
                             · 
                             
                               
                                 y 
                                 . 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                            
                         
                          
                         
                           A 
                            
                           
                             ( 
                             
                               s 
                               , 
                               η 
                             
                             ) 
                           
                         
                       
                     
                     ∈ 
                     
                       
                         S 
                         m 
                       
                        
                       
                         ( 
                         
                           R 
                           3 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where s,t,x, and η are the functions of ξ obtained by solving the system equation (36), and A(s,η) is interpreted as 0 if no solution exists. e −iG(ξ) {tilde over (g)}(ξ) is polyhomogeneous if A is. 
         [0076]    Similarly to equations (8) and (9), it is found that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       g 
                       ~ 
                     
                      
                     
                       ( 
                       σξ 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         
                           
                             ( 
                             
                               2 
                                
                               π 
                             
                             ) 
                           
                           3 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         V 
                       
                        
                       
                         
                           ∫ 
                           
                             R 
                             3 
                           
                         
                          
                         
                           
                             ∫ 
                             I 
                           
                            
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                              
                             
                               
                                 
                                   ϕ 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                
                                               
                                                 x 
                                                 - 
                                                 
                                                   y 
                                                    
                                                   
                                                     ( 
                                                     s 
                                                     ) 
                                                   
                                                 
                                               
                                                
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         
                                           x 
                                           - 
                                           
                                             y 
                                              
                                             
                                               ( 
                                               s 
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             t 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               η 
                                               · 
                                               
                                                 y 
                                                  
                                                 
                                                   ( 
                                                   s 
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       O 
                                        
                                       
                                         ( 
                                         
                                            
                                           η 
                                            
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               × 
                               
                                 A 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     η 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                    
                                    
                                   
                                     [ 
                                     
                                       
                                         H 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             , 
                                             η 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         η 
                                         · 
                                         
                                           ( 
                                           
                                             
                                               y 
                                                
                                               
                                                 ( 
                                                 s 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               t 
                                                
                                               
                                                 ( 
                                                 
                                                   x 
                                                   - 
                                                   
                                                     y 
                                                      
                                                     
                                                       ( 
                                                       q 
                                                       ) 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         σξ 
                                         · 
                                         x 
                                       
                                     
                                     ] 
                                   
                                 
                               
                                
                               
                                  
                                 t 
                               
                                
                               
                                  
                                 s 
                               
                                
                               
                                  
                                 η 
                               
                                
                               
                                 
                                    
                                   x 
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Changing variables η→ση and factoring out σ, the same expression as the one in brackets in equation (37). Equation (43) implies that the signature of the matrix in equation (41) is a multiple of 4. Applying the same technique as in described in “The analysis of linear partial differential operators”, to the integral with respect to η in equation (45) and then using the stationary phase method. 
         [0077]    This method allows us to generalize the notion of visible singularities to the dynamic case. The singularity (z,η)εWF(f(s,•)) is visible, i.e. can be stably detected from the data, if (z−y(s))·η=0 and η·{dot over (y)}(s)≠0. Comparing the results with equations (8) and (11), it is concluded that in both the static and dynamic cases the visible singularities of f are reconstructed with the same resolution. In particular, there is no smearing due to motion in the dynamic case. 
         [0078]    As previously described, non-local artifacts arise in the dynamic case and the strength of these artifacts is the same as in the static case. 
         [0000]    (z (0) =H η (s 0 ,η (0) ),η (0) )εWF(f) is selected such that the plane Π(z (0) ,η (0) ) is tangent to C at y(s 0 ) and H s (s 0 ,η (0) )=0. Fix x (0) εL + (s 0 ,z (0) ), x (0) ∉{z (0) ,y(s 0 )} and (s 0 ,x (0) )εsuppφ. It is again assumed that f is of the form previously described and the Gaussian curvature of the surface z(η)=H′ η (s 0 ,η),ηεΩ, is nonzero near z (0) . Subtracting equation (21) from equation (12) produces the same expression as in equation (20), but the function x 3   (0)  is now given by 
         [0000]        x   3   (0) ( s,t )= h ( s,ŷ ( s )+ t ( {circumflex over (x)}−ŷ ( s ))).  (46) 
       Here z 3 =h(s,z 1 ,z 2 ) is determined by solving 
       [0079]        H   j ( s ,({circumflex over (ξ)},1))= z   j ( s,t ), j= 1,2,  (47) 
         [0000]    for {circumflex over (ξ)}={circumflex over (ξ)}(s,t) (cf. (17)), and then substituting {circumflex over (ξ)}(s,t) into H′ 3  (cf. (16) and (21)): 
         [0000]        h ( s,z   1   ,z   2 ):= H   3 ( s ,({circumflex over (ξ)}( s,t ),1)).  (48) 
       By assumption, 
       [0080]        h   1 ( s   0   ,{circumflex over (z)}   (0) )= h   2 ( s   0   ,{circumflex over (z)}   (0) )=0.  (49) 
       Again assuming x=x (0) , we get a=0 and 
       [0081]        a   s   =h   s +( h   1   {dot over (y)}   1   +h   2   {dot over (y)}   2   −{dot over (y)}   3 )(1 −t )=0, 
         [0000]        a   t   =h   1 ( x   1   (0)   −y   1 )+ h   2 ( x   2   (0)   −y   2 )+ y   3 =0  (50) 
         [0000]    when s=s 0 ,t=t 0 :=−c/(b−c). The additional condition H s (s 0 ,η (0) )=0 is used which implies h s (s 0 ,{circumflex over (z)} (0) )=0. The analogue of equation (25) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         a 
                         ss 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         h 
                         ss 
                       
                       + 
                       
                         2 
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                               
                             
                             + 
                             
                               
                                 h 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                          
                         
                           b 
                           
                             b 
                             - 
                             c 
                           
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               b 
                               
                                 b 
                                 - 
                                 c 
                               
                             
                             ) 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 11 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                                 2 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 h 
                                 12 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 h 
                                 22 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           
                             y 
                             ¨ 
                           
                           3 
                         
                          
                         
                           b 
                           
                             b 
                             - 
                             c 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         a 
                         st 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           h 
                           
                             s 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           ( 
                           
                             b 
                             - 
                             c 
                           
                           ) 
                         
                       
                       + 
                       
                         b 
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 11 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                               
                             
                             + 
                             
                               
                                 h 
                                 12 
                               
                                
                               
                                 
                                   y 
                                   . 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         a 
                         tt 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                           , 
                           
                             t 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           h 
                           11 
                         
                          
                         
                           ( 
                           
                             b 
                             - 
                             c 
                           
                           ) 
                         
                       
                       2 
                     
                   
                   , 
                   
                     
 
                   
                    
                   so 
                   , 
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     det 
                      
                     
                       ( 
                       
                         
                           a 
                           ″ 
                         
                          
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                             , 
                             
                               t 
                               0 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             b 
                             - 
                             c 
                           
                           ) 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             
                               h 
                               ss 
                             
                              
                             
                               h 
                               11 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 h 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                        
                       
                         b 
                          
                         
                           ( 
                           
                             b 
                             - 
                             c 
                           
                           ) 
                         
                       
                        
                       
                         
                           
                             y 
                             . 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             
                               
                                 h 
                                 11 
                               
                                
                               
                                 h 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 h 
                                 
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 h 
                                 12 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         b 
                          
                         
                           [ 
                           
                             
                               b 
                                
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   2 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         h 
                                         11 
                                       
                                        
                                       
                                         h 
                                         22 
                                       
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         
                                           ( 
                                           
                                             h 
                                             12 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   y 
                                   ¨ 
                                 
                                 3 
                               
                                
                               
                                 
                                   h 
                                   11 
                                 
                                  
                                 
                                   ( 
                                   
                                     b 
                                     - 
                                     c 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
         [0000]    There are at most two more points on the ray L + (s 0 ,z (0) ) (i.e., two values of b) where det(a″) can be zero. Thus, under the assumption that det(a″)≠0, the same expression as in equation (29) is derived for the non-local artifact in the dynamic case. 
       Numerical Implementation and Experiments: 
       [0082]    Since the LT function in equation (2) stays the same regardless of whether f depends on time or not, implementation of equation (2) is identical in both the static and dynamic cases. The following description describes how equation (2) is computed when the source trajectory is a helix: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         1 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       R 
                        
                       
                           
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             s 
                             - 
                             
                               s 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       
                         y 
                         2 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       R 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             s 
                             - 
                             
                               s 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       
                         y 
                         3 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                     = 
                     
                       
                         y 
                         3 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         
                           h 
                           
                             2 
                              
                             π 
                           
                         
                          
                         
                           s 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
       Here 
       [0083]    R is the radius of the helix; 
         [0084]    h is the pitch of the helix; 
         [0085]    s 0  and y 3   (0)  are the parameters that determine the initial point on the helix; and 
         [0086]    (y 1 (s),y 2 (s),y 3 (s)) are the coordinates of a point on the helix. 
         [0087]    Following the curved detector geometry described in F. Noo, J. Pack and D. Heuscher, “Exact helical reconstruction using native cone-beam Geometries”, Physics in Medicine and Biology, vol. 48, (2003), pp. 3787-3818, let α and w be the conventional detector coordinates: α is the polar angle (with the origin at the source), and w is the vertical coordinate. Hence the cone beam data are given by D f (s,(α,w)). Assuming that the detector contains the axis of rotation, let DP(s) denote the surface of the detector, which corresponds to the source located at y(s). In the numerical experiments it is tacitly assumed that the detector is large enough, e.g. that it contains at least the Tam window. In general, however, this is not required for LT. Let (α(q,s,x), w(q,s,x)) be the coordinates of the point where the ray z=y(s)+tβ(q,x),t&gt;0, intersects DP(s). Similarly, let (α(s,x), w(s,x)) be the coordinates of the projection of x onto DP(s). Clearly, 
         [0000]      α( q,s,x )| q=s =α( s,x ), w ( q,s,x )| q=s   =w ( s,x ).  (54) 
         [0000]    Using the chain rule: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           g 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ∫ 
                               I 
                             
                              
                             
                               
                                 ϕ 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     x 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∂ 
                                   2 
                                 
                                 
                                   ∂ 
                                   
                                     q 
                                     2 
                                   
                                 
                               
                                
                               
                                 
                                   D 
                                   f 
                                 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     
                                       ( 
                                       
                                         
                                           α 
                                            
                                           
                                             ( 
                                             
                                               q 
                                               , 
                                               s 
                                               , 
                                               x 
                                             
                                             ) 
                                           
                                         
                                         , 
                                         
                                           w 
                                            
                                           
                                             ( 
                                             
                                               q 
                                               , 
                                               s 
                                               , 
                                               x 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                            
                           
                              
                             
                               q 
                               = 
                               s 
                             
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             ≅ 
                               
                              
                             
                               
                                 ∫ 
                                 I 
                               
                                
                               
                                 
                                   ϕ 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         c 
                                         α 
                                         2 
                                       
                                        
                                       
                                         
                                           ∂ 
                                           2 
                                         
                                         
                                           ∂ 
                                           
                                             α 
                                             2 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                         c 
                                         α 
                                       
                                        
                                       
                                         c 
                                         w 
                                       
                                        
                                       
                                         
                                           ∂ 
                                           2 
                                         
                                         
                                           
                                             ∂ 
                                             α 
                                           
                                            
                                           
                                             ∂ 
                                             w 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         c 
                                         w 
                                         2 
                                       
                                        
                                       
                                         
                                           ∂ 
                                           2 
                                         
                                         
                                           ∂ 
                                           
                                             w 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     D 
                                     f 
                                   
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       
                                         ( 
                                         
                                           α 
                                           , 
                                           w 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               
                                 a 
                                 = 
                                 
                                   a 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 w 
                                 = 
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             s 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
         [0000]    is obtained from equation (2) where 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                       α 
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             α 
                              
                             
                               ( 
                               
                                 q 
                                 , 
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           q 
                         
                       
                        
                       
                          
                         
                           q 
                           = 
                           s 
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       c 
                       w 
                     
                     = 
                     
                       
                         
                           ∂ 
                           
                             w 
                              
                             
                               ( 
                               
                                 q 
                                 , 
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           q 
                         
                       
                        
                       
                          
                         
                           q 
                           = 
                           s 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and ≈ denotes equality up to the leading singular term. The terms omitted on the right in equation (55) contain, at most, the first order derivatives of D f , so that the results previously obtained apply to the new function. 
         [0088]    Using the approach described in id., for α(q,s,x) and w(q,s,x): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         tan 
                          
                         
                             
                         
                          
                         
                           α 
                            
                           
                             ( 
                             
                               q 
                               , 
                               s 
                               , 
                               x 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 y 
                                 1 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 - 
                                 
                                   
                                     y 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 y 
                                 2 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 - 
                                 
                                   
                                     y 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 1 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 - 
                                 
                                   
                                     y 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 y 
                                 2 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 - 
                                 
                                   
                                     y 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     q 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         w 
                          
                         
                           ( 
                           
                             q 
                             , 
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                       = 
                       
                         R 
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                 x 
                                 3 
                               
                               - 
                               
                                 
                                   y 
                                   3 
                                 
                                  
                                 
                                   ( 
                                   q 
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         1 
                                       
                                       - 
                                       
                                         
                                           y 
                                           1 
                                         
                                          
                                         
                                           ( 
                                           q 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       - 
                                       
                                         
                                           y 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           q 
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           . 
                           
                             
 
                           
                            
                           Recall 
                         
                          
                         
                             
                         
                          
                         that 
                          
                         
                           : 
                         
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                        
                       
                         ( 
                         
                           s 
                           , 
                           x 
                         
                         ) 
                       
                     
                     := 
                     
                       R 
                       - 
                       
                         
                           
                             
                               x 
                               1 
                             
                              
                             
                               
                                 y 
                                 1 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               x 
                               2 
                             
                              
                             
                               
                                 y 
                                 2 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                         R 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       tan 
                        
                       
                           
                       
                        
                       
                         α 
                          
                         
                           ( 
                           
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             x 
                             2 
                           
                            
                           
                             
                               y 
                               1 
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             x 
                             1 
                           
                            
                           
                             
                               y 
                               2 
                             
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                       
                       
                         RV 
                          
                         
                           ( 
                           
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       w 
                        
                       
                         ( 
                         
                           s 
                           , 
                           x 
                         
                         ) 
                       
                     
                     = 
                     
                       R 
                        
                       
                           
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         α 
                          
                         
                           ( 
                           
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                        
                       
                         
                           
                             
                               x 
                               3 
                             
                             - 
                             
                               
                                 y 
                                 3 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           
                             V 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
       Differentiating equation (57) with respect to q and using equations (58) and (54), after some transformations, it is found that 
       [0089]    
       
         
           
             
               
                 
                   
                     
                       c 
                       α 
                     
                     = 
                     
                       - 
                       
                         
                           R 
                            
                           
                               
                           
                            
                           
                             cos 
                             2 
                           
                            
                           
                             α 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           V 
                            
                           
                             ( 
                             
                               s 
                               , 
                               x 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       c 
                       w 
                     
                     = 
                     
                       
                         
                           R 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           V 
                            
                           
                             ( 
                             
                               s 
                               , 
                               x 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     x 
                                   
                                   ) 
                                 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 α 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     x 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               h 
                               
                                 2 
                                  
                                 π 
                               
                             
                           
                           ) 
                         
                         . 
                         
                           □ 
                           * 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
         [0090]    In general, there is a significant flexibility in selecting the function φ(s,x). By choosing the appropriate φ(s,x), a LT can be adopted for practically any scan geometry, including truncated projections, incomplete or segmented source trajectory, etc. A possible candidate, which guarantees the reconstruction of all visible singularities and is efficient from the computational point of view is determined. Let w=w bot (α) and w=w top (α) be the equations of the top and bottom boundaries of the Tam window on the detector. Choose any χεC 0   ∞ (R) such that χ≡1 on [0,1], and define 
         [0000]    
       
         
           
             
               
                 
                   
                     ϕ 
                      
                     
                       ( 
                       
                         s 
                         , 
                         x 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       χ 
                        
                       
                         ( 
                         
                           
                             
                               
                                 w 
                                 top 
                               
                                
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                             - 
                             w 
                           
                           
                             
                               
                                 w 
                                 top 
                               
                                
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                             - 
                             
                               
                                 w 
                                 bot 
                               
                                
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           a 
                           = 
                           
                             a 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           w 
                           = 
                           
                             w 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    By construction, φ(s,x)&gt;0 whenever sεI π (x) (the π-interval of x). This property guarantees that the visible singularities of f are detected (see equations (11) and (44)). The advantage of equation (60) is that φ(s,x) only depends on the projection of x onto DP(s), so the function is written in the form φ(α(s,x), w(s,x)). Equation (59) is used to rewrite equation (55) in the form 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         g 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       = 
                       
                         
                           ∫ 
                           I 
                         
                          
                         
                           
                             
                               R 
                               2 
                             
                             
                               
                                 V 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             Φ 
                              
                             
                               ( 
                               
                                 
                                   s 
                                   ; 
                                   
                                     α 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         , 
                                         x 
                                       
                                       ) 
                                     
                                   
                                 
                                 , 
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             s 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         Φ 
                          
                         
                           ( 
                           
                             
                               s 
                               ; 
                               α 
                             
                             , 
                             w 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ϕ 
                            
                           
                             ( 
                             
                               α 
                               , 
                               w 
                             
                             ) 
                           
                         
                          
                         
                           cos 
                           2 
                         
                          
                         
                           α 
                            
                           
                             [ 
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                  
                                 α 
                                  
                                 
                                   
                                     ∂ 
                                     2 
                                   
                                   
                                     ∂ 
                                     
                                       α 
                                       2 
                                     
                                   
                                 
                               
                               - 
                               
                                 2 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                    
                                   
                                     ( 
                                     
                                       
                                         w 
                                          
                                         
                                             
                                         
                                          
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         α 
                                       
                                       - 
                                       
                                         h 
                                         / 
                                         
                                           ( 
                                           
                                             2 
                                              
                                             π 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∂ 
                                     2 
                                   
                                   
                                     
                                       ∂ 
                                       α 
                                     
                                      
                                     
                                       ∂ 
                                       w 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         w 
                                          
                                         
                                             
                                         
                                          
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         α 
                                       
                                       - 
                                       
                                         h 
                                         / 
                                         
                                           ( 
                                           
                                             2 
                                              
                                             π 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     ∂ 
                                     2 
                                   
                                   
                                     ∂ 
                                     
                                       w 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           
                             
                               D 
                               f 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 
                                   ( 
                                   
                                     α 
                                     , 
                                     w 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
         [0000]    As is seen, equation (61) admits a very efficient filtered-backprojection implementation. Moreover, the filtering step consists only of computing derivatives, i.e. is local. 
         [0091]    For comparison purposes the LT function is computed which is analogous to the one described in A. K. Louis and P. Maass, “Contour reconstruction in 3-D X-ray CT”, IEEE Transaction on Medical Imaging, vol. 12 (1993) pp. 764-769 as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             Λ 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             I 
                           
                            
                           
                             
                               ϕ 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                              
                             
                               Δ 
                               x 
                             
                              
                             
                               
                                 D 
                                 f 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   
                                     ( 
                                     
                                       
                                         α 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             , 
                                             x 
                                           
                                           ) 
                                         
                                       
                                       , 
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             , 
                                             x 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               s 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≅ 
                           
                          
                         
                           
                             ∫ 
                             I 
                           
                            
                           
                             
                               ϕ 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                        
                                       
                                         
                                           ∇ 
                                           x 
                                         
                                          
                                         
                                           α 
                                            
                                           
                                             ( 
                                             
                                               s 
                                               , 
                                               x 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                     
                                     2 
                                   
                                    
                                   
                                     
                                       ∂ 
                                       2 
                                     
                                     
                                       ∂ 
                                       
                                         α 
                                         2 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     ∇ 
                                     x 
                                   
                                    
                                   
                                     α 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         , 
                                         x 
                                       
                                       ) 
                                     
                                   
                                 
                                 · 
                                 
                                   
                                     ∇ 
                                     x 
                                   
                                    
                                   
                                     w 
                                      
                                     
                                       ( 
                                       
                                         s 
                                         , 
                                         x 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 
                                   ∂ 
                                   α 
                                 
                                  
                                 
                                   ∂ 
                                   w 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                                 
                                
                               
                                 
                                   
                                      
                                     
                                       
                                         ∇ 
                                         x 
                                       
                                        
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             , 
                                             x 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                   
                                   2 
                                 
                                  
                                 
                                   
                                     ∂ 
                                     2 
                                   
                                   
                                     ∂ 
                                     
                                       w 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 D 
                                 f 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   
                                     ( 
                                     
                                       α 
                                       , 
                                       w 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           
                              
                             
                               
                                 a 
                                 = 
                                 
                                   a 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 w 
                                 = 
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             s 
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
       Using equation (58) produces 
       [0092]    
       
         
           
             
               
                 
                   
                     
                       
                         ∇ 
                         x 
                       
                        
                       
                         α 
                          
                         
                           ( 
                           
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             α 
                           
                           RV 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   y 
                                   1 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                               - 
                               
                                 
                                   y 
                                   2 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                             
                             , 
                             
                               
                                 
                                   y 
                                   1 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                               + 
                               
                                 
                                   y 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             a 
                             = 
                             
                               a 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                           
                           
                             V 
                             = 
                             
                               V 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         ∇ 
                         x 
                       
                        
                       
                         w 
                          
                         
                           ( 
                           
                             s 
                             , 
                             x 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             α 
                           
                           RV 
                         
                          
                         
                           ( 
                         
                          
                         w 
                          
                         
                           ( 
                         
                          
                         
                           y 
                           1 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         α 
                       
                       + 
                       
                         
                           y 
                           2 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         α 
                          
                         
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
                       w 
                        
                       
                         ( 
                       
                     
                     - 
                     
                       
                         y 
                         1 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       α 
                     
                     + 
                     
                       
                         y 
                         2 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       α 
                        
                       
                         ) 
                       
                     
                   
                   , 
                   
                     
                       
                         R 
                         2 
                       
                        
                       
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           a 
                           = 
                           
                             a 
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                         
                           
                             V 
                             = 
                             
                               V 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                           
                           
                             w 
                             = 
                             
                               w 
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   x 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
       After simple calculations, equation (63) implies 
       [0093]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           
                             ∇ 
                             x 
                           
                            
                           α 
                         
                          
                       
                       2 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             α 
                           
                           V 
                         
                         ) 
                       
                       2 
                     
                   
                   , 
                   
                     
                       
                         
                           ∇ 
                           x 
                         
                          
                         α 
                       
                       · 
                       
                         
                           ∇ 
                           x 
                         
                          
                         w 
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
                       
                          
                         
                           
                             ∇ 
                             x 
                           
                            
                           w 
                         
                          
                       
                       2 
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               α 
                             
                             V 
                           
                           ) 
                         
                         2 
                       
                        
                       
                         
                           ( 
                           
                             
                               R 
                               2 
                             
                             + 
                             
                               w 
                               2 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
       Finally, substitution of equation (64) into equation (62) yields 
       [0094]    
       
         
           
             
               
                 
                   
                     
                       
                         g 
                         Λ 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         I 
                       
                        
                       
                         
                           1 
                           
                             
                               V 
                               2 
                             
                              
                             
                               ( 
                               
                                 s 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                          
                         
                           
                             Φ 
                             Λ 
                           
                            
                           
                             ( 
                             
                               
                                 s 
                                 ; 
                                 
                                   α 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       , 
                                       x 
                                     
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     , 
                                     x 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           s 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         Φ 
                         Λ 
                       
                        
                       
                         ( 
                         
                           
                             s 
                             ; 
                             α 
                           
                           , 
                           w 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ϕ 
                          
                         
                           ( 
                           
                             α 
                             , 
                             w 
                           
                           ) 
                         
                       
                        
                       
                         cos 
                         2 
                       
                        
                       
                         α 
                         [ 
                         
                           
                             
                               ∂ 
                               2 
                             
                             
                               ∂ 
                               
                                 α 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   R 
                                   2 
                                 
                                 + 
                                 
                                   w 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 ∂ 
                                 2 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           
                             D 
                             f 
                           
                            
                           
                             ( 
                             
                               s 
                               , 
                               
                                 ( 
                                 
                                   α 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
         [0095]      FIG. 1  is a flow diagram of the steps for the cone beam local tomography according to the present invention. In step  10  the current cone beam projection is loaded into computer memory with the assumption that the CB projection loaded into computer memory corresponds to the x-ray source located at y(s). Using the CB projection data D f (y(s),(α,w)) in step  20 , the combination of derivatives that will result in suppression of the artifacts of the image is determined, wherein the combination of derivatives corresponds to the differentiation of the data along the direction tangent to the curve. Based on the determined combination direction, the cone beam projection data is differentiated. In this example, the derivatives 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                 
                   ∂ 
                   
                     α 
                     2 
                   
                 
               
                
               
                 
                   D 
                   f 
                 
                  
                 
                   ( 
                   
                     s 
                     , 
                     
                       ( 
                       
                         
                           α 
                           i 
                         
                         , 
                         
                           w 
                           j 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 
                   ∂ 
                   2 
                 
                 
                   
                     ∂ 
                     α 
                   
                    
                   
                     ∂ 
                     w 
                   
                 
               
                
               
                 
                   D 
                   f 
                 
                  
                 
                   ( 
                   
                     s 
                     , 
                     
                       ( 
                       
                         
                           α 
                           i 
                         
                         , 
                         
                           w 
                           j 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 
                   ∂ 
                   2 
                 
                 
                   ∂ 
                   
                     w 
                     2 
                   
                 
               
                
               
                 
                   D 
                   f 
                 
                  
                 
                   ( 
                   
                     s 
                     , 
                     
                       ( 
                       
                         
                           α 
                           i 
                         
                         , 
                         
                           w 
                           j 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    are found for all (α i ,w j ) on the detector. The computation can be accomplished using interpolation and finite differences. Thus, step  20  includes finding the combination of derivatives that will result in reduced artifacts and then computing the derivatives, in other words, it is local. 
         [0096]    In step  30 , the derivative results are processed to compute the determined combination of the derivatives. In this example, the derivatives are multiplied by the quantities cos 2  α, −2 cos α(w sin α−h/(2π)), and (w sin α−h/(2π)) 2 , respectively, for all (α i , w j ) on the detector. Then the results are multiplied by φ(α i , w j ) cos 2  α i  to produce the function Φ(s; α i , w j ) according to equation (61), where the function φ(α, w) is determined using equation (60) and the relation φ(α(s,x), w(s,x))=φ(s,x). By construction, φ(s, x)&gt;0 whenever sεI π (x) (the π-interval of x). This property guarantees that the visible singularities of f are detected. The advantage of equation (60) is that φ(s, x) only depends on the projection of x onto DP(s), so the function is written in the form φ(α(s,x), w(s,x)). 
         [0097]    In backprojection step  40 , for each reconstruction point x, the data found in step  30  is back-projected according to the first line of equation (61).  FIG. 2  is a seven substep flow diagram for backprojection, which corresponds to step  40  of  FIG. 1 . In step  41 , a reconstruction point x is fixed, which represents a point inside the patient where it is required to reconstruct the image. The reconstruction point x is projected onto the detector in step  42  by computing α(s,x), w(s,x) by formulas given in equation (58). In step  43  it is determined if φ(α(s,x), w(s,x))=0. If φ(α(s,x), w(s,x))=0, then the preprocessed CB data is not used for image reconstruction at x and step  41  is repeated for a different reconstruction point. If φ(α(s,x), w(s,x))≠0, then the filtered CB data affects the image at x and steps  44 - 47  are performed. 
         [0098]    In step  44 , the value Φ(s;α(s,x), w(s,x)) is estimated by interpolation from a few values Φ(s;α i ,w j ) for α i ,w j  close to α(s,x), w(s,x). Φ(s;α(s,x), w(s,x)) is multiplied by 
         [0000]    
       
         
           
             
               R 
               2 
             
             
               
                 V 
                 2 
               
                
               
                 ( 
                 
                   s 
                   , 
                   x 
                 
                 ) 
               
             
           
         
       
     
         [0000]    in step  45 , where V(s,x) is computed by equation (58). The result of step  45  is added to the image being reconstructed at the point x in step  46  according to a pre-selected scheme (for example, the Trapezoidal scheme) for approximate evaluation of the integral in equation (61). In step  47 , go to step  41  and select a different reconstruction point x. After repeating steps  41 - 47  for each reconstruction point inside the area of interest, the process advances to step  50 . 
         [0099]    Referring back to  FIG. 1 , steps  10 - 40  are repeated until the scan is finished or when the image reconstruction process is complete for the required points of the area of interest to reconstruct the useful features. At step  50 , the process returns to step  10  shown in  FIG. 1  to load the next CB projection data into computer memory. The image can be displayed at all reconstruction points x for which the image reconstruction process has been completed, that is, when subsequent CB projections are not necessary for reconstructing the image at those points. The CB projection that have already been processed is no longer needed for image reconstruction and is discarded from the computer memory. The algorithm concludes when the scan is finished or the image reconstruction process has completed at all the required points. 
         [0100]    For LT function testing, experiments were performed. Table 1 provides simulation and reconstruction parameters used for the experiments. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Simulation and reconstruction parameters 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 R (radius of the spiral) 
                 600 mm 
               
               
                   
                 h (pitch of the spiral) 
                  66 mm 
               
               
                   
                 detector pixel size 
                 10 −3  rad × 0.6 mm 
               
               
                   
                 (as projected to isocenter) 
               
               
                   
                 number of detector rows 
                 128 
               
               
                   
                 number of detectors per row 
                 950 
               
               
                   
                 number of source positions per rotation 
                 1000  
               
               
                   
                 voxel size in each direction 
                  1.0 mm 
               
               
                   
                   
               
             
          
         
       
     
         [0101]      FIGS. 3   a  and  3   b  show the results of reconstructing the conventional clock phantom as described in A. Katsevich, Samit Basu, and Jiang Hsieh, “Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography”, Physics in Medicine and Biology, vol. 49 (2004) pp. 3089-3103. The original clock phantom, which is slightly different from the one used here is described in H. Turbell and P. E. Danielsson, “Helical cone beam tomography”, Int. Journal of Imaging System and Technology, vol. 11 (2000), pp. 91-100. 
         [0102]      FIG. 4   a  shows the LT function g according to the present invention and  FIG. 4   b  shows the LT function g Λ  according to the prior art.  FIG. 4   a  shows the results of reconstructing the phantom, which consists of three elongated ellipsoids with half-axes 100×100×10 (all the sizes and coordinates are given in mm). The locations of their centers are (0,0,−40), (0,0,0), and (0,0,40), and the cross-section |x 1 |≦150,x 2 =0,|x 3 |≦60 is shown. 
         [0103]      FIGS. 5   a  and  5   b  show the results for another phantom according to another experiment. The phantom in this embodiment consists of two identical balls having radius  40  centered at approximately (−80,0,0) and approximately (80,0,0), respectively. As is seen from the figures, the non-local artifacts in g are weaker than those in g Λ . Reconstruction of the clock phantom with the cross-section |x 1 |≦255.5,|x 2 |≦255.5,x 3 =0 is shown in  FIG. 3   a .  FIG. 3   a  shows the LT function g according to the present invention and  FIG. 3   b  shows the LT function g Λ  according to the prior art. 
         [0104]    Reconstructions in the dynamic case are shown in  FIGS. 6 and 7  wherein the phantom is represented by a single ellipsoid 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 x 
                                 1 
                               
                               - 
                               
                                 
                                   x 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               a 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 x 
                                 2 
                               
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                               b 
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         ( 
                         
                           
                             
                               x 
                               3 
                             
                             - 
                             
                               
                                 x 
                                 3 
                               
                                
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                           
                             c 
                              
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     ≤ 
                     1 
                   
                   , 
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein the half-axes (a(s), b(s), c(s)) and the center (x 1 (s),x 2 (s),x 3 (s))—depend on time. For  FIG. 6   a:    
         [0000]        a ( s )= b ( s )= c ( s )=50, 
         [0000]        x   1 ( s )= x   2 ( s )=0 ,x   3 ( s )=10 sin( s/ 3).  (67) 
       For FIG. 6 b:    
       [0105]        a ( s )= b ( s )= c ( s )=50, 
         [0000]        x   1 ( s )=10 cos( s/ 3), x   2 ( s )= x   3 ( s )=0.  (68) 
       For FIG. 7 a:    
       [0106]        a ( s )= b ( s )=50(1+0.2 cos( s/ 3)), c ( s )=50(1+0.2 sin( s/ 3)), 
         [0000]        x   1 ( s )= x   2 ( s )= x   3 ( s )=0.  (69) 
       For FIG. 7 b:    
       [0107]        a ( s )= b ( s )=50(1+0.2 cos( s/ 3)), c ( s )=50(1+0.2 sin( s/ 3)), 
         [0000]        x   1 ( s )=10 cos( s/ 3), x   2 ( s )= x   3 ( s )=0.  (70) 
         [0000]    The results of dynamic reconstructions confirm that the visible singularities are reconstructed with about the same resolution as in the static case, and that no additional artifacts arise because of the changes in the phantom. As soon as some singularity of f becomes “invisible” due to motion, the corresponding singularity in g ends at that point, but this does not cause any streaks across the image. Equivalently we can say that LT is quite stable with respect to inconsistencies in the data caused by motion. 
         [0108]    The images shown in  FIGS. 6 and 7  were obtained by ignoring the dynamic nature of the phantom during reconstruction. Clearly, it is theoretically impossible to stably reconstruct an accurate 4D image of f (3D+time) from the cone beam data with only three degrees of freedom. On the other hand, in many cases one can use additional information for improving image quality. Consider, for example, cardiac imaging based on a circular source trajectory, which is of significant importance in medical CT. 
         [0109]    Using the electro cardiogram (ECG) of the patient, which is measured concurrently with cone beam projections, the data corresponding to a fixed cardiac phase (e.g., when the heart is at rest) is accumulated from multiple source rotations and then LT reconstruction is performed from that data. Moreover, the problem of 4D CT imaging can be approximately solved. Using again the ECG data, the complete cardiac cycle is split into several segments, accumulates the cone beam data for each segment from multiple source rotations, and then applies LT separately for each segment. See F. Koo, H. Kudo and L. Zeng, Proceedings of the VIIIth International Conference on Fully Three-Dimensional Reconstruction in Radiology and Nuclear Medicine, Salt Lake City, Utah, University of Utah, (2005), where similar techniques are used together with more conventional inversion algorithms. 
         [0110]    Because of the flexibility of LT, its relative stability with respect to inconsistencies in the data (compared with “global” algorithms) and the ability to accurately reconstruct edges inside objects, it is expected that LT can become a valuable tool, which provides important information complementing well-established inversion techniques. 
         [0111]    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.