Completely Convex Formulation of the Chan-Vese Image Segmentation Model
Completely Convex Formulation of the Chan-Vese Image Segmentation Model
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DOI：
10.1007/s11263-011-0499-y
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发表时间：
2012-05-01
影响因子：
19.5
通讯作者：
Bresson, Xavier
中科院分区：
文献类型：
--
作者：
Brown, Ethan S.;Chan, Tony F.;Bresson, Xavier
Chan和Vese的无边缘活动轮廓模型（IEEE Transactions on Image Processing 10（2）：266-277，2001）是一种基于分段常数Mumford-Shah模型计算将图像分割成两个阶段的流行方法。最小化问题是非凸的，即使最佳区域常数是已知的先验。在（SIAM Journal of Applied Mathematics 66（5）：1632-1648，2006）中，Chan、Esedoalu和Nikolova通过显示可以从凸松弛获得解来提供计算全局极小值的方法。在本文中，我们提出了一个凸松弛方法来解决的情况下，分割和最佳常数是未知的两个阶段和多个阶段。换句话说，我们提出了一个凸松弛流行的K-means算法。我们的方法基于Goldstein等人（UCLA CAM Report 09-77，2009）和Brown等人（UCLA CAM Report 10-43，2010）开发的向量值松弛技术。这个想法是考虑的最佳常数的功能受到约束的梯度。虽然建议的松弛技术不能保证找到原问题的精确全局极小值，我们的实验表明，我们的方法计算最优解的紧密近似。特别地，我们提供了数值例子，在这些例子中，我们的方法找到了比Chan等人提出的方法更好的解（SIAM Journal of Applied Mathematics 66（5）：1632-1648，2006），其解的质量取决于初始条件的选择。
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, Esedoalu, 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.