  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.
