4.1
What does it mean by discontinuities and outliers? Unfortunately, their definitions are usually ambiguous. What we are certain of is that the likelihood of a discontinuity between a pair of neighboring pixels is related to the difference in pixel labels (such as pixel values) and an outlier is related to the distance between the location of the datum and the estimated value. Where the label difference is very large, there is likely to be a discontinuity between the pixels; and where the datum is very far from the cluster, it is likely an outlier.
A more concrete comparison can be made by analyzing the adaptation (to discontinuities) and the robustness (to outliers) in mathematical terms [Li 1995a]. The adaptation is realized as follows: The interaction between related ( e.g. neighboring) points must be decreased as the violation of the relational bond between them is increased, and prohibited in the limit. This is true of both the MRF and the robust models. We give the necessary condition for such adaptation and then, based on this condition, a definition of a class of adaptive interaction functions for both models. The definition captures the essence of the adaptation ability and is general enough to offer in theory infinitely many suitable choices of such functions.
The problem of discontinuities and outliers also exists in other areas. In model based object recognition, for example, there are two related sub-problems: first, separating the scene into different parts, each being due to a single object; and second, finding feature correspondences between each separate part of the scene and an object. The two sub-problems somewhat have to be solved simultaneously. The process of matching while separating is similar to reconstruction with discontinuities and estimation with outliers. Indeed, matches to one object can be considered as outliers w.r.t. matches to a different object. Different groups of matches should not be constraining to each other. The separation can be done as inserting ``discontinuities'' between different groups. This view can be regarded as a generalization of the weak constraint [Hinton 1978 ; Blake and Zisserman 1987].