Abstract:
A nuclear medical imaging apparatus receives an associated object ( 18 ). A radiation detector ( 12 ) is equipped with a slat collimator ( 14 ) including a plurality of spaced apart slats ( 114 ) separating individual detecting elements of an essentially linear array of detecting elements ( 116 ). The slat collimator produces planar collimation and results in projection data which is weighted inversely with distance in the projection direction. An image reconstruction processor ( 34 ) converts the projection data obtained by the detector ( 12 ) into an image, including correction for the inverse distance weighting. The image reconstruction processor ( 34 ) includes a memory, a preconditioning operator P ( 36 ), a projection operator S ( 38 ), and an iterative loop operator ( 40 ) which applies the preconditioning operator P ( 36 ) and the projection operator S ( 38 ) to the memory contents to calculate updated memory contents. The iterative loop operator ( 40 ) iteratively corrects for the inverse distance dependence of the projection data.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    The present invention relates to the art of diagnostic imaging. It finds particular application in conjunction with rotating one-dimensional (1D) slat-collimated gamma cameras and single photon emission computed tomography (SPECT), and will be described with particular reference thereto. However, it is to be appreciated that the present invention is also amenable to other like applications and other diagnostic imaging modes such as, e.g., positron emission tomography (PET).  
           [0002]    In diagnostic nuclear imaging, one or more radiation detectors are mounted on a movable gantry to view an examination region which receives a subject therein. Typically, one or more radiopharmaceuticals or radioisotopes such as  99m Tc or  18 F-Fluorodeoxyglucose (FDG) capable of generating emission radiation are introduced into the subject. The radioisotope preferably travels through a portion of the circulating system or accumulates in an organ of interest whose image is to be produced. The detectors scan the subject along a selected path or scanning trajectory and radiation events are detected on each detector.  
           [0003]    In a traditional SPECT Anger camera, the detector includes a scintillation crystal that is viewed by an array of photomultiplier tubes. A collimator which includes a grid- or honeycomb-like array of radiation absorbent material is located between the scintillation crystal and the subject to limit the angle of acceptance of radiation which will be received by the scintillation crystal. The relative outputs of the photomultiplier tubes are processed and corrected to generate an output signal indicative of the position and energy of the detected radiation. A detector of this type isolates a scintillation event as originating along an approximate ray or line of view, or more precisely along a narrow-angle cone of view. Because radiation events along a spatial line are projected through an opening of the collimator array grid or honeycomb, the collected data is often referred to as projection data. The projection data is then reconstructed into a three-dimensional image of a region of interest by a reconstruction processor.  
           [0004]    A rotating laminar emission camera, also known as the rotating laminar radionuclide camera, has linear collimators usually formed by mounting parallel collimating plates or slats between a line of individual detectors. Alternately, individual detector areas of a large-area detector are defined and isolated through the placement of slats. The slat collimator isolates planar spatial projections; whereas, the grid collimator of traditional scintillation detectors isolates essentially linear spatial projections. The detector-collimator assembly of a slat camera is typically rotated about an axis perpendicular to the detector face in order to resolve data for accurate two-dimensional image projection. Again, projection data collected at angular orientations around the subject are reconstructed into a three-dimensional volume image representation.  
           [0005]    While maintaining certain advantages, such as a better sensitivity-resolution compromise, over, e.g., traditional Anger cameras, slat detectors are burdened by some other undesirable limitations. For example, the one dimensional collimation or slat geometry used by slat detectors complicates the image reconstruction process. The slat geometry results in a plane integral reconstruction as opposed to the line integral reconstruction that is generally encountered in traditional Anger camera applications. Moreover, the geometry produces a plane integral only in a first approximation.  
           [0006]    In actuality, the plane integral should have a weighting factor introduced thereto to account for the fact that the detector&#39;s sensitivity has a 1/r dependence, where r represents the distance between a detected radiation event occurring in the object under consideration and the detection point on the detector. That is to say, the detector is generally more sensitive to relatively close objects and less sensitive to far away objects.  
           [0007]    Reconstruction of linear projection data obtained using conventional Anger cameras usually incorporates backprojection using a form of the inverse Radon transform R −1 . Reconstruction of the planar projection data obtained from a slat-type camera is complicated in two respects. First, the integrations are planar integrations rather than line intregrals. Second, the 1/r term which occurs in projection data obtained by a slat detector reduces the spatial symmetry of the projection data. The reduced symmetry prevents the use of mathematical methods which are typically employed to implement the Radon transform R and its inverse R −1 .  
           [0008]    Most previous reconstruction methods for projection data acquired by a slat detector merely disregard or ignore the 1/r weighting factor in solving the reconstruction problem. This approximation results in degradation of the reconstructed image. This type of image degradation could be reduced or even eliminated by a new or improved reconstruction algorithm which accounts for the 1/r dependence.  
           [0009]    The present invention contemplates a new and improved reconstruction technique which overcomes the above-referenced problems and others.  
         SUMMARY OF THE INVENTION  
         [0010]    In accordance with one aspect of the present invention, a nuclear medical imaging apparatus is disclosed. An object is received in a receiving region. A radiation detector has a side facing the receiving region. The detector includes a collimator fabricated from radiation attenuative material arranged on the detector side facing the receiving region. The collimator includes a plurality of spaced apart slats. The detector also includes an essentially linear array of detecting elements, the detecting elements being disposed between the slats on the detector side facing the receiving region. The imaging apparatus further includes an image reconstruction processor which converts the projection data from the detector into an image representation. The image reconstruction processor includes a memory, a preconditioning operator P, a projection operator S, and an iterative loop operator which applies the preconditioning operator P and the projection operator S to the memory contents to calculate updated memory contents. Preferably, the preconditioning operator P applies an inverse Radon transform operator R −1 . In one embodiment, the memory stores projection data, and the iterative loop operator applies the preconditioning operator P to the projection data stored in the memory, and then applies the projection operator S to produce a second set of projection data. The projection operator preferably incorporates a plurality of Radon transforms R, each Radon transform being applied to an image weighted by a weighting factor selected such that the projection operator approximates the projection transform physically implemented by the radiation detector, the approximating including at least approximating a 1/r dependence of the projection data generated by the radiation detector.  
           [0011]    In accordance with another aspect of the present invention, a diagnostic imaging system is disclosed. A scanner generates projection data that is weighted inversely with distance in a projection direction. A backprojector backprojects the generated projection data into an image memory without compensating for the inverse weighting with distance to reconstruct an artifacted image representation. A forward projector forward projects the artifacted image to generate reprojected data. A correction circuit (i) compares the generated projection data and the reprojection data, and (ii) generates a correction factor in accordance with a deviation between the generated projection data and the reprojection data. The scanner preferably includes a one-dimensional array of radiation detectors, a collimator which collimates received radiation into planes, and a rotor for rotating radiation planes around an axis perpendicular to a face of the detector array.  
           [0012]    In accordance with another aspect of the present invention, an image reconstruction process for generating a final image from measured projection data acquired by a slat detector is disclosed. The measured projection data is preconditioned using a preconditioning operator P to obtain an image. The image is iteratively improved to obtain a final image. The iterative process includes projecting the image with a projection operator S to generate reprojected data, comparing the reprojected data with the measured projection to generate correction data, and backprojecting one of the correction data, the reprojection data corrected with the correction data, and the projection data corrected with the correction data, using the preconditioning operator P to obtain an improved image. Preferably, the preconditioning operator P incorporates an inverse Radon transform R −1 .  
           [0013]    In accordance with yet another aspect of the present invention, a diagnostic imaging process is disclosed. Projection data is generated which is weighted inversely with distance in a projection direction. The projection data is backprojected into an image representation without compensating for the inverse weighting to reconstruct a flawed image representation. The flawed image representation is forward projected to form reprojected data. The reprojected data is compared with the projection data. A correction is generated from a deviation between the reprojection data and the projection data. A backprojecting of one of: (i) the correction data into the flawed image representation; and (ii) the correction data combined with one of the projection and reprojection data is performed to generate a less flawed image representation. Steps starting with the forward projecting are iteratively repeated until the comparing step meets a preselected closeness criteria. Preferably, the backprojecting step uses an inverse Radon transform. The projection data generating step preferably includes: introducing a radiation source into a subject; collimating radiation from the source into planes; detecting the radiation in each plane; and rotating the collimation planes around a first axis parallel to at least one of the planes. Additionally, the data generating step may include rotating the collimation planes about a second axis through the subject and transverse to the first axis.  
           [0014]    One advantage of the present invention is that it corrects for the 1/r dependence of the data.  
           [0015]    Another advantage of the present invention is that it properly reconstructs data obtained from slat cameras.  
           [0016]    Another advantage of the present invention is that it efficiently transforms the measured projection data into image space without neglecting the 1/r dependence of the slat detector projection data.  
           [0017]    Yet another advantage of the present invention is that it utilizes the Radon transform R, which is often implemented in reconstruction algorithms for SPECT, PET, and other nuclear imaging methods, as a preconditioner.  
           [0018]    Still further advantages and benefits of the present invention will become apparent to those of ordinary skill in the art upon reading and understanding the following detailed description.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0019]    The invention may take form in various components and arrangements of components, and in various steps and arrangements of steps. The drawings are only for the purposes of illustrating preferred embodiments and are not to be construed as limiting the invention.  
         [0020]    [0020]FIG. 1 is a diagram of a nuclear imaging system in accordance with the invention;  
         [0021]    [0021]FIG. 2(A) is a side view of a slate detector in accordance with the present invention;  
         [0022]    [0022]FIG. 2(B) is a top view of the slat detector of FIG. 2(A);  
         [0023]    [0023]FIG. 3 is a block diagram of an implementation of the slat detector projection transform S in accordance with the invention;  
         [0024]    [0024]FIG. 4 is a block diagram of the reconstruction processor of FIG. 1; and  
         [0025]    [0025]FIG. 5 is an alternate embodiment of the reconstruction processor. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0026]    With reference to FIG. 1, a nuclear camera system  10  includes a plurality of slat-type detector heads  12 , in the preferred embodiment three slat detector heads  12   1 ,  12   2 , and  12   3 . Of course, other numbers of detector heads can also be utilized. Each of the detector heads includes a slat-type collimator  14   1 ,  14   2 , and  14   3  and a linear array of detectors  16   1 ,  16   2 , and  16   3 . The collimators collimate incoming radiation from a subject  18  to parallel planes with an embedded 1/r dependence. Each camera detector-collimator unit is rotatable by a motor  20   1 ,  20   2  and  20   3  about an axis perpendicular to the detector face and approximately centered thereon. Angular orientation monitors  22   1 ,  22   2 , and  22   3  track the angular orientation of the respective collimator/detector units.  
         [0027]    A rotating gantry  24  rotates all three detector heads about the subject  18 . An angular orientation monitor  26  determines the angular orientation of each of the heads at each angular data collection position. Each of the detector heads records radiation events in terms of which detector element received the radiation, angle of the collimator-detector unit within the head, and angle of the head around the subject.  
         [0028]    A data acquisition system  30  receives the projection data along with the corresponding positional and angular parameters. This data is stored in a projection memory  32 . The projection data is processed by reconstruction processor  34  which reconstructs an image from the projection data. The reconstruction processor includes at least a preconditioning processor  36  denoted by P and a projection processor  38  denoted by S which are used in conjunction with an iterative loop processor  40  to perform the reconstruction. The reconstructed image is preferably stored in an image memory  42  from which it is retrieved by a video processor  44  and displayed on a video monitor  46 . Of course other output devices, such as a color printer, projector display, CCD display, active matrix display, or the like may also be used.  
         [0029]    With reference to FIG. 2(A) and FIG. 2(B), each collimator  14  includes a plurality of collimator slats  114 . The collimator slats are preferably perpendicular to a face of detector array  16 , although uniformly or non-uniformly tilted slats are also contemplated. A plurality of detector elements  116  are located on the detector face  112  in the gaps between the slats. Detector elements  116  may be discrete detectors, or may be regions of a large area detector which are isolated by the slats.  
         [0030]    Referring back to FIG. 1, the reconstruction processor  34  accepts projection data from projection memory  32  and calculates an image which is subsequently stored in image memory  42 . The reconstruction processor includes a preconditioning processor  36  denoted by P and a projection processor  38  denoted by S which are used in conjunction with an iterative loop processor  40  to perform the reconstruction.  
         [0031]    The reconstruction processor solves the equation Sf(θ, s)=g(θ, s). If a preconditioner P is applied, this may be written as: 
           PSf (θ, s )= Pg  (θ, s )  (1) 
         [0032]    or equivalently, 
           SPh=g,Ph=f   (2) 
         [0033]    where g(θ, s) is the measured projection data, f(θ, s) is the image to be reconstructed, S is the slat detector projection operator  38 , P is a preconditioner operator  36 , and h is an intermediate variable. From equation (1) it is clear that the forward projection operator S acts to project the image f. S is therefore the mathematical equivalent of the projection transform physically implemented by the radiation detection system  10  and associated data acquisition system  30 . To calculate the reconstructed image f, the reconstruction processor effectively implements a backprojection operator S −1 . Alternatively, equation (2) indicates that if (PS) −1  and P are implementable, then h and subsequently f may be calculated from the measured projection data set g. Furthermore, if PS is close to unity, a Neumann series representation may be employed,  
                 (     P                 S     )       -   1       =       ∑   k            (     I   -     P                 S       )     k               (   3   )                               
 
         [0034]    where I is the identity operator. Thus, to implement the slat detector backprojection operator S −1 , the invention employs implementable expressions for the slat detector projection S and for a preconditioner P as described below, with the further restriction that PS be close to unity so that the Neumann series is appropriate. The preferred preconditioner P, as detailed next, is the inverse Radon or filtered backprojection transform R −1 .  
         [0035]    The imaged area is contained within an imaging volume which in the case of a slat detector is a spherical volume of radius R. The Radon transform R is defined by:  
               R                 f     =       ∫       x   ·   θ     =   s              f        (   x   )                          x                 (   4   )                               
 
         [0036]    where f is the image contained within the imaging volume.  
         [0037]    The slat detector projection is described by transform S as,  
                 S                 f     =       ∫       x   ·   θ     =   s                f        (   x   )              x   -   a                    x           ,     a   =       s                 θ     +       R   s          θ     1   ,   ⊥                     (   5   )                               
 
         [0038]    using the parameterizations,  
                     θ        (     ϕ   ,   ψ     )       =                  (           cos                 ϕ                 sin                 ψ               sin                 ϕ                 sin                 ψ               cos                 ψ           )     =     (           sin                 ψ                   θ        (   ϕ   )                   cos                 ψ           )                       θ     1   ,   ⊥            (     ϕ   ,   ψ     )       =                  (             -   sin                   ϕ               cos                 ϕ             0         )     =                (                          θ   ⊥          (   ϕ   )                 0         )                       θ     2   ,   ⊥            (     ϕ   ,   ψ     )       =                  (           cos                 ϕ                 cos                 ψ               sin                 ϕ                 cos                 ψ                 -   sin                   ψ           )     =     (           cos                 ψ                   θ        (   ϕ   )                     -   sin                   ψ           )                       θ        (   ϕ   )       =                (           cos                 ϕ               sin                 ϕ           )       ,         θ   ⊥          (   ϕ   )       =       (             -   sin                   ϕ               cos                 ϕ           )     =     θ        (     ϕ   +     Π   2       )                         (   6   )                               
 
         [0039]    where R S  is the distance from the detector to the object.  
         [0040]    For the limit R S →∞, the approximation Rf (θ, s)˜R S Sf(θ, s) holds. This implies that a suitable preconditioning operator P is the filtered backprojection operator, i.e. the inverse filtered Radon transform R −1 . However, in applying the inverse Radon transform R −1  to slat detector data a complication arises due to the reduced symmetry of the slat detector data versus conventional Anger camera data. For the Radon transform, the following symmetries apply, 
           Rf (φ,ψ, s )= Rf (φ,ψ+π,− s )= Rf (φ+π,−ψ, s )= Rf (φ+π,π−ψ,− s )  (7). 
         [0041]    Thus, the backprojection needs to be performed on the data for φ and ψ in the range [0, π] only. For the slat detector projection transform S, since the weight depends on the value of φ, the symmetries are reduced to, 
           Sf (ψ,ψ, s )− Sf (ψ,−ψ,− s )  (8) 
         [0042]    without further relations, so non-redundant projection data occurs for φ in the range [0, 2π]. Thus, when applying the inverse Radon transform R −1  to projection data obtained by a slat detector, the projection data for φ and −φ are averaged to account for the lack of redundancy.  
         [0043]    Turning next to the implementation of the slat detector projection operator S, it is shown that S can be implemented by applying a plurality of Radon transforms R, each Radon transform being applied to the image weighted by a weighting factor. The approach is therefore to develop a fast Radon transform, and then to extend this to implement the slat detector projection operator S.  
         [0044]    The Radon transform R can be written as,  
                     R                 f     =                  ∫       x   ·   θ     =   s              f        (   x   )               x                     =                ∫     ∫       f        (       s                 θ     +     u                   θ     1   ,   ⊥         +     v                   θ     2   ,   ⊥           )               u             v                       =                ∫     ∫       f   (                        s                 cos                 ϕ                 sin                 ψ     -     u                 sin                 ϕ     +     v                 cos                 ϕ                 cos                 ψ                   s                 sin                 ϕ                 sin                 ψ     +     u                 cos                 ϕ     +     v                 sin                 ϕ                 cos                 ψ                   s                 cos                 ψ     -     v                 sin                 ψ                        )             u             v                       =                ∫     ∫       f        (               (       s                 sin                 ψ     +     v                 cos                 ψ       )                     θ        (   ϕ   )         +     u                     θ   ⊥          (   ϕ   )                       s                 cos                 ψ     -     v                 sin                 ψ             )               u               v     .                         (   9   )               Defining   ,     
              g   ϕ          (       x   1     ,     x   3       )       =       ∫   u            f        (               x   1          θ        (   ϕ   )         +     u                     θ   ⊥          (   ϕ   )                     x   3           )                          u                   (   10   )                               
 
         [0045]    we have 
           Rf (θ, s )=∫ g   ψ ( s  sin ψ+ v  cos ψ, s  cos ψ− v  sin ψ) dv   (11). 
         [0046]    The resulting algorithm for the evaluation of Rf is: (1) tabulate g φ (x 1 , x 3 ) for φ in the range [0, 2π] and x 1 , x 3  satisfying x 1   2 +x 3   2 ≦R; and (2) compute Rf from g. Using a step size of O(1/N) for all variables, the computational effort is O(N 4 ). Since the algorithm only requires us to compute line integrals in 2D, it is readily implemented.  
         [0047]    To extend the fast Radon transform to a fast slat detector projection transform S, it is first recognized that the integrals carry an additional weight. Direct separation of the integrand is not possible, since the weight depends upon both u and v, as  
               S                   f        (     θ   ,   s     )         =       ∫   u            ∫   v              f        (       s                 θ     +     u                   θ     1   ,   ⊥         +     v                   θ     2   ,   ⊥           )               (     u   -     R   s       )     2     +     v   2                   u               v     .                   (   12   )                               
 
         [0048]    Assume that the weight can be approximated by a sum of the following form,  
               w        (     u   ,   v     )       =       1           (     u   -     R   s       )     2     +     v   2           ∼       ∑     v   =   0     M              A   v          (     u   -     R   s       )              B   v          (   v   )                     (   13   )                               
 
         [0049]    for arbitrary functions A ν  and B ν . Thus, w is now a sum of separable functions in u and v. Then,  
               S                   f        (     θ   ,   s     )         =       ∑     v   =   0     M            ∫   v              B   v          (   v   )              ∫   u            f        (       s                 θ     +     u                   θ     1   ,   ⊥         +     v                   θ     2   ,   ⊥           )              A   v          (     u   -     R   s       )               u             v                       (   14   )                               
 
         [0050]    and g is defined analogous to the Radon transform case,  
                 g     v   ,   ϕ            (       x   1     ,     x   3       )       =       ∫   u            f        (               x   1          θ        (   ϕ   )         -     u                     θ   ⊥          (   ϕ   )                     x   3           )              A   v          (     u   -     R   s       )               u                 (   15   )                               
 
         [0051]    so that the slat detector transform S is given by,  
               S                   f        (     θ   ,   s     )         =       ∑     v   =   0     M            ∫   v              g     v   ,   ϕ            (         s                 sin                 ψ     +     v                 cos                 ψ       ,       s                 cos                 ψ     -     v                 sin                 ψ         )              B   v          (   v   )                 v     .                   (   16   )                               
 
         [0052]    Comparing equation (16) with equation (11), it will be observed that the Radon transform R and the slat detector projection transform S differ by the B ν (v) weighting.  
         [0053]    Referring back to equation (11), since uθ 1,⊥ +θ 2,⊥  is a circle of radius R and R S &gt;R, R S −u is positive and the weight w(u,v) can be written as,  
               w        (     u   ,   v     )       =       1       R   s     -   u       ·       1       1   +       (     v   /     (       R   s     -   u     )       )     2           .               (   17   )                               
 
         [0054]    Taking the second fraction as a function of v/(R S −u), it can be approximated by a polynomial with coefficients α ν . Weight w is then approximated in the desired form by  
               w        (     u   ,   v     )       =       ∑     v   =   0     M                       (         α   ν          (     1       R   s     -   u       )         ν   +   1       )                       (     v   ν     )     .                 (   18   )                               
 
         [0055]    To find the proper polynomial, the size of the argument must be estimated. As u 2 +v 2 ≦R, the argument is smaller than (R 2 −u 2 ) ½ /(Rs−u) with u in the interval [−R, R]. Since R S &gt;R, the maximum u max  of that function is located at R 2 /R S  with the maximum value R/(R S   2 −R 2 ) ½ . For the typical choices R S =1.2 and R=1, a maximum value of about 1.51 is obtained.  
         [0056]    The polynomial chosen should give a minimal absolute error for u in the range [0, u max ]. For the values R S =1.2 and R=1 it is preferably found by the Remez algorithm to be 
         1.003815469+(−0.0627153542+(−0.3875254049+0.1541203805 ·x )· x )· x   (19) 
         [0057]    with a maximum error of 0.004.  
         [0058]    The preferred implementation of the fast slat detector projection operator S (element  38  in FIG. 1) according to equation (16) is shown in FIG. 3. The B ν  weighting factors  200  are applied  202  to image f  204  and subsequently operated upon by Radon transforms  206 . The summed result  208  yields the desired fast slat detector projection transform Sf  210 .  
         [0059]    Having identified a preferred preconditioner P as the inverse Radon transform R −1  with the symmetry correction discussed previously, as well having obtained a fast slat detector projection transform S as exemplified by equation (16) and as illustrated in FIG. 3, a preferred algorithm implemented by reconstruction processor  34  will now be disclosed. Combining equation (1) with the Neumann series of equation (3) yields the iterative algorithm, 
           f   k+1   =f   k   +p   k+1 , 
         [0060]    where 
           p   k+1 =( I−PS ) p   k   (20) 
         [0061]    with the initial conditions, 
           f   0   =p   0   ,p   0   =Pg   (21). 
         [0062]    With reference to FIG. 4, each view of projection data g from the projection memory  32  is backprojected  36  with the inverse Radon transform R −1  into the image memory  42 . Because the 1/r weighting error has been ignored, the reconstructed image is inaccurate. When the image is forward projected  38  along each of the original projection planes, the forward projected data g′ deviates from the corresponding original projection data f. Corresponding forward projected and original projection data are compared  300  to determine a correction factor, e.g. subtracted. The set of correction factors are backprojected  36  into the image memory  42 . With each iteration, the deviation becomes smaller as the reconstructed image converges on the precise reconstruction. Once the comparison  300  determines that the corrections are sufficiently small, the iterative correction process is terminated.  
         [0063]    A second preferred algorithm is derived by combining equations (2) and (3) to yield: 
           f   k+1   −Ph   k+1 , 
         [0064]    where 
           h   k+1   =h   k   +p   k+1 , 
         [0065]    and 
           p   k+1 =( I−SP ) p   k   (22) 
         [0066]    with the starting conditions, 
           f   0   =Ph   0   ,h   0   =p   0   ,p   0   =g.   (23) 
         [0067]    The reconstruction processor algorithm of equations (22) and (23) is illustrated in FIG. 5. Measured slat detector projection data g stored in projection memory  32  of the nuclear imaging system (FIG. 1) is directly input into the initially zeroed loop projection memory  410  during the zeroeth iteration, in accordance with equation (23). Each loop iteration thereafter adds  412  an improved correction to the loop projection memory  410 . The correction is formed as h k+1  in accordance with equation (22) combining the difference between the original data g o  and re-projected data g r  with the last iteration corrected data g c . In the zeroeth iteration the original projection data g o  is moved into a loop projection memory  410 . The projection data g o  is backprojected  436  into the image memory  42 . The image is re-projected  438  and its negative determined  418 . The previous iteration of projection data g c  is combined with the difference between the original projection data g o  and the most recent iteration re-projected data g r  to update the corrected projection data set. The updated projection data set g c  is backprojected  436  using the inverse Radon transform R −1  to generate a corrected image by subtracting  418  from the projection of the previous iteration  416  the projection formed by first backprojecting the loop projection memory using preconditioner  436 , which once again is preferably an inverse Radon transform R −1 , and then re-projecting  438  the obtained image  420  using the slat detector projection operator S. The loop projection memory is updated  424 . When the loop converges to a solution, the final image may be extracted from image memory  42  for further processing in accordance with FIG. 1.  
         [0068]    The invention has been described with reference to the preferred embodiments. Obviously, modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.