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“Statistical Image Processing: Restoration and Reconstruction”
by Thomas James Hebert
April 1989
This dissertation develops statistical approaches to image restoration and reconstruction. A statistic-based stopping criterion for iterative least squares and maximum likelihood image reconstruction is developed to regularize the solution by effecting a trade-off between the maximum likelihood or least squares solution and a maximally smooth uniform image. Bayesian estimation is embraced as a probabilistic method of incorporating qualitative information regarding the local smoothness of images. Markov random field priors in the form of Gibbs distributions are formulated to incorporate local correlations and smoothness yet allow abrupt changes to occur across a boundary between two regions in an image. A computationally efficient algorithm for Bayesian image reconstruction based on Gibbs distribution priors is derived. This algorithm combines the complete/incomplete data formulation of the expectation - maximization approach, coordinate ascent, the optimal step-size of a maximum likelihood algorithm derived along parallel lines, and a method of adjusting the step size if necessary. A statistic-based approach to selection of the Gibbs prior parameter is developed and its potential for selecting the optimal parameter with respect to L1 and L2 restoration error is evaluated. The resulting solution is set in a control systems framework and a feedback algorithm for use in conjunction with any deterministic MAP algorithm for simultaneous parameter selection and image restoration or reconstruction is presented. Some problems in emission tomography are also addressed. An intermediate polar pixel representation for the unknown 3-D image is used to reduce the computational requirements of iterative reconstruction algorithms. An efficient method of directly including attenuation in iterative reconstruction algorithms is suggested. An experimental emission imaging system with a spatially varying point source response is examined. A probabilistically based factorization of the system matrix for the experimental system is introduced allowing the computational demands of a class of iterative algorithms to be reduced by several orders of magnitude. A modification to a well known maximum likelihood algorithm is introduced to increase the rate of convergence of the likelihood by a factor of five.