Markov

Markov

Markov Random Fields and Images
by
Patrick P_erez

At the intersection of statistical physics and probability theory, Markov random_elds and Gibbs distributions have emerged in the early eighties as powerful tools for modeling images and coping with high-dimensional inverse problems from low-level vision. Since then, they have been used in many studies from the image processing and computer vision community. A brief and simple introduction to the basics of the domain is proposed.

1. Introduction and general framework
With a seminal paper by Geman and Geman in 1984 [18], powerful tools known for long by physicists [2] and statisticians [3] were brought in a com-prehensive and stimulating way to the knowledge of the image processing and computer vision community. Since then, their theoretical richness, their prac-tical versatility, and a number of fruitful connections with other domains, have resulted in a profusion of studies. These studies deal either with the mod-eling of images (for synthesis, recognition or compression purposes) or with the resolution of various high-dimensional inverse problems from early vision (e.g., restoration, deblurring, classi_cation, segmentation, data fusion, surface reconstruction, optical ow estimation, stereo matching, etc. See collections of examples in [11, 30, 40]).
The implicit assumption behind probabilistic approaches to image analysis is that, for a given problem, there exists a probability distribution that can capture to some extent the variability and the interactions of the di_erent sets of relevant image attributes. Consequently, one considers the variables of the problem as random variables forming a set (or random vector) X = (Xi)ni=1 with joint probability distribution PX 1.
1 PX is actually a probability mass in the case of discrete variables, and a probability density
function when the Xi's are continuously valued. In the latter case, all summations over
states or con_gurations should be replaced by integrals.