Discontinuities in MRF's

Smoothness is a generic assumption underlying a wide range of physical phenomena. It characterizes the coherence and homogeneity of matter within a scope of space (or an interval of time). It is one of the most common assumptions in computer vision models, in particular, those formulated in terms of Markov random fields (MRF’s) (Geman and Geman 1984; Elliott et al. 1984; Marroquin 1985) and regularization (Poggio et al. 85a). Its applications are seen widely in image restoration, surface reconstruction, optical flow and motion, shape from X, texture, edge detection, region segmentation, visual integration, and so on.

This chapter1 presents a systematic study on smoothness priors involving discontinuities. Through an analysis of the Euler equation associated with the energy minimization in MRF and regularization models, it is identified that the fundamental difference among different models for dealing with discontinuities lies in their ways of controlling the interaction between neighboring points. Thereby, an important necessary condition is derived for any regularizers or MRF prior potential functions to be able to deal with discontinuities.