Abstract:
Data collected from a cone beam scanner is reconstructed into a volumetric image representation by defining a plurality of oblique surfaces which are reconstructed into a cylinder. An interpolator identifies non-redundant rays of radiation passing through the surfaces. Rays of radiation intersecting a center point of each oblique surface are identified along with rays tangent to surface rings on each surface. Data from the identified non-redundant rays is weighted by a first processor. A second processor convolves the weighted data and passes it to a backprojector which backprojects it into an image memory. The oblique surface reconstruction technique facilitates use of conventional two-dimensional convolution and backprojection techniques that enjoy relative computational simplicity and efficiency as well as three-dimensional reconstruction techniques that use a minimum number of projections. Additionally, the technique facilitates accurate reconstruction of cone beam projections with over ten times the area of current multi-ring scanners.

Description:
This application is Continuation Patent Application of pending prior U.S. patent application Ser. No. 09/595,858 filed on Jun. 16, 2000 (now U.S. Pat. No. 6,343,108) which claims the benefit of Provisional application Ser. No. 60/140,050, filed Jun. 18, 1999. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to the art of image reconstruction. It finds particular application in conjunction with reconstructing x-ray transmission data from computed tomography (CT) scanners which move a cone-beam or wedge beam of radiation along a helical trajectory, and will be described with particular reference thereto. However, it is to be appreciated that the present invention will also find application in conjunction with the reconstruction of data from CT scanners, nuclear cameras, and other diagnostic scanners that process data representing nonparallel trajectories. 
     Conventionally, spiral or helical CT scanners include an x-ray source which projects a thin slice or beam of penetrating radiation. The x-ray source is mounted for rotational movement about a subject that is translated along the axis of rotation. An arc or ring of radiation detectors receive radiation which has traversed the subject. Data from the radiation detectors represents a single spiraling slice through the subject. The data from the detectors is reconstructed into a three-dimensional image representation. 
     Current helical CT scanners with two or three detector rings improve data acquisition speed and permit thin slab scanning. Several 3-D image reconstruction techniques for reconstructing data from helical cone or wedge beam systems have been suggested. For example, commonly assigned U.S. Pat. No. 5,625,660 to Tuy discloses an image reconstruction technique for helical partial cone-beam data in which the data stream is divided into two parts which are processed separately and then recombined. In addition, other similar reconstruction techniques process a single data stream. These three-dimensional reconstruction techniques generally involve increased computational load and complexity. This is due to the fact that these reconstruction techniques perform “true 3D reconstruction,” involving a 3D backprojection of convolved projections. 
     In contrast, current 2D helical reconstruction techniques enjoy decreased computational load and simplicity. However, current 2D helical reconstruction techniques limit the number of rings or cone angle over which accurate reconstructions can be obtained. An additional difficulty with reconstruction of spiral cone or wedge beam data is the elimination of redundant rays of data. 
     The present invention contemplates a new and improved image reconstruction technique which overcomes the above-referenced problems and others. 
     SUMMARY OF THE INVENTION 
     In accordance with one aspect of the present invention, a method of volumetric image reconstruction includes collecting partial cone beam data in two-dimensional arrays at a plurality of sampling positions, where the collected data corresponds to rays of radiation which diverge in two dimensions from a common vertex as the vertex travels along a continuous path. A plurality of two-dimensional oblique surfaces are defined throughout a region of interest and rays of radiation which pass through the plurality of oblique surfaces are defined. The data from the identified rays is reconstructed into a reconstruction cylinder having an axis along a z-direction. A volume data set is generated from the reconstructed oblique surface data. 
     In accordance with another aspect of the present invention, a method of diagnostic imaging includes generating penetrating radiation and the receiving the penetrating radiation with two-dimensional radiation detectors along a plurality of divergent rays, where the rays are focused at a common origin vertex and diverge in two dimensions. The vertex is rotated along at least an arc segment of a helical path. The radiation detectors are sampled at a plurality of angular increments along the helical arc segment to generate a plurality of two-dimensional projection views, where each view includes a two-dimensional array of data values and each data value corresponds to one of the divergent rays. A first and last oblique surface are defined, where the first and last oblique surfaces are formed by the intersection of a cone beam of penetrating radiation and the region of interest. A plurality of additional oblique surfaces are defined, where the plurality of oblique surfaces are at least one of rotated and translated with respect to the first oblique surface. Projection views corresponding to each oblique surface are weighed and a two-dimensional convolution of the projection view data is computed. The convolved projection data corresponding to each oblique surface is two-dimensionally backprojected into a volumetric image memory. 
     In accordance with another aspect of the present invention, a method of selecting non-redundant rays of penetrating radiation during a computed tomography scan is provided where the non-redundant rays from a plurality of oblique surfaces for two-dimensional reconstruction into a volumetric image representation. The method includes at each angular orientation about an examination region, selecting detected rays of penetrating radiation that intersect a geometric center point of each oblique surface. The method further comprises interatively identifying rays of penetrating radiation which are tangent to surface rings extending outward from the geometric center point of each oblique surface. 
     In accordance with a more limited aspect of the present invention, the method includes where penetrating radiation data does not exist, interpolating closest adjacent rays tangent to the oblique surface rings between the center point and an outer radius of the region of interest. 
     In accordance with another aspect of the present invention, a computed tomography scanner includes a first gantry which defines an examination region and a rotating gantry mounted to the first gantry for rotation about the examination region. A source of penetrating radiation is arranged on the rotating gantry for rotation therewith. The source of penetrating radiation emit a cone-shaped beam of radiation that passes through the examination region as the rotating gantry rotates. A subject support holds a subject being examined at least partially within the examination region, wherein at least one of the first gantry and the subject support is translated such that the subject passes through the examination region while the rotating gantry is rotated and the source of penetrating radiation follows a helical path relative to the subject. A two dimensional array of radiation detectors is arranged to receive the radiation emitted from the source of penetrating radiation after it has traversed the examination region. A reconstruction processor reconstructs images of the subject from data collected by the two-dimensional array of radiation detectors. The reconstruction processor includes a control processor which defines a plurality of oblique surfaces, where the oblique surfaces are defined by the intersection of the cone-shaped beam of radiation and a portion of the subject, and an interpolator which identifies non-redundant rays of penetrating radiation that pass through the oblique surfaces. A first data processor weights the data corresponding to the identified non-redundant rays and a second data processor receives data from the first data processor and performs a two-dimensional convolution on the data. A backprojector receives the data from the second data processor and two-dimensionally backprojects it into an image memory. The CT scanner further includes a human-viewable display which accessed an image memory to display to display reconstructed images of the subject. 
     One advantage of the present invention is increased efficiency in both the scan and reconstruction process. 
     Another advantage of the present invention is that it provides accurate reconstruction for larger area cone beam projections. 
     Another advantage of the present invention is that it enjoys greater computational simplicity. 
     Yet another advantage of the present invention resides in the ability to achieve a three-dimensional volumetric reconstruction using conventional two-dimensional reconstruction techniques. 
     Other benefits and advantages of the present invention will become apparent to those skilled in the art upon a reading and understanding of the preferred embodiments. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     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 purposes of illustrating preferred embodiments and are not to be construed as limiting the invention. 
     FIG. 1 is a diagrammatic illustration of a computerized tomographic (CT) diagnostic system in accordance with aspects of the present invention; 
     FIG. 2A is an expanded sagittal view of a reconstruction cylinder having a radius r c  and a length z c  in accordance with the present invention; 
     FIG. 2B is an expanded three-dimensional view of a reconstruction cylinder of FIG. 2A with a cone beam of radiation superimposed at three locations along the z-direction; 
     FIG. 2C is an axial view of the reconstruction cylinder shown in FIGS. 2A and 2B; 
     FIG. 2D is a diagrammatic illustration of the oblique surface derivation in accordance with the present invention; 
     FIG. 3 is a flow chart illustrating the interpolation method in accordance with the present invention; and, 
     FIGS. 4A-4F are flow charts illustrating alternate embodiments of the oblique surface reconstruction method in accordance with the present invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     With reference to FIG. 1, a computed tomography (CT) scanner  10  includes a stationary gantry  12  which defines an examination region  14 . A rotating gantry  16  is mounted on the stationary gantry  12  for rotation about the examination region  14 . A source of penetrating radiation  20 , such as an x-ray tube, is arranged on the rotating gantry  16  for rotation therewith. The source of penetrating radiation is collimated to produce a cone-shaped beam of radiation  22  that passes through the examination region  14  as the rotating gantry  16  rotates. A collimator and shutter assembly  24  shapes the beam of radiation  22  and selectively gates the beam  22  on and off. Alternately, the radiation beam  22  is gated on and off electronically at the source  20 . 
     A subject support  30 , such as a couch or the like, suspends or otherwise holds a subject being examined or imaged at least partially within the examination region  14 . Moreover, as the rotating gantry  16  rotates, the support  30 , and consequently the subject thereon, are translated along a central horizontal axis of the examination region  14 . In this manner, the source  20  follows a helical path relative to the subject. Optionally, in an alternative embodiment, the support  30  remains stationary while the “stationary gantry”  12  is translated or otherwise moved relative to the subject such that the source  20  follows a helical path relative thereto. 
     In the illustrated fourth generation CT scanner, a plurality of rings of radiation detectors  40  are mounted peripherally around the examination region  14  on the stationary gantry  12 . Alternately, in a preferred embodiment, a third generation CT scanner is employed with a two-dimensional array of the radiation detectors  40  mounted on the rotating gantry  16  on a side of the examination region  14  opposite the source  20  such that they span the area defined by the cone-shaped beam of radiation  22 . Regardless of the configuration, the radiation detectors  40  are arranged such that a two-dimensional array thereof receive the radiation emitted from the source  20  after it has traversed the examination region  14 . 
     In a source cone geometry, an array of detectors which span the radiation  22  emanating from the source  20  are sampled concurrently at short time intervals as the source  20  rotates behind the examination region  14  to generate a source view. In a detector cone geometry, each detector is sampled a multiplicity of times in each of a plurality of revolutions as the source  20  rotates behind the examination region  14  to generate a detector view. The path between the source  20  and each of the radiation detectors  40  is denoted as a ray. 
     The radiation detectors  40  convert the detected radiation into electronic data. That is to say, each of the radiation detectors produces an output signal which is proportional to the intensity of received radiation. Optionally, a reference detector may detect radiation which has not traversed the examination region  14 . A difference between the magnitude of radiation received by the reference detector and each active radiation detector  40  provides an indication of the amount of radiation attenuation or absorption along a corresponding ray of sampled radiation. 
     In detector view geometry, each view or two-dimensional array of data represents a cone of rays having its vertex at one of the radiation detectors  40  collected over one or more short periods of time as the source  20  rotates behind the examination region  14  from the detectors. In source view geometry collected by a concurrent sampling of detectors, each view has a complete two-dimensional array of data representing a cone of rays having a vertex at the source  20  and a ray from the source to every detector. In a source view, the resolution is limited by the detector size and the number of views is limited by the sampling rate. In a detector view, the resolution is limited by the sampling rate and the number of views is determined by the detector size. 
     A gantry acquisition memory board  42  receives sampled data from the radiation detectors  40  and gantry geometry parameters, such as helix pitch. The gantry acquisition memory board  42  organizes the data into a selected one of a detector cone geometry and a source cone geometry and performs a ripple filtering operation before passing the data to a control processor  44  which defines a plurality of oblique surfaces, the operation of which is described more fully below. A reconstruction processor  50  processes the collected and manipulated data from the gantry acquisition memory board  42  and control processor  44  and ultimately two-dimensionally backprojects it into an image memory  60  for access and display on a human-viewable display  62 , such as a video monitor. 
     More particularly, the reconstruction processor  50  includes an interpolator  52  which interpolates oblique surface projections based on a defined first oblique surface, S 1 (r, θ, z). As is described more fully below, a plurality of these oblique surfaces, each rotated and translated with respect to the first oblique surface, make up the full volume of a reconstruction cylinder C. FIGS. 2A and 2B show a sagittal view and a corresponding three-dimensional view of the reconstruction cylinder C. The reconstruction cylinder C represents the volume of interest or scanned volume for the particular diagnostic study being performed. In other words, the portion of the subject which is being examined lies within the boundaries of the reconstruction cylinder. As shown in FIG. 2A, the reconstruction cylinder has a radius r c  and a length z c . 
     As described above, the radiation source  20  moves along a helical trajectory H with respect to the subject disposed within the examination region  14 . The helix H with radius r s  and pitch p extends from a start angle of Φ=−φ−π/2 to Φ=2πz c /p+φ+π/2 for a total angle of 2πz c /p+2φ+π, where φ=sin −1  (r c /r s ) is the additional rotation angle beyond π for completely reconstructing the oblique surface S 1 (r, θ, z) and is shown in the axial view of the reconstruction cylinder (FIG.  2 C). The minimum rotation angle to reconstruct each oblique surface is 2φ+π or 180° plus the cone angle or fan angle of the radiation source, where the fan angle is defined by the diameter of the reconstruction cylinder C. While the present invention is described in terms of a source moving along helical trajectory, it is to be appreciated that it is applicable to scanners employing other source trajectories. 
     With reference to FIG.  2 D and continuing reference to FIG. 1, once the oblique surfaces are defined  44  based on control parameters, such as helical pitch and radius, the interpolator  52  identifies non-redundant rays of radiation which pass through each oblique surface. The oblique two-dimensional surface S(r, θ, z) is defined by iteratively aligning the detected rays of radiation by interpolation of the cone beam projections to equal the mean (weighted) z-axis position of all rays tangent to the cylinder (r, θ), extending to the outer radius r c  of the reconstruction cylinder C. This iterative process is expressed as:          S        (       r   n     ,   θ   ,     z   0       )       =         ∑     i   =       0                 …                 n     -   1                W        (     θ   T     )              Z   T          (       r   i     ,     r   n     ,   θ   ,     z   0       )          Δ                   θ        (       r   i     ,     r   n       )               ∑     i   =       0                 …                 n     -   1              Δ                   θ        (       r   i     ,     r   n       )                                    
     for r n =0 . . . r c , 
     where: 
     W(θ T )=180° plus fan weighting for rays (T) tangent to the cylinder of radius r i , 
     Z T (r i , r n , θ, z 0 )=z-axis location at (r n , θ) of rays (T) tangent to the cylinder of radius r i , and 
     Δθ(r i , r n )=angular increment between rays intersection at (r n , θ). 
     With reference to FIG. 3, the interpolator  52  identifies all rays of radiation that intersect a center point (0, θ, z) of each oblique surface  70 . The interpolator  52  then iteratively identifies pixels at least one pixel out from the center point of each oblique surface  72 , where these pixels form a plurality of surface rings which extend from the center point to the outer radius r c  of the reconstruction cylinder C, as is described mathematically above. The interpolator then iteratively identifies rays of radiation that are tangent to each of the plurality of surface rings  74 , interpolating closest adjacent rays  76 , where necessary. 
     This process defines a plurality of oblique surfaces throughout the reconstruction cylinder C, which are stored  78 , along with the corresponding ray positions, in an intermediate memory as S(x,y,z(x,y)). Further, the iterative process results in a plurality of oblique surfaces with minimum dispersion, i.e. minimal z-axis deviations of the projected rays from each surface S(r, θ, z). Each of the plurality of oblique surfaces is rotated and translated with respect to the first oblique surface S 1 (r, θ, z) to make up the full volume of the reconstruction cylinder C, given by: S(r, θ, z)=S(r, θ+2π(z−z 0 )/p, z 0 )+z−z 0 . Otherwise, the oblique surfaces will change along the source trajectory, with each surface definition requiring a source path segment of at least 180° plus the fan angle. 
     In order to adequately sample the volume, the spacing between the reconstructed surfaces along the z-axis is selected to be no more than one-half the effective z-axis resolution (≦Δz/2). The maximum dispersion occurs at the radius r c . A “dispersion factor” ρ is defined as: 
     
       
         (maximum z-axis deviation from the surface at radius r c )/Δz,  
       
     
     where Δz is the effective z-axis resolution. It should be appreciated that maximum dispersion tolerances are of lesser importance in embodiments where three-dimensional reconstruction algorithms are employed because of the far greater accuracy present in the three-dimensional embodiments. 
     The interpolated rays defining each reconstructed surface are weighted W(θ) by an appropriately smooth weighting function, such that the weights of all opposing rays sum to unity, with all other rays having a weighting factor of one. That is, rays which travel substantially the same trajectory in opposite directions are replaced by an average of the two. Because the dispersion factor increases with helical pitch, a maximum value of the dispersion factor can be used to define the maximum pitch of the helix that accurately reconstructs the cylinder of radius r c . 
     Once the end surfaces of the reconstruction cylinder are defined, the leading and trailing collimation of the cone beam are also defined. All non-zero weighted projections, required to reconstruct each oblique surface, map to a detector surface  82  that encompasses the reconstruction cylinder C and extends along the z-axis over a distance (projected at r=0) of approximately p/2. The detector surface is slightly oblique to the normal of the z-axis, depending on the helical pitch p. 
     It should be appreciated that a thin slab, such as 3-6 substantially parallel rows, may be defined instead of a two-dimensional oblique surface. In this embodiment, once the rays or radiation passing through a surface are identified, the adjoining rays, fore and aft, in the appropriate direction are considered as passing through two parallel surfaces of a slab. 
     Referring again to FIG. 1, a first data processor  54  weights the data from the identified rays passing through each oblique surface. In one embodiment, conventional plus-region weighting along Z(θ) is applied based on 180° oblique surface projections and end projections. In addition, cone beam angle weighting is optionally applied to account for angular distribution of backprojected rays. A second data processor  56  receives the data from the first data processor  54  and performs a conventional two-dimensional convolution on the data. A backprojector  58  receives the convolved data from the second data processor  56  and two-dimensionally backprojects the oblique surface projections over 180° after combining complimentary views. In another embodiment, the backprojector  58 , performs a V-weighted backprojection of oblique surface projections over the full 180° plus fan. In another embodiment, a three-dimensional backprojection is performed for sets of projections with the Z interpolation of the projections corresponding to a linear position along the oblique surface image matrix. 
     It should be appreciated that the interpolated projections corresponding to each surface may be interpolated into either fan-beam or parallel-beam projections, depending on the density of views collected in one revolution of the helix. 
     Once the oblique surfaces are reconstructed, the reconstruction cylinder C may be re-sampled onto an orthogonal matrix C(x,y,z) by interpolating the reconstructed surfaces S(x,y,z) for each (x,y) value along the z-direction. In other words C(x,y,z) is mapped from the reconstructed surface matrix S(x,y,z(x,y)). 
     It should be appreciated that tradeoffs may be made between the dispersion factor and helical pitch. For example, for the same dispersion at radius r c =125 mm versus r c =250 mm, scans with almost twice the helical pitch may be used. 
     With reference to FIGS. 4A-4F, six alternate embodiments for the reconstruction of oblique surfaces are provided. FIG. 4A is preferred for cone beam scans where the cone angle is sufficiently narrow to permit accurate reconstruction using simple 2D backprojection of oblique surfaces. More particularly, data is collected  100  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  105  or up-sampled by a factor of two using high-order interpolation. Optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  110  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  115  in both the azimuthal and radial directions. 
     Following rebinning  115 , a plus-region weighting algorithm is applied  120  along Z(θ) based on 180° oblique surface projections and end projections. In order to account for the angular distribution of backprojected rays, a cone beam angle weighting algorithm  125  is applied. Elements in each row have the same cosine weight value because the rows are equally distributed along the z-axis. After the projections along rows are completed, a FFT convolution  130  is performed on all of the rows. Following the convolution, the oblique surface projections are interpolated  135  over each detector surface into oblique surface projection rows and columns. 
     The oblique surface projections are two-dimensionally backprojected  140  over 180° after combining complimentary views. A rectangular (x,y,z) volumetric data set is generated  145  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  150  along the z-direction to define the effective z-axis resolution. 
     FIG. 4B provides an alternate embodiment which provides tradeoffs in image quality and requires the addition of linear depth weighting during the two-dimensional backprojection. More particularly, data is collected  200  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  205  or up-sampled by a factor of two using high-order interpolation. Optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  210  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  215  in both the azimuthal and radial directions. 
     Following rebinning  215 , the oblique surface projections are interpolated  220  over the extent of the detector surface, where the skew is tangent to two points along the end rows. An oblique surface projection weighting  225  is applied proportional to the inverse distance between interpolated samples along U. It should be appreciated that U and V are projection space directions, as illustrated in FIG.  2 C. After the projections along rows are completed, a FFT convolution  230  is performed on all of the rows. Following the convolution, a plus-region weighting algorithm is applied  235  along Z(θ) based on 180° oblique surface projections and end projections. 
     The oblique surface projections are V-weighted, two-dimensionally backprojected  240  over the full 180° plus angle. The V-weighting is applied using a linear weighted set of values between W maxv  (U) and W minv  (U). The V-weighting compensates for convolving dispersed rays in each oblique surface projection. A rectangular (x,y,z) volumetric data set is generated  245  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  250  along the z-direction to define the effective z-axis resolution. 
     FIG. 4C illustrates an alternate embodiment which is particularly accurate for large cone angles. More particularly, data is collected  300  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  305  or up-sampled by a factor of two using high-order interpolation. Optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  310  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  315  in both the azimuthal and radial directions. 
     Following rebinning  315 , the oblique surface projections are interpolated  320  over the extent of the detector surface, where the skew is tangent to two points along the end rows. An oblique surface projection weighting  325  is applied proportional to the inverse distance between interpolated samples along U. After the projections along rows are completed, a FFT convolution  330  is performed on all of the rows. Following the convolution, dZ(U) required to backproject onto a surface at z=z 0  is identified  335  around each oblique surface. A plus-region weighting is applied  340  for all samples in a given region within each view. The samples are interpolated to stretch the data  345  along U bounded by dZ(U) to projections bounded by dZ max . V-weighting is applied  350  using a linear weighted set of values between W maxv (U) and W minv (U) to the region bounded by dZ max . The V-weighting compensates for convolving dispersed rays in each oblique surface projection. A three-dimensional backprojection is applied  355  to the sets of projections with the Z interpolation of the projections corresponding to a linear position in V along the oblique surface image matrix. A rectangular (x,y,z) volumetric data set is generated  360  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  365  along the z-direction to define the effective z-axis resolution. 
     FIG. 4D illustrates an alternate embodiment which is analogous to the embodiment illustrated in FIG. 4C where the oblique surfaces are defined as planar surfaces. More particularly, data is collected  400  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  405  or up-sampled by a factor of two using high-order interpolation. optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  410  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  415  in both the azimuthal and radial directions. 
     Following rebinning  415 , the oblique planar projections are interpolated  420  over the extent of the detector surface, where the skew is tangent to two points along the end rows. An oblique planar projection weighting  425  is applied proportional to the inverse distance between interpolated samples along U. It is to be appreciated that for oblique planar projections, the oblique planar projection weighting  425  corresponds to a constant value for each projection. After the projections along rows are completed, a FFT convolution  430  is performed on all of the rows. 
     Following the convolution, the dZ(U) required to backproject onto the plane at z=z 0  is identified  435  around each oblique planar projection. A plus-region weighting is applied  440  for all samples in a given region within each view. The samples are interpolated to stretch the data  445  along U bounded by dZ(U) to projections bounded by dZ max . V-weighting is applied  450  using a linear weighted set of values between W maxv (U) and W minv (U) to the region bounded by dZ max . The V-weighting compensates for convolving dispersed rays in each oblique planar projection. It is to be appreciated that for oblique planar projections, dZ and dZ max  are larger than for oblique surface projections due to the larger dispersion factor. A three-dimensional backprojection is applied  455  to the sets of projections with the Z interpolation of the projections corresponding to a linear position in V along the oblique planar image matrix. A rectangular (x,y,z) volumetric data set is generated  460  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  465  along the z-direction to define the effective z-axis resolution. 
     FIG. 4E illustrates an alternate embodiment which is analogous to the embodiment illustrated in FIG. 4C where views are rebinned into wedge projections. More particularly, data is collected  500  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  505  or up-sampled by a factor of two using high-order interpolation. Optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  510  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  515  in both the azimuthal and radial directions into wedge projections. 
     Following rebinning  515 , a plus-region weighting algorithm is applied  520  along Z(θ) based on 180° oblique surface projections and end projections. In order to account for the angular distribution of backprojected rays, a cone beam angle weighting algorithm  525  is applied. Elements in each row have the same cosine weight value because the rows are equally distributed along the z-axis. After the projections along rows are completed, a FFT convolution  530  is performed on all of the rows. Following the convolution, the oblique surface projections are interpolated  535  over each detector surface into oblique surface projection rows and columns. 
     Following the interpolation, dZ(U) required to backproject onto a surface at z=z 0  is identified  540  around each oblique surface. The samples are interpolated to stretch the data  545  along U bounded by dZ(U) to projections bounded by dZ max . A three-dimensional backprojection is applied  550  to the sets of projections with the Z interpolation of the projections corresponding to a linear position in V along the oblique surface image matrix. A rectangular (x,y,z) volumetric data set is generated  555  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  560  along the z-direction to define the effective z-axis resolution. 
     FIG. 4F illustrates aln alternate embodiment which is analogous to the embodiment illustrated in FIG. 4E where the oblique surfaces are defined as planar surfaces. More particularly, data is collected  600  in the native sampling geometry, such as flat, source arc, or scan center arc. Columns and rows are pre-interpolated  605  or up-sampled by a factor of two using high-order interpolation. Optionally, the first and last rows are duplicated, if necessary. Alternately, for embodiments in which the number of rows is small, the 180° data points from the first row are used to interpolate the points at the end of the last row. The row and column elements are interpolated  610  to generate equiangular data which is orthogonal to the z-axis with the rows and partial rows equally spaced along the z-axis. The views are rebinned  615  in both the azimuthal and radial directions into wedge projections. 
     Following rebinning  615 , a plus-region weighting algorithm is applied  620  along Z(θ) based on 180° oblique surface projections and end projections. In order to account for the angular distribution of backprojected rays, a cone beam angle weighting algorithm  625  is applied. Elements in each row have the same cosine weight value because the rows are equally distributed along the z-axis. After the projections along rows are completed, a FFT convolution  630  is performed on all of the rows. Following the convolution, the oblique planar projections are interpolated  635  over each detector surface into oblique surface projection rows and columns. 
     Following the interpolation, the dZ(U) required to backproject onto the plane at z=z 0  is identified  640  around each oblique planar projection. It should be appreciated that for oblique planar projections dZ and dZ max  are larger than for oblique surface projections due to the larger dispersion factor. The samples are interpolated to stretch the data  645  along U bounded by dZ(U) to projections bounded by dZ max . A three-dimensional backprojection is applied  650  to the sets of projections with the Z interpolation of the projections corresponding to a linear position in V along the oblique planar image matrix. A rectangular (x,y,z) volumetric data set is generated  655  by interpolating the oblique surface matrices along the z-direction. It should be appreciated that the sequence of reconstructions along the z-axis is separated by no more that the half-width of a single radiation detector. The resulting volumetric data set is filtered  660  along the z-direction to define the effective z-axis resolution. 
     It should be appreciated that the oblique surface derivation is the same for embodiments where two-dimensional convolution and backprojection algorithms are applied as for embodiments where corresponding three-dimensional reconstruction algorithms are applied. Further, applying a three-dimensional backprojection algorithm to the defined surfaces provides greater accuracy and requires a minimal set of projections for complete image reconstruction. 
     The invention has been described with reference to the preferred embodiment. Modifications and alterations will occur to others upon a reading and understanding of 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.