Reconstruction of SPECT images using generalized matrix inverses

Generalized matrix inverses are used to estimate source activity distributions from single photon emission computed tomography (SPECT) projection measurements. Image reconstructions for a numerical simulation and a clinical brain study are examined. The photon flux from the source region and photon detection by the gamma camera are modeled by matrices which are computed by Monte Carlo methods. The singular value decompositions (SVDs) of these matrices give considerable insight into the SPECT image reconstruction problem and the SVDs are used to form approximate generalized matrix inverses. Tradeoffs between resolution and error in estimating source voxel intensities are discussed, and estimates of these errors provide a robust means of stabilizing the solution to the ill-posed inverse problem. In addition to its quantitative clinical applications, the generalized matrix inverse method may be a useful research tool for tasks such as evaluating collimator design and optimizing gamma camera motion.

[1]  N. Pelc,et al.  An attenuated projector-backprojector for iterative SPECT reconstruction. , 1985, Physics in medicine and biology.

[2]  E. Levitan,et al.  A Maximum a Posteriori Probability Expectation Maximization Algorithm for Image Reconstruction in Emission Tomography , 1987, IEEE Transactions on Medical Imaging.

[3]  William H. Press,et al.  Numerical recipes , 1990 .

[4]  C E Metz,et al.  Quantitative effects of stationary linear image processing on noise and resolution of structure in radionuclide images. , 1974, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[5]  K. Lange,et al.  EM reconstruction algorithms for emission and transmission tomography. , 1984, Journal of computer assisted tomography.

[6]  R. Jaszczak,et al.  On Bayesian image reconstruction from projections: uniform and nonuniform a priori source information. , 1989, IEEE transactions on medical imaging.

[7]  J Llacer,et al.  Feasible images and practical stopping rules for iterative algorithms in emission tomography. , 1989, IEEE transactions on medical imaging.

[8]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[9]  L. Shepp,et al.  Maximum Likelihood Reconstruction for Emission Tomography , 1983, IEEE Transactions on Medical Imaging.

[10]  C E Floyd,et al.  Inverse Monte Carlo as a unified reconstruction algorithm for ECT. , 1986, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[11]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[12]  R. Wiggins,et al.  The general linear inverse problem - Implication of surface waves and free oscillations for earth structure. , 1972 .

[13]  J. Llacer,et al.  A fast Bayesian reconstruction algorithm for emission tomography with entropy prior converging to feasible images. , 1990, IEEE transactions on medical imaging.

[14]  E. Veklerov,et al.  Stopping Rule for the MLE Algorithm Based on Statistical Hypothesis Testing , 1987, IEEE Transactions on Medical Imaging.

[15]  Ronald J. Jaszczak,et al.  Analysis of SPECT including Scatter and Attenuation Using Sophisticated Monte Carlo Modeling Methods , 1982, IEEE Transactions on Nuclear Science.

[16]  R. Jaszczak,et al.  Inverse Monte Carlo: A Unified Reconstruction Algorithm for SPECT , 1985, IEEE Transactions on Nuclear Science.

[17]  Eugene Veklerov,et al.  The feasibility of images reconstructed with the methods of sieves , 1990 .

[18]  J. Ollinger.  Iterative reconstruction-reprojection and the expectation-maximization algorithm. , 1990, IEEE transactions on medical imaging.

[19]  D. Gilland,et al.  Implementation of simultaneous attenuation and detector response correction in SPECT , 1988 .

[20]  B. C. Penney,et al.  Restoration of combined conjugate images in SPECT: comparison of a new Wiener filter and the image-dependent Metz filter , 1990 .

[21]  Gengsheng Larry Zeng,et al.  Spectral decomposition of the exponential radon transform , 1990, Optics & Photonics.

[22]  Lee-Tzuu Chang,et al.  Attenuation Correction and Incomplete Projection in Single Photon Emission Computed Tomography , 1979, IEEE Transactions on Nuclear Science.

[23]  J R Perry,et al.  Correction of nonuniform attenuation in cardiac SPECT imaging. , 1989, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[24]  Barbara Y. Croft,et al.  Single-photon emission computed tomography , 1986 .

[25]  Gene H. Golub,et al.  Matrix computations , 1983 .

[26]  C E Floyd,et al.  Convergence of the maximum likelihood reconstruction algorithm for emission computed tomography. , 1987, Physics in medicine and biology.

[27]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[28]  Jorge Llacer,et al.  Matrix-Based Image Reconstruction Methods for Tomography , 1985, IEEE Transactions on Nuclear Science.

[29]  L. J. Thomas,et al.  Noise and Edge Artifacts in Maximum-Likelihood Reconstructions for Emission Tomography , 1987, IEEE Transactions on Medical Imaging.

[30]  Ronald J. Jaszczak,et al.  Weighted backprojection implemented with a non-uniform attenuation map for improved SPECT quantitation , 1988 .

[31]  R. Ricardo Brechner,et al.  Non-uniform attenuation and scatter correction in SPECT , 1988 .

[32]  Joel Franklin,et al.  Well-posed stochastic extensions of ill-posed linear problems☆ , 1970 .

[33]  Jorge Llacer.  Theory of Imaging with a Very Limited Number of Projections , 1979, IEEE Transactions on Nuclear Science.

[34]  Michael I. Miller,et al.  The Use of Sieves to Stabilize Images Produced with the EM Algorithm for Emission Tomography , 1985, IEEE Transactions on Nuclear Science.

[35]  J. Llacer.  Tomographic Image Reconstruction by Eigenvector Decomposition: Its Limitations and Areas of Applicability , 1982 .

[36]  Richard M. Leahy,et al.  Fast MLE for SPECT using an intermediate polar representation and a stopping criterion , 1988 .

[37]  G. Backus,et al.  Uniqueness in the inversion of inaccurate gross Earth data , 1970, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[38]  B. C. Penney,et al.  Two-dimensional filtering of SPECT images using the Metz and Wiener filters. , 1984, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[39]  Jack J. Dongarra,et al.  Matrix Eigensystem Routines — EISPACK Guide Extension , 1977, Lecture Notes in Computer Science.

[40]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[41]  R. A. Brooks,et al.  Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging , 1976, Physics in medicine and biology.