# MathSciDoc: An Archive for Mathematician ∫

#### Optimization and Controlmathscidoc:1705.27004

SIAM Journal on Imaging Sciences, 6, (1), 368-390, 2013
The Mumford–Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two-stage segmentation method based on the Mumford–Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford–Shah model. Once g is obtained, then in the second stage the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle–Pock method. We prove that our method is convergent and that the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any K-phase segmentations by choosing (K − 1) thresholds after g is found in the first stage, and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images.
image segmentation, Mumford–Shah model, split-Bregman, total variation
```@inproceedings{xiaohao2013a,
title={A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding},
author={Xiaohao Cai, Raymond Chan, and Tieyong Zeng},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530133020344201760},
booktitle={SIAM Journal on Imaging Sciences},
volume={6},
number={1},
pages={368-390},
year={2013},
}
```
Xiaohao Cai, Raymond Chan, and Tieyong Zeng. A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding. 2013. Vol. 6. In SIAM Journal on Imaging Sciences. pp.368-390. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530133020344201760.