Abstract:
A method for interpolating at least one oblique line of response ray representing nuclear image projection data through a rectangular volume and a system for using the method. The method consists of steps of interpolating all the direct rays in a rectangular volume, making a projected ray by projecting the oblique ray onto a surface of the rectangular volume, matching the projected ray to a coinciding interpolated direct ray, shearing the rectangular volume to match the projected ray, and interpolating the oblique ray in the sheared volume.

Description:
TECHNICAL FIELD 
       [0001]    The current invention is in the field of medical imaging, and in particular pertains to reconstruction of tomographic images from acquired projection data obtained by an imaging apparatus. 
       BACKGROUND OF THE INVENTION 
       [0002]    Medical imaging is one of the most useful diagnostic tools available in modern medicine. Medical imaging allows medical personnel to non-intrusively look into a living body in order to detect and assess many types of injuries, diseases, conditions, etc. Medical imaging allows doctors and technicians to more easily and correctly make a diagnosis, decide on a treatment, prescribe medication, perform surgery or other treatments, etc. 
         [0003]    There are medical imaging processes of many types and for many different purposes, situations, or uses. They commonly share the ability to create an image of a bodily region of a patient, and can do so non-invasively. Examples of some common medical imaging types are nuclear medical (NM) imaging such as positron emission tomography (PET) and single photon emission computed tomography (SPECT). Using these or other imaging types and associated apparatus, an image or series of images may be captured. Other devices may then be used to process the image in some fashion. Finally, a doctor or technician may read the image in order to provide a diagnosis. 
         [0004]    A PET camera works by detecting pairs of gamma ray photons in time coincidence. The two photons arise from the annihilation of a positron and electron in the patient&#39;s body. The positrons are emitted from a radioactive isotope that has been used to label a biologically important molecule (a radiopharmaceutical). Hundreds of millions such decays occur per second in a typical clinical scan. Because the two photons arising from each annihilation travel in opposite directions, the rate of detection of such coincident pairs is proportional to the amount of emission activity, and hence the molecule, along the line connecting the two detectors at the respective points of gamma ray interaction. In a PET camera the detectors are typically arranged in rings around the patient. By considering coincidences between all appropriate pairs of these detectors, a set of projection views can be formed, each element of which represents a line integral, or sum, of the emission activity in the patient&#39;s body along a well defined path. These projections are typically organized into a data structure called a sinogram, which contains a set of plane parallel projections at uniform angular intervals around the patient. A three dimensional image of the radiopharmaceutical&#39;s distribution in the body then can be reconstructed from these data. 
         [0005]    A SPECT camera functions similarly to a PET camera, but detects only single photons rather than coincident pairs. For this reason, a SPECT camera must use a lead collimator with holes, placed in front of its detector panel, to pre-define the lines of response in its projection views. One or more such detector panel/collimator combinations rotates around a patient, creating a series of planar projections each element of which represents a sum of the emission activity, and hence biological tracer, along the line of response defined by the collimation. As with PET, these data can be organized into sinograms and reconstructed to form an image of the radiopharmaceutical tracer distribution in the body. 
         [0006]    The purpose of the reconstruction process is to retrieve the spatial distribution of the radiopharmaceutical from the projection data. A conventional reconstruction step involves a process known as back-projection. In simple back-projection, an individual data sample is back-projected by setting all the image pixels along the line of response pointing to the sample to the same value. In less technical terms, a back-projection is formed by smearing each view back through the image in the direction it was originally acquired. The back-projected image is then taken as the sum of all the back-projected views. Regions where back-projection lines from different angles intersect represent areas which contain a higher concentration of radiopharmaceutical. 
         [0007]    While back-projection is conceptually simple, it does not by itself correctly solve the reconstruction problem. A simple back-projected image is very blurry; a single point in the true image is reconstructed as a circular region that decreases in intensity away from the center. In more formal terms, the point spread function (PSF) of back-projection is circularly symmetric, and decreases as the reciprocal of its radius. 
         [0008]    Filtered back-projection (FBP) is a technique to correct the blurring encountered in simple back-projection. Each projection view is filtered before the back-projection step to counteract the blurring point spread function. That is, each of the one-dimensional views is convolved with a one-dimensional filter kernel (e.g. a “ramp” filter) to create a set of filtered views. These filtered views are then back-projected to provide the reconstructed image, a close approximation to the “correct” image. 
         [0009]    The inherent randomness of radioactive decay and other processes involved in generating nuclear medical image data results in unavoidable statistical fluctuations (noise) in PET or SPECT data. This is a fundamental problem in clinical imaging that is dealt with through some form of smoothing of the data. In FBP this is usually accomplished by modifying the filter kernel used in the filtering step by applying a low-pass windowing function to it. This results in a spatially uniform, shift-invariant smoothing of the image that reduces noise, but may also degrade the spatial resolution of the image. A disadvantage of this approach is that the amount of smoothing is the same everywhere in the image although the noise is not. Certain regions, e.g. where activity and detected counts are higher, may have relatively less noise and thus require less smoothing than others. Standard windowed FBP cannot adapt to this aspect of the data. 
         [0010]    There are several alternatives to FBP for reconstructing nuclear medical data. In fact, most clinical reconstruction of PET images is now based on some variant of regularized maximum likelihood (RML) estimation because of the remarkable effectiveness of such algorithms in reducing image noise compared to FBP. In a sense, RML&#39;s effectiveness stems from its ability to produce a statistically weighted localized smoothing of an image. These algorithms have some drawbacks however: they are relatively expensive because they must be computed iteratively; they generally result in poorly characterized, noise dependent, image bias, particularly when regularized by premature stopping (unconverged); and the statistical properties of their image noise are difficult to determine. 
         [0011]    In a class of algorithms for calculating projections known as the Square Pixel Method, the basic assumption is that the object considered truly consists of an array of N×N square pixels, with the image function ƒ(x, y) assumed to be constant over the domain of each pixel. The method proceeds by evaluating the length of intersection of each ray with each pixel, and multiplying the value of the pixel (S). 
         [0012]    The major problem of this method is the unrealistic discontinuity of the model. This is especially apparent for rays whose direction is exactly horizontal or vertical, so that relatively large jumps occur in S values as the rays cross pixel boundaries. 
         [0013]    A second class of algorithms for calculating projections is the forward projection method. This method is literally the adjoint of the process of “back projection” of the FBP reconstruction algorithm. The major criticism of this algorithm is that the spatial resolution of the reprojection is lessened by the finite spacing between rays. Furthermore, increasing the number of pixels does not contribute to a reduction in this spacing, but does greatly increase processing time. 
         [0014]    A third algorithm for calculating projections based on line-integral approximation, developed by Peter M. Joseph and described in the paper entitled  An Improved Algorithm for Reprojecting Rays Through Pixel Images , IEEE Transactions on Medical Imaging, Vol. MI-1, No. 3, pp. 192-196, November 1982 (hereinafter, “Joseph&#39;s Method”), incorporated by reference herein in its entirety, is similar to the structure of the square pixel method. Each given ray K is specified exactly as a straight line. The basic assumption is that the image is a smooth function of x and y sampled on the grid of positions. The line integral desired is related to an integral over either x or y depending on whether the ray&#39;s direction lies closer to the x or y axis. While this algorithm produces a much clearer image than the other two methods, it is slower than either method, especially when interpolating oblique segments. When interpolating oblique segments, an interpolation is required in both the transaxial and axial directions for each ray, further slowing the process. 
         [0015]    Therefore, there exists a need in the art to have a method for calculating projections that has the clarity of Joseph&#39;s Method yet takes less processing time. 
       SUMMARY OF THE INVENTION 
       [0016]    Provided is a method for reconstructing a tomographic image from projection data by interpolating an oblique ray or line of response (LOR) through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, creating a projected ray by projecting the oblique ray onto a surface of the rectangular volume, matching the projected ray to a coinciding interpolated direct ray, shearing the rectangular volume to match the projected ray, and interpolating the oblique ray in the sheared volume. 
         [0017]    Further provided is a method for interpolating a number of oblique rays through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, creating a plurality of projected rays for each oblique ray by projecting the oblique rays onto a surface of the rectangular volume, matching each projected ray to a coinciding interpolated direct ray, creating a plurality of sheared volumes by shearing the rectangular volume to match the projected rays, and interpolating each oblique ray in each sheared volume. 
         [0018]    Further provided is a method for interpolating at least two oblique rays of opposite polar angle through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, projecting the of oblique rays of opposite polar angle onto a surface of the rectangular volume, matching the projected rays to a coinciding interpolated direct ray, creating sheared volumes for each projected ray by shearing the rectangular volume to match the projected rays, interpolating one oblique ray of opposite polar angle in its corresponding sheared volume, and applying the interpolated value to the rest of the oblique rays of opposite polar angle. 
         [0019]    Further provided is a system for reconstructing tomographic images from projection data by interpolating oblique rays or LORs. The system includes a medical imaging device, a processor, and software running on the processor that executes the methods of the present invention. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0020]    The invention will now be described in greater detail in the following by way of example only and with reference to the attached drawings, in which: 
           [0021]      FIG. 1  is a representation of Joseph&#39;s Method for two dimensional interpolation. 
           [0022]      FIGS. 2A-C  are three dimensional, front, and side views, respectively, of a oblique ray in a rectangular volume. 
           [0023]      FIG. 3  is a representation of Joseph&#39;s Method for three dimensional interpolation. 
           [0024]      FIG. 4A-B  are front and side views, respectively, of the three dimensional interpolation of  FIG. 3 . 
           [0025]      FIG. 5  is a front view of a sheared space for the three dimensional interpolation of  FIG. 3  in accordance with the present invention. 
           [0026]      FIG. 6  is a three dimensional space with two opposite polar angle rays passing through it. 
           [0027]      FIGS. 7A and 7B  are front and side views, respectively, of the three dimensional space of  FIG. 6 . 
           [0028]      FIGS. 8A and 8B  are the front views of the sheared space for the two rays in  FIG. 6 . 
           [0029]      FIG. 9  is a flow chart of a method according to the present invention. 
           [0030]      FIG. 10  is a system using the methods of the present invention. 
           [0031]      FIGS. 11A and 11B  are top and cross-sectional views, respectively, of a cylindrical PET scanner with multiple detector rings, which is applicable to the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0032]    As required, disclosures herein provide detailed embodiments of the present invention; however, the disclosed embodiments are merely exemplary of the invention that may be embodied in various and alternative forms. Therefore, there is no intent that specific structural and functional details should be limiting, but rather the intention is that they provide a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present invention. 
         [0033]    Joseph&#39;s Method is a method for reprojecting rays through pixel images using line integrals. The basic assumption is that the image is a smooth function of x and y sampled on a grid of points in (x,y) space.  FIG. 1  is a representation of Joseph&#39;s Method in two dimensional space  110 . Each ray  120  passing through space  110 , is specified as a straight line, using either: 
         [0000]        y ( x )=−cot (θ) x+y   0    
         [0000]      or 
         [0000]        x ( y )=− y  tan (θ)+ x   0 . 
         [0034]    The line integral desired is related to an integral over either x or y depending on whether ray  120 &#39;s direction lies closer to the x or y axis, that is 
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         [0000]    The above two equations are related in the interchange of x and y as independent and dependent variables. 
         [0035]    In each case, the one dimensional integral is approximated by a simple sum, such as a Riemann sum; for example, the x-direction integral becomes 
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         [0000]    where the terms T 1  and T N  represents the first and last pixel on the line and are treated separately, and λ n  is the fractional number defined by 
         [0000]        n ′=integer part of  y ( x   n ) 
         [0000]      λ n   =y ( x   n )− n′.    
         [0036]    Interpolation enters in two senses: 1) explicitly, in the use of fraction λ n  to estimate the value of 
         [0000]        f ( x   n   ,y ( x   n ))≅(1−λ n ) P   n,n′ +λ n   P   n,n′+1    
         [0000]    and 2) implicitly in the sense that the summation above is the application of the trapezoidal rule to numerically estimate the one dimensional (x) integral. 
         [0037]    The treatment of the endpoints T 1  and T N  depend on the application. In some situations, they may be taken to be zero if outside the object images. For applications to heart-isolating algorithms, it is necessary to make them proportional to the length of intersection of the ray with the first and last pixels. 
         [0038]    Looking at  FIG. 1 , in two dimensions, Joseph&#39;s Method can be summarized as follows: For a given line or row yin two-dimensional space  110 , each ray  120  receives information from the two nearest pixels  130 A and  130 B. The distances  160 A and  160 B between the centers of pixels  130 A and  130 B and the point  150  where ray  120  intersects the horizontal line  140  passing through the center of pixels  130 A and  130 B define the interpolation coefficients. 
         [0039]    When there is translational symmetry in the axial (z) direction, the interpolation coefficients are the same for all the rays which differ only by their axial coordinate. This is shown in  FIGS. 11A-11B , which is a schematic representation of a cylindrical PET scanner  1101 , and its cross-section, respectively. The PET scanner  1101  includes multiple detector rings, such as rings  1102 - 1105 . Oblique rays  1106  and  1107  correspond to various non-zero ring difference. For example, ray  1106  extends between rings  1104  and  1105 , while ray  1107  extends between rings  1103  and  1104 . Rays  1106  and  1107  have the same transaxial coordinates (in the x-y plane) as direct rays  1108 , which extends within the same detector ring  1102 . There is also an axial translation symmetry for all rays with the same ring difference. 
         [0040]      FIG. 2A  is an example of an oblique segment ray  220  in three dimensional space  210 . Oblique segment ray  220  receives information from the four nearest voxels (i.e., volume elements or three dimensional pixels)  215 A,  215 B,  215 C and  215 D in an (x,y,z) image volume: the four voxels can be broken down into four pixels, two pixels  230 A and  230 B in the x direction ( FIG. 2B ), and two pixels  231 A and  231 B in the axial or z direction ( FIG. 2C ). 
         [0041]    In order to interpolate oblique ray  220 , interpolations over both the x direction and the z direction must be made. As in the two dimensional case, the distances  260 A and  260 B between the centers of pixels  230 A and  230 B and the point  250  where the ray  220  intersects the horizontal line  240  passing through the center of pixels  230 A and  230 B define the interpolation coefficients in the x direction. Likewise, the distances  261 A and  261 B between the centers of pixels  231 A and  231 B and the point  251  where the ray  220  intersects the horizontal line  241  passing through the center of pixels  231 A and  231 B, define the interpolation coefficients in the axial direction. 
         [0042]      FIG. 3  shows an example of an oblique ray  320  in a rectangular image volume  310  for a full three-dimensional reconstruction. If one were to interpolate based on Joseph&#39;s Method as described above, both front (i.e. xy) surface  410 A and side (i.e. yz) surface  410 B projections of the oblique ray  320  (see  FIGS. 4A and 4B ) would be necessary for each such oblique ray  320 , thus creating a front ray projection  420 A and a side ray projection  420 B. 
         [0043]    However, front ray projection  420 A of oblique ray  320  on front surface  410 A may coincide with the projection of a direct (i.e. two dimensional) ray on the same plane. Therefore, the interpolation coefficients in the x direction may be the same for front ray projection  420 A of oblique ray  320  and the direct two-dimensional ray. The pixel interpolation values for the direct rays thus could be reused on front ray projection  420 A. 
         [0044]    An efficient way to use such interpolated pixel values over the whole image volume would be to compute a sheared volume  510  (see  FIG. 5 ). In sheared volume  510 , in each row from volume  310  of  FIG. 3 , the vertical edges of the voxels may be skewed so that they are aligned with front ray projection  420 A on the xy surface. 
         [0045]    By so shearing the volume space to create sheared volume  510  to match the direction of ray projection  420 A, the two interpolations otherwise needed for oblique ray  320  may be reduced to a single interpolation of oblique ray  320  in sheared volume  510 . When there is translational symmetry in the z direction as shown in  FIG. 11 , the interpolation coefficients may be the same for all the rays which differ only by their x coordinate. Therefore, only one interpolation coefficient can be used for all voxels of one axial row in the sheared volume. This coefficient may be different for each plane. 
         [0046]      FIG. 6  shows a three dimensional space  610  through which model ray  620  and model ray  630  pass. Model rays  620  and  630  have opposite polar angles (i.e. opposite angles in the y-z plane). When rays  620  and  630  are projected onto the xy side surface  710 B (see  FIG. 7B ), it can be seen that they both have the identical xy side surface projection  740 . Yet, when model rays  620  and  630  are projected onto the yz front surface  710 A (see  FIG. 7A ), it can be seen that they have opposite or mirrored yz front surface projections  720 A and  730 A. 
         [0047]      FIGS. 8A and 8B  show front views of sheared volumes  810 A and  810 B for front projections  720 A and  730 A in accordance with the present invention. While sheared volumes  810 A and  810 B are different, each front projection  720 A and  730 A may coincide with a projection of a direct ray on the same plane. In practice, the same sheared volume may be used for both positive and negative polar angles, such that only one of the volumes  810 A and  810 B is actually necessary. 
         [0048]    Once the sheared volumes  810 A or  810 B are matched to the direct rays, the interpolation may reduce to a single interpolation of oblique model rays  620  and  630  in the sheared volume  810 A or  810 B, respectively. Since both model rays  620  and  630  have the same side projection  740 , both rays can be interpolated in the same single interpolation. 
         [0049]    For example, an oblique ray in a positive segment uses the following one dimensional axial interpolation: 
         [0000]        P   positive segment =value= w   z *shearedvoxel( ρ,y,z )+(1− w   z )* shearedvoxel(ρ ,y,z+ 1) 
         [0000]    While the same ray in the negative segment reuses the coefficients as: 
         [0000]        P   negative segment =(1− w   z )*shearedvoxel(ρ ,y,z )+ w   z *shearedvoxel(ρ ,y,z+ 1)=shearedvoxel(ρ ,y,z )+shearedvoxel( ρ,y,z+ 1)−value 
         [0050]    This excludes multiplication when calculating rays for one of the segments for the voxels in the sheared volume belonging to the intersection of the two segments. The algorithm may be thus summarized as follows. 
         [0051]    The equations for Joseph&#39;s method can be rewritten for the 3D case as: 
         [0000]    
       
         
           
             
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                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   
                      
                     
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                      
                   
                   ≥ 
                   
                     1 
                     
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0052]    Where θ is the azimuthal and φ is the polar angle. 
         [0053]    For the case where 
         [0000]    
       
         
           
             
                
               
                 sin 
                  
                 
                     
                 
                  
                 θ 
               
                
             
             ≥ 
             
               1 
               
                 2 
               
             
           
         
       
     
         [0000]    
       
         
           
             S 
             = 
             
               
                 1 
                 
                    
                   
                     sin 
                      
                     
                         
                     
                      
                     θ 
                      
                     
                         
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     φ 
                   
                    
                 
               
               [ 
               
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       2 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       μ 
                       n 
                     
                      
                     
                       ( 
                       
                         
                           P 
                           
                             n 
                             , 
                             
                               n 
                               ′ 
                             
                             , 
                             
                               n 
                               ″ 
                             
                           
                         
                         + 
                         
                           
                             λ 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 P 
                                 
                                   n 
                                   , 
                                   
                                     
                                       n 
                                       ′ 
                                     
                                     + 
                                     1 
                                   
                                   , 
                                   
                                     n 
                                     ″ 
                                   
                                 
                               
                               - 
                               
                                 P 
                                 
                                   n 
                                   , 
                                   
                                     n 
                                     ′ 
                                   
                                   , 
                                   
                                     n 
                                     ″ 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         μ 
                         n 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         P 
                         
                           n 
                           , 
                           
                             n 
                             ′ 
                           
                           , 
                           
                             
                               n 
                               ″ 
                             
                             + 
                             1 
                           
                         
                       
                       + 
                       
                         
                           λ 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               P 
                               
                                 n 
                                 , 
                                 
                                   
                                     n 
                                     ′ 
                                   
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   
                                     n 
                                     ″ 
                                   
                                   + 
                                   1 
                                 
                               
                             
                             - 
                             
                               P 
                               
                                 n 
                                 , 
                                 
                                   n 
                                   ′ 
                                 
                                 , 
                                 
                                   
                                     n 
                                     ″ 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where 
         [0054]    n′=integer part of y(x n ) 
         [0055]    λ n =y(x n )−n′. 
         [0056]    n″=integer part of z(x n ) 
         [0057]    μ n =z(x n )−n″ 
         [0000]    where P n,n′,n″ =f (n, n′, n″). 
         [0058]    For each azimuthal angle, a sheared volume is calculated using a 1 D transaxial interpolation in the original volume. Because of the transaxial symmetry, the original and sheared volumes are stored with axial index first. An array of depth coordinates d is also computed, as such coordinates are used when computing interpolation factors for oblique segments. Projection rays are also stored with axial index first. The storage of the axial index as the first index is very important from a hardware point of view, as all operations are applied in axial direction first. Thus, having the axial index as the first index facilitates an efficient use of the memory cache and enables use of hardware parallelization. This results in fast computing. The projections for 2D segments are calculated at the same time as the sheared volumes. The projections for all oblique segments are then obtained by a 1 D axial interpolation in the sheared volume. 
         [0059]      FIG. 9  shows an embodiment of a method  900  in accordance with the present invention. The first step  910  is to interpolate all the direct (i.e. planar) rays in the image volume. Once there are a number of direct rays, in step  920  the front surface ray projections of the oblique rays may be matched to the direct rays. The voxel space may then be sheared at step  930  to align with the matched front ray projections. Finally, the oblique rays may be interpolated at step  940  in the sheared volumes. 
         [0060]      FIG. 10  is a system  1000  for using method  900 . System  1000  may be comprised of a medical imaging device  1010 , i.e. a PET scanner, a SPECT scanner or similar device capable of acquiring a medical image. Medical imaging device  1010  may be attached to a processor  1020  for receiving the data. Processor  1020  may have software running on it that executes a method of the present invention and outputs a fully three dimensional reconstruction of the object scanned. 
         [0061]    The invention having been thus described, it will be apparent to those skilled in the art that the same may be varied in many ways without departing from the spirit of the invention. Any and all such variations are intended to be covered within the scope of the following claims. For example, the method can be extended to a so-called LOR projection geometry when the transverse distance between rays is not a constant, as in a ring scanner. In such case, the method requires only a scanner with axial translation symmetry. The method also can be extended in the case of an unmatched back-projector. In such case, a different shear procedure would be used where each voxel receives contributions from two nearest projection rays in the transverse direction. This is important when the transverse voxel size is significantly smaller than the transverse projection size.