Numerical Integration of Riemannian Gradient Flows for Image Labeling
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 was proposed  by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. In this paper, we adopt an approach from the literature on uncoupled replicator dynamics and extend it to the geometric labeling flow, that couples the dynamics through Riemannian averaging over spatial neighborhoods. As a result, the gradient flow on the assignment manifold transforms to a flow on a vector space of matrices, such that parallel numerical update schemes can be derived by established numerical integration. A quantitative comparison of various schemes reveals a superior performance of the adaptive scheme originally proposed, regarding both the number of iterations and labeling accuracy.
KeywordsImage labeling Assignment manifold Riemannian gradient flow Replicator equation Multiplicative updates
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