Completely convex formulation of the Chan-Vese image segmentation model.

*(English)*Zbl 1254.68271Summary: The active contours without edges model of T. F. Chan and L. A. Vese [IEEE Trans. Image Process. 10, No. 2, 266–277 (2001; Zbl 1039.68779)] is a popular method for computing the segmentation of an image into two phases, based on the piecewise constant Mumford-Shah model. The minimization problem is non-convex even when the optimal region constants are known a priori. In [SIAM J. Appl. Math. 66, No. 5, 1632–1648 (2006; Zbl 1117.94002)], the second author, S. Esedoḡlu and M. Nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation. In this paper, we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases. In other words, we propose a convex relaxation of the popular \(K\)-means algorithm. Our approach is based on the vector-valued relaxation technique developed in [{T. Goldstein}, the third author and S. Osher, Geometric applications of the split Bergmann method: segmentation and surface reconstruction. UCLA CAM Report 09-77 (2009)] and [the authors, A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Shah segmentation. UCLA CAM Report 10-43 (2010)]. The idea is to consider the optimal constants as functions subject to a constraint on their gradient. Although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem, our experiments show that our method computes tight approximations of the optimal solutions. Particularly, we provide numerical examples in which our method finds better solutions than the method proposed in [the second author et al., loc. cit.], whose quality of solutions depends on the choice of the initial condition.

##### Keywords:

image segmentation; Chan-Vese model; convex relaxation; level set method; vector-valued functional lifting; \(K\)-means
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\textit{E. S. Brown} et al., Int. J. Comput. Vis. 98, No. 1, 103--121 (2012; Zbl 1254.68271)

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