Bayesian Rationale for the Variational Formulation

Abstract

One of the primary goals of low-level vision is to segment the domain D of an image I into the parts D _i on which distinct surface patches, belonging to distinct objects in the scene, are visible, Although this sometimes requires high level knowledge about the shape and surface appearance of various classes of objects, there are many low-level clues about the appearance of the individual surface patches and the boundaries between them. For example, the surface patches usually have characteristic albedo patterns, textures, on them, and these textures often change sharply as you cross a boundary between two patches. Therefore, one approach to the segmentation problem has been to try to merge all the low-level clues for splitting and merging different parts of the domain D and come up with probability measures p({D _i }) of how likely a given segmentation {D _i } is on the basis of all available low-level information, and what is the most likely segmentation. Alternately, one sets E({D _i }) = − log(p({Di})), which one calls the ‘energy’ of the segmentation, and seeks the segmentation with the minimum energy, In general, these models have two parts: a prior model of possible scene segmentations, possibly including variables to describe other scene structures that are relevant (e.g. depth relationships), and a data model of what images are consistent with this prior model of the scene. If we write w for the variables used to describe the scene, e.g. the subsets D _i or the set of all their boundary points Γ, then the prior model is some probability space (Ω _w,d p), where Ω _w,d , is the set of all possible values of w. The model is specified by giving the probability distribution p(w) on all these values. The data model is a larger probability space (Ω _w,d , p), where Ω _w,d is the set of all possible values of w and of all possible observed images I.