Patent Number: 
Section: claims

1. A method for determining a system matrix for a medical imaging system, the method comprising:using a closed form expression to determine a penetration term for a collimator of the medical imaging system;determining a point spread function of the collimator based on the penetration term; andcalculating the system matrix for the medical imaging system based on the determined point spread function. 2. A method in accordance with claim 1 further comprising convolving the collimator point spread function with a detector response for a detector of the medical imaging system. 3. A method in accordance with claim 1 further comprising convolving the collimator point spread function with a projection of a voxel onto a detector of the medical imaging system. 4. A method in accordance with claim 1 wherein determining a penetration term comprises analytically deriving the penetration term. 5. A method in accordance with claim 1 further comprising reconstructing an image based on the system matrix, wherein the system matrix is pre-calculated for use during image reconstruction. 6. A method in accordance with claim 1 further comprising performing an element by element multiplication of system matrix terms to incorporate attenuation factors to the system matrix. 7. A method in accordance with claim 1 wherein the collimator comprises a pinhole collimator. 8. A method in accordance with claim 1 wherein the collimator comprises a focusing pinhole collimator. 9. A method in accordance with claim 1 wherein the medical imaging system comprises a single photon emission computed tomography (SPECT) imaging system. 10. A method in accordance with claim 1 wherein the closed form expression defines a path length ΔL of a photon through a collimator as:            Δ      ⁢                          ⁢      L        =                            Δ          ⁢                                          ⁢          t                          sin          ⁢                                          ⁢                      θ            a                              =                                                  d                              f                ⁢                                                                                        ⁢            tan            ⁢                                                  ⁢            α            ⁢                                                  ⁢                          (                                                N                  ⁢                                                                          ⁢                  sin                  ⁢                                                                          ⁢                  γ                                ⁢                                                                  +                                  cos                  ⁢                                                                          ⁢                  γ                                            )                                -                                    (                                                Q                  1                  2                                -                                  PR                  1                                            )                                      1              /              2                                -                                    (                                                Q                  2                  2                                -                                  PR                  2                                            )                                      1              /              2                                                P          (                                                    csc                2                            ⁢              θ                        -                          2              ⁢              ρ              ⁢                                                          ⁢              cot              ⁢                                                          ⁢              θ              ⁢                                                          ⁢                                                cos                  ⁡                                      (                                          β                      -                                              ϕ                        /                        h                                            +                                                                        ρ                          2                                                /                                                  h                          2                                                                                      )                                                                                        -                    1                                    /                  2                                                                          ,whereP=M2+AN2−2BN+C; M=cot θ cos φ−ρ cos β/h N=cot θ sin φ−ρ sin β/h; A=cos2 γ−sin2 γ tan2 α;B=cos γ sin γ(1+tan2 α); C=sin2 γ=cos2 γ tan2 α;R1=ρ2(cos2 β+A sin2 β)−ρdf sin β sin γ tan α−0.25df2;R2=ρ2(cos2 β+A sin2 β)+ρdf sin β sin γ tan α−0.25df2;Q1=Mρ cos β+ρ sin β(AN−B)−0.5df tan α(N sin γ+cos γ);Q2=Mρ cos β+ρ sin β(AN−B)+0.5df tan α(N sin γ+cos γ). 11. A method in accordance with claim 1 further comprising using a precomputed system matrix from the calculating and incorporating a table translation into image reconstruction. 12. A method in accordance with claim 1 further comprising pre-computing the system matrix for voxels within a volume of interest (VOI) at a higher resolution and computing the system matrix for voxels outside the VOI as a lower resolution. 13. A method in accordance with claim 1 further comprising performing angular sampling with a variable step size and a variable translation step size for adjacent views of the medical imaging system. 14. A method for determining a system matrix for a medical imaging system, the method comprising:determining a penetration term for a collimator of the medical imaging system without performing any measurements using the medical imaging system;determining a sensitivity term, including a geometric term and a penetration term, for shape of a point spread function for the collimator based on the penetration term; andcalculating the system matrix for the medical imaging system based on the determined point spread function. 15. A method in accordance with claim 14 wherein determining the shape of the point spread function comprises using a closed form expression to model the collimator. 16. A method in accordance with claim 14 wherein the determined penetration term comprises an analytically derived term. 17. A method in accordance with claim 14 wherein the collimator comprises a pinhole collimator. 18. A method in accordance with claim 14 further comprising adding attenuation factors to the system matrix using matrix multiplication. 19. A method in accordance with claim 14 further comprising convolving a distance driven based function, and calibration parameters and sensitivity terms, with the system matrix. 20. A method for determining a system matrix for a medical imaging system, the method comprising:determining parametric values where a plurality of planes that contain a voxel in an image space intersect a collimator and a detector of the imaging system;marking a location where the plurality of planes intersect a surface of the collimator as end points on the detector; andcalculating the system matrix for the medical imaging system based on an inner most shape through which photons from a point source pass through the collimator and are detected. 21. A method in accordance with claim 20 wherein the penetration of the photon through the collimator is modeled by determining locations where edges of the collimator intersect a plane containing a point source and passes through a detector of the medical imaging systems. 22. A method in accordance with claim 21 wherein calculating a path length of photons through a material of the collimator from a shape of the collimator for each location on a surface of the detector lying between the projection of the edges of the collimator. 23. A method for reducing the size of a system matrix for a medical imaging system, the method comprising:calculating the system matrix for one or more geometric configurations of a collimator and detector of a medical imaging system, wherein the calculated system matrix for all locations in an image space based on one of linear and non-linear transformations is used; andprecomputing a reduced system matrix for all angular views at one position of a table of the medical imaging system. 24. A method in accordance with claim 23 wherein the precomputing comprises using a variable-pitch-helical acquisition of the medical imaging system, wherein a collimator and gantry are rotated to varying view angles located as far apart as possible and a patient table is moved. 25. A method in accordance with claim 24 further comprising computing the system matrix for different table translations using a distance of the table motion. 26. A method in accordance with claim 23 further comprising precomputing the system matrix and storing the system matrix for voxels within a volume of interest (VOI), and for voxels outside the VOI calculating element for the system matrix using one of a fast approximation and precomputing for a larger interval between the voxels than initially used to precomputed the system matrix. 27. A method in accordance with claim 23 further comprising approximating values of different elements of the system matrix using a set of parameters, including at least one of a combination of geometric shapes and physical parameters, which include one of a radius, offset values, an aspect ratios and an amplitude. 28. A method in accordance with claim 23 further comprising sequentially sampling the image in a same order during generation of the system matrix and reconstruction. 29. A method in accordance with claim 23 further comprising storing one of system matrix elements and parameters by saving multiple values in a same memory location element. 30. A medical imaging system comprising:a plurality of nuclear medicine imaging detectors;a plurality of pinhole collimators attached to the plurality of nuclear medicine imaging detectors; andan image reconstruction processor configured to reconstruct an image using a system matrix calculated based on an analytically derived pinhole penetration term. 31. A medical imaging system in accordance with claim 30 wherein the analytically derived pinhole penetration term is calculated using a closed form expression. 32. A medical imaging system in accordance with claim 30 wherein the plurality of nuclear medicine imaging detectors comprise gamma cameras and the plurality of pinhole collimators comprise focusing pinhole collimators.