Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images

The authors describe conjugate gradient algorithms for reconstruction of transmission and emission PET images. The reconstructions are based on a Bayesian formulation, where the data are modeled as a collection of independent Poisson random variables and the image is modeled using a Markov random field. A conjugate gradient algorithm is used to compute a maximum a posteriori (MAP) estimate of the image by maximizing over the posterior density. To ensure nonnegativity of the solution, a penalty function is used to convert the problem to one of unconstrained optimization. Preconditioners are used to enhance convergence rates. These methods generally achieve effective convergence in 15-25 iterations. Reconstructions are presented of an (18)FDG whole body scan from data collected using a Siemens/CTI ECAT931 whole body system. These results indicate significant improvements in emission image quality using the Bayesian approach, in comparison to filtered backprojection, particularly when reprojections of the MAP transmission image are used in place of the standard attenuation correction factors.

[1]  G. Krishna,et al.  Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography , 1992, IEEE Trans. Medical Imaging.

[2]  Federico Girosi,et al.  Parallel and Deterministic Algorithms from MRFs: Surface Reconstruction , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  K. Lange,et al.  A Theoretical Study of Some Maximum Likelihood Algorithms for Emission and Transmission Tomography , 1987, IEEE Transactions on Medical Imaging.

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

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

[6]  J. M. Ollingery.  The use of maximum a posteriori and maximum likelihood transmission images for attenuation correction in PET , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[7]  Harrison H. Barrett,et al.  Information processing in medical imaging , 1994, Image Vis. Comput..

[8]  Alan C. Evans,et al.  A Technique To Reject Scattered Radiation In PET Transmission Scans , 1986, Other Conferences.

[9]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[10]  Jeffrey A. Fessler.  Tomographic Reconstruction Using Information-Weighted Spline Smoothing , 1993, IPMI.

[11]  Anand Rangarajan,et al.  Bayesian Reconstruction for Emissiom Tomography via Deterministic Annealing , 1993, IPMI.

[12]  J. H. Hubbell,et al.  Photon mass attenuation and energy-absorption coefficients , 1982 .

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

[14]  Jolyon A. Browne,et al.  Developments with maximum likelihood X-ray computed tomography , 1992, IEEE Trans. Medical Imaging.

[15]  T J Hebert,et al.  Maximum Likelihood Preprocessing for Improved Filtered Back‐Projection Reconstructions , 1994, Journal of computer assisted tomography.

[16]  D. Snyder,et al.  Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography. , 1991, IEEE transactions on medical imaging.

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

[18]  J. M. Ollinger,et al.  Model based scatter correction in three dimensions (positron emission tomography) , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[19]  Tinsu Pan,et al.  Preconditioning methods for improved convergence rates in iterative reconstructions , 1993, IEEE Trans. Medical Imaging.

[20]  Alvaro R. De Pierro,et al.  On methods for maximum a posteriori image reconstruction with a normal prior , 1992, J. Vis. Commun. Image Represent..

[21]  Alvaro R. De Pierro,et al.  A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography , 1995, IEEE Trans. Medical Imaging.

[22]  J.A. Fessler.  Segmented attenuation correction for PET , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[23]  Simon R. Cherry,et al.  A statistical approach to transmission image reconstruction from ring source calibration measurements in PET , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[24]  Linda Kaufman,et al.  Implementing and Accelerating the EM Algorithm for Positron Emission Tomography , 1987, IEEE Transactions on Medical Imaging.

[25]  Magnus Dahlbom,et al.  PET system calibrations and corrections for quantitative and spatially accurate images , 1989 .

[26]  Ken D. Sauer,et al.  A generalized Gaussian image model for edge-preserving MAP estimation , 1993, IEEE Trans. Image Process..

[27]  C. Pelizzari,et al.  Retrospective geometric correlation of MR, CT, and PET images. , 1988, Radiology.

[28]  Ken D. Sauer,et al.  Bayesian estimation of transmission tomograms using segmentation based optimization , 1992 .

[29]  Stephen M. Moore,et al.  An Evaluation of the Use of Sieves for Producing Estimates. Of Radioactivity Distributions with the EM Algorithm for PET , 1986, IEEE Transactions on Nuclear Science.

[30]  Richard M. Leahy,et al.  Incorporation of Anatomical MR Data for Improved Dunctional Imaging with PET , 1991, IPMI.

[31]  S R Cherry,et al.  Attenuation correction using count-limited transmission data in positron emission tomography. , 1993, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[32]  Wesley E. Snyder,et al.  Mean field annealing: a formalism for constructing GNC-like algorithms , 1992, IEEE Trans. Neural Networks.

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

[34]  T. Hebert,et al.  A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. , 1989, IEEE transactions on medical imaging.

[35]  R.E. Carson,et al.  Scatter correction in maximum-likelihood reconstruction of PET data , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[36]  Robert M. Lewitt,et al.  Accelerated Iterative Reconstruction for Positron Emission Tomography Based on the EM Algorithm for Maximum Likelihood Estimation , 1986, IEEE Transactions on Medical Imaging.