Approximate maximum likelihood hyperparameter estimation for Gibbs priors

Zhenyu Zhou, Richard M. Leahy, Jinyi Qi

Research output: Contribution to journalArticlepeer-review

102 Scopus citations


The parameters of the prior, the hyperparameters, play an important role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, β, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of β from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of β from incomplete data is intractable for most image processing. Here we present an approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate. The algorithm is based on a mean field approximation technique through which multidimensional Gibbs distributions are approximated by a separable function equal to a product of one-dimensional (1-D) densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to optimize the approximation for a restricted class of problems. We present the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration.

Original languageEnglish (US)
Pages (from-to)844-861
Number of pages18
JournalIEEE Transactions on Image Processing
Issue number6
StatePublished - 1997
Externally publishedYes

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Computer Graphics and Computer-Aided Design
  • Software
  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Computer Vision and Pattern Recognition


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