A Variational Perspective on the Assignment Flow

Fabrizio Savarino, Christoph Schnörr

Abstract: The image labeling problem can be described as assigning to each pixel a single element from a finite set of predefined labels. Recently, a smooth geometric approach for inferring such label assignments was proposed by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. Due to the specific Riemannian structure, this results in a coupled replicator dynamic incorporating local spatial geometric averages of lifted data-dependent distances. However, in this framework an approximation of the flow is necessary in order to arrive at explicit formulas. We propose an alternative variational model, where lifting and averaging are decoupled in the objective function so as to stay closer to established approaches and at the same time preserve the main ingredients of the original approach: the overall smooth geometric setting and regularization through geometric local averages. As a consequence the resulting flow is explicitly given, without the need for any approximation. Furthermore, there exists an interesting connection to graphical models.