#### Technical Report USC-SIPI-281

“Maximum Likelihood Hyperparameter Estimation for Gibbs Priors from Incomplete Data with Applications in Image Processing”

by Zhenyu Zhou

April 1995

In this dissertation we first discuss proper forms of Markov random fields (MRFs) image prior models characterized by Gibbs distributions which model both smooth regions and sharp boundaries. We then survey and compare several deterministic optimization algorithms for computing maximum a posterior (MAP) estimates associated with various forms of MRF image priors.

The dissertation concentrates on the problem of the selection of hyperparameter of Gibbs Prior distributions from degraded measurements. The choice of hyperparameters, plays a critical role in Bayesian methods for solving ill-posed inverse problems. Of particular importance for the case of Gibbs priors is the global hyperparameter which multiplies the Hamiltonian. The maximum likelihood (ML) estimate of hyperparameters from degraded measurements can be formulated in terms of parameter estimation from incomplete data in the sense of EM algorithm. The incomplete data are the observed measurements and the complete data are the unobserved images. Computing the exact ML estimate of the hyperparameter from incomplete data is intractable for most image processing problems due to the complexity and high dimensionality of the joint probability densities involved. Here, we develop an approximate ML estimator for this global hyperparameter, which is computed simultaneously with the MAP image. The new algorithm relies mostly on an approximation closely related to the mean field theory of statistical mechanics. Through mean field theory, a complicated large dimensional Gibbs distribution can be approximated by a separable function equal to a product of one dimensional density functions. In essence, this reduction in complexity is achieved by approximating the influence of the neighbors of each pixel over their entire sample space, by their mean field. We examines the bias and variance of this estimator for the problem of image restoration for cases where the true value of the global hyperparameter is known using Monte Carlo simulations.

Several applications of methods presented here are discussed with the focus on the optical flow computation and image reconstruction in positron emission tomography (PET). We have made extensive quantitative studies of the performance of these methods for the problem of MAP PET image reconstruction.

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