A Geometric Approach to Image Labeling

Abstract

We introduce a smooth non-convex approach in a novel geometric framework which complements established convex and non-convex approaches to image labeling. The major underlying concept is a smooth manifold of probabilistic assignments of a prespecified set of prior data (the “labels”) to given image data. The Riemannian gradient flow with respect to a corresponding objective function evolves on the manifold and terminates, for any $\delta > 0$ δ > 0 , within a $\delta $ δ -neighborhood of an unique assignment (labeling). As a consequence, unlike with convex outer relaxation approaches to (non-submodular) image labeling problems, no post-processing step is needed for the rounding of fractional solutions. Our approach is numerically implemented with sparse, highly-parallel interior-point updates that efficiently converge, largely independent from the number of labels. Experiments with noisy labeling and inpainting problems demonstrate competitive performance.