Markov Random Fields, Stochastic Quantization and Image Analysis

Abstract

Markov random fields based on the lattice Z 2 have been extensively used in image analysis in a Bayesian framework as a-priori models for the intensity field and on the dual lattice (Z 2) as models for boundaries. The choice of these models has usually been based on algorithmic considerations in order to exploit the local structure inherent in Markov fields. No fundamental justification has been offered for the use of Markov random fields (see, for example, GEMAN-GEMAN [1984], MARROQUIN-MITTER-POGGIO [1987]). It is well known that there is a one-one correspondence between Markov fields and Gibbs fields on a lattice and the Markov Field is simulated by creating a Markov chain whose invariant measure is precisely the Gibbs measure. There are many ways to perform this simulation and one such way is the celebrated Metropolis Algorithm. This is also the basic idea behind Stochastic Quantization. We thus see that if the use of Markov Random fields in the context of Image Analysis can be given some fundamental justification then there is a remarkable connection between Probabilistic Image Analysis, Statistical Mechanics and Lattice-based Euclidean Quantum Field Theory. We may thus expect ideas of Statistical Mechanics and Euclidean Quantum Field Theory to have a bearing on Image Analysis and in the other direction we may hope that problems in image analysis (especially problems of inference on geometrical structures) may have some influence on statistical physics.