Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Fast Global Minimization of the Active Contour/Snake Model
Journal of Mathematical Imaging and Vision
Efficient MRF deformation model for non-rigid image matching
Computer Vision and Image Understanding
A Convex Formulation of Continuous Multi-label Problems
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part III
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Convex Multi-class Image Labeling by Simplex-Constrained Total Variation
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Graph cut optimization for the Mumford-Shah model
VIIP '07 The Seventh IASTED International Conference on Visualization, Imaging and Image Processing
Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction
Journal of Scientific Computing
Convex relaxation for multilabel problems with product label spaces
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach
International Journal of Computer Vision
SIAM Journal on Imaging Sciences
Global Solutions of Variational Models with Convex Regularization
SIAM Journal on Imaging Sciences
Exact optimization for Markov random fields with convex priors
IEEE Transactions on Pattern Analysis and Machine Intelligence
IEEE Transactions on Image Processing
A binary level set model and some applications to Mumford-Shah image segmentation
IEEE Transactions on Image Processing
Coupling Image Restoration and Segmentation: A Generalized Linear Model/Bregman Perspective
International Journal of Computer Vision
Computer Vision and Image Understanding
Information Sciences: an International Journal
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The active contours without edges model of Chan and Vese (IEEE Transactions on Image Processing 10(2):266---277, 2001) 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 Journal of Applied Mathematics 66(5):1632---1648, 2006), Chan, Esedo驴lu, and 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 by Goldstein et al. (UCLA CAM Report 09-77, 2009) and Brown et al. (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 by Chan et al. (SIAM Journal of Applied Mathematics 66(5):1632---1648, 2006), whose quality of solutions depends on the choice of the initial condition.