中文摘要：

提出了一种隐式开曲面上多相图像分割的变分模型并设计了相应的对偶方法和Split Bregman算法。任意拓扑结构的开曲面用符号距离函数的零水平集与二值标记函数的交集表达,曲面上的多区域划分用n-1个二值标记函数划分n个区域的方案,借助内蕴梯度、内蕴散度等概念建立了隐式开曲面上多相图像分割的变分模型。在对标记函数交替优化的过程中,通过凸松弛技术将离散标记函数松弛为有界连续函数,从而将原优化问题转化为对不同标记函数的凸优化问题。通过引进对偶变量设计了对标记函数优化的对偶方法,并通过引进辅助变量和Bregman迭代参数设计了对标记函数优化的Split Bregman算法。通过多个数值实验对所提出的模型和算法的正确性及计算效率进行了验证。

英文摘要：

This paper presents a variational model of multiphase image segmentation on implicit open surfaces and designs corresponding dual method and Split Bregman algorithm. The open surface with arbitrary topology is expressed using the in- tersection of zero level set of a continuous signed distance function and a binary label function. The multiple region division on a surface adopts the scheme of using n - 1 binary label functions to partition n regions. Variational model for multiphase image segmentation on implicit open surfaces is established using the concepts of intrinsic gradient and intrinsic diver- gence. In the alternate optimization process of label functions ,the convex relaxation technique is used to relax the discrete label functions as bounded continuous functions, so that the original optimization problem is transformed into the convex optimization problem of different label functions. In order to improve the computation efficiency, the dual method that is used to optimize the label functions is designed via introducing dual variables ;the Split Bregman algorithm that is used to optimize the label functions is designed via introducing auxiliary variables, and Bregman iterative parameters. Various nu- merical experiments validate the correctness and computation efficiency of the proposed model and algorithm.