Non-unique solutions for a convex TV - L-1 problem in image segmentation

Abstract

One important task in image segmentation is to find a region of interest, which is, in general, a solution of a nonlinear and nonconvex problem. The authors of Chan and Esedoglu (Aspects of total variation regularized (Formula presented.) function approximation. SIAM J. Appl. Math. 2005;65:1817-1837) proposed a convex (Formula presented.) problem for finding such a region Σ and proved that when a binary input f is given, a solution (Formula presented.) to the convex problem gives rise to other solutions (Formula presented.) for a.e. (Formula presented.) from which they raised a question of whether or not (Formula presented.) must be binary. The same is to ask if the two-phase Mumford-Shah model in image processing has a unique solution with a binary input. In this paper, we will discuss how to construct a non-binary solution that provides a negative answer to the question through a connection of two ideas, one from the two-phase Mumford-Shah model in image segmentation and the other from mean curvature motions discussed in some geometric problems (e.g. Alter, Caselles, Chambolle. A characterization of convex calibrable sets in (Formula presented.). Math. Ann. 2005;322:329-366; Chambolle. An algorithm for mean curvature motion. Interfaces Free Boundaries 2004;6:195-218) revealing the nature of non-uniqueness in image segmentation.