Chapter 3
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 (MRFs) [Geman and Geman 1984 ; Elliott et al. 1984 ; Marroquin 1985] and regularization [Poggio et al. 1985]. 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.
The assumption of the uniform smoothness implies the smoothness everywhere. However, improper imposition of it can lead to undesirable, oversmoothed, solutions. This occurs when the uniform smoothness is violated, for example, at discontinuities where abrupt changes occur. It is necessary to take care of discontinuities when using smoothness priors. Therefore, how to apply the smoothness constraint while preserving discontinuities has been one of the most active research areas in low level vision (see, e.g. [Blake 1983 ; Terzopoulos 1983b ; Geman and Geman 1984 ; Grimson and Pavlidis 1985 ; Marroquin 1985 ; Mumford and Shah 1985 ; Terzopoulos 1986b ; Blake and Zisserman 1987 ; Lee and Pavlidis 1987 ; Koch 1988 ; Leclerc 1989 ; Shulman and Herve 1989 ; Li 1990b ; Nordstrom 1990 ; Geiger and Girosi 1991 ; Li 1991]).
This chapter presents a systematic study on smoothness priors involving discontinuities. The results are based on an analysis of the Euler equation associated with the energy minimization in MRF and regularization models. Through the analysis, 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.
Based on these findings, a so-called discontinuity adaptive (DA)
smoothness model is defined in terms of the Euler
equation constrained by a class of adaptive
interaction functions (AIFs). The
DA solution is 
continuous, allowing arbitrarily large but bounded
slopes. Because of the continuous nature, it is stable to changes in
parameters and data. This is a good property for regularizing
ill-posed problems. The results provide principles for the selection of
a priori clique potential functions in stochastic MRF models and
regularizers in deterministic regularization models. It is also shown
that the DA model includes as special instances most of the existing
models, such as the line process (LP) model
[Geman and Geman 1984 ; Marroquin 1985], weak string and membrane
[Blake and Zisserman 1987], approximations of the LP model
[Koch et al. 1986 ; Yuille 1987], minimal description length
[Leclerc 1989], biased anisotropic diffusion [Nordstrom 1990], and mean field theory approximation
[Geiger and Girosi 1991].
The study of discontinuities is most sensibly carried out in terms of analytical properties, such as derivatives. For this reason, analytical regularization, a special class of MRF models, is used as the platform for it. If we consider that regularization contains three parts [Boult 1987]: the data, the class of solution functions and the regularizer, the present work addresses mainly the regularizer part. In Section 3.1, regularization models are reviewed in connection to discontinuities. In Section 3.2, a necessary condition for the discontinuity adaptivity is made explicit; based on this, the DA model is defined and compared with other models. In Section 3.3, an algorithm for finding the DA solution is presented, some related issues are discussed and experimental results are shown. Finally, conclusions are drawn.