3.1
In MRF vision modeling, the smoothness assumption can be encoded into an energy via one of the two routes: analytic and probabilistic. In the analytic route, the encoding is done in the regularization framework [Poggio et al. 1985 ; Bertero 1988]. From the regularization viewpoint, a problem is said to be ``ill-posed'' if it fails to satisfy one or more of the following criteria: the solution exists, is unique and depends continuously on the data. Additional, a priori, assumptions have to be imposed on the solution to convert an ill-posed problem into a well-posed one. An important assumption of such assumptions is the smoothness [Tikhonov and Arsenin 1977]. It is incorporated into the energy function whereby the cost of the solution is defined.
From the probabilistic viewpoint, a regularized solution corresponds to the maximum a posteriori (MAP) estimate of an MRF [Geman and Geman 1984 ; Marroquin et al. 1987]. Here, the prior constraints are encoded into the a priori MRF probability distribution. The MAP solution is obtained by maximizing the posterior probability or equivalently minimizing the corresponding energy.
The MRF model is more general than the regularization model in (1) that it can encode prior constraints other than the smoothness and (2) that it allows arbitrary neighborhood systems other than the nearest ones. However, the analytic regularization model provides a convenient platform for the study of smoothness priors because of close relationships between the smoothness and the analytical continuity.