The Assignment Manifold: A Smooth Model for Image Labeling

Abstract

We introduce a novel geometric approach to the image labeling problem. 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. No tuning parameters are involved, except for two parameters setting the spatial scale of geometric averaging and scaling globally the numerical range of features, respectively. Our geometric variational approach can be implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.