While the majority of competitive image segmentation methods are based on energy minimization, only few allow to efficiently determine globally optimal solutions. A graph-theoretic algorithm for finding globally optimal segmentations is given by the Minimum Ratio Cycles, first applied to segmentation in [8]. In this paper we show that the class of image segmentation problems solvable by Minimum Ratio Cycles is significantly larger than previously considered. In particular, they allow for the introduction of higher-order regularity of the region boundary. The key idea is to introduce an extended graph representation, where each node of the graph represents an image pixel as well as the orientation of the incoming line segment. With each graph edge representing a pair of adjacent line segments, edge weights can depend on the curvature. This way arbitrary positive functions of curvature can be introduced into globally optimal segmentation by Minimum Ratio Cycles. In numerous experiments we demonstrate that compared to length-regularity the integration of curvature-regularity will drastically improve segmentation results. Moreover, we show an interesting relation to the Snakes functional: Minimum Ratio Cycles provide a way to find one of the few cases where the Snakes functional has a meaningful global minimum.
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