随着信息科学和图像处理技术的发展，基于Mumford-Shah(MS)模型的分割问题研究已成为分割领域内的一个重要研究课题。本项目针对MS型分割模型中存在的缺陷和数值困难，利用图像的区域信息、边缘信息、统计特征和先验信息，结合分块常数区域内的梯度信息、适当的光滑逼近函数和聚类分析方法，提出有效的分块常数区域和分块光滑(非齐次)区域分割模型，并综合利用数值优化和偏微分方程数值解理论，改进水平集方法和图割方法；通过建立适当的约束空间和预处理技术，改进凸松弛方法和极大流/极小割方法，并对提出的算法进行稳定性和收敛性分析。本项目的完成将为MS型分割模型的课题研究提供更有效的分割模型和数值算法，为图像分割问题的实际应用提供更坚实的理论保障和技术支撑。

With the development of information science and image processing technology, the segmentation problem based on the Mumford-Shah(MS) model becomes one of the most important research topics in this field. However, many challenging issues on mathematical understanding and numerical implementations are still well worthy of deep investigation. This project aims to surmount some of the obstacles for Mumford-Shah-type models. More precisely, we extract some information of regions and edges, statistic structures and a priori information of the image, and integrate the gradient information in the piecewise regions with the clustering techniques to develop efficient segmentation models for piecewise constant images and piecewise smooth (inhomogeneous) image segmentation. We search for new level set method and the graph cut method by using the related theory of numerical solution of partial differential equation and numerical optimization methods. We improve the convex relaxation method and max-flow/min-cut method based on proposing some suitable constrained spaces and preconditioning methods. We also attempt to analyze the stability and the convergence of proposed algorithms. The success of the project will lead to novel effective segmentation models and numerical methods. It will also provide more theoretical and technical foundation for the practical application of image segmentation problem.