Image Labeling by Assignment

TL;DR

This paper presents a geometric variational method for image labeling that models the problem on a manifold of stochastic matrices, using Riemannian gradient flows and replicator equations to find optimal label assignments.

Contribution

It introduces a novel geometric framework and a smooth non-convex variational approach for image labeling, utilizing Riemannian gradient flows and efficient sparse interior-point numerics.

Findings

Converges efficiently to global maxima representing optimal labelings.

Provides a smooth non-convex inner approximation to the labeling problem.

Utilizes parallel multiplicative updates for numerical implementation.

Abstract

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.