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A variety of competing algorithms exist for segmentation of both single-channel and multichannel synthetic aperture radar (SAR) images. Among the most successful of these algorithms is the approach presented by Stewart et al. This algorithm defines a cost which is a weighted sum of a likelihood term that estimates the statistical likelihood of the membership of pixels to neighboring segments and a shape term that is intended to provide a smoothing constraint on segment boundaries. The shape term in the original implementation of the Stewart algorithm was rather rudimentary, and in this paper, we explore the performance of a shape term based on Sethian-Osher curvature-flow theory. We demonstrate the performance of the refined curvature-cost (CC) shape term on a set of simulated images as well as an ASAR scene. We assess the segmentation performance using a hybridized shape metric and on the number of segments produced. We find that the CC shape term significantly improves the performance of the Stewart segmentation algorithm, particularly for high-contrast edges. In spite of this success, we argue that further improvements to the algorithm will be difficult due to the architecture of the system.