An ADMM Approach of a Nonconvex and Nonsmooth Optimization Model for Low-Light or Inhomogeneous Image Segmentation

Abstract

In this paper, we propose a novel nonconvex and nonsmooth optimization model for low-light or inhomogeneous image segmentation which is a hybrid of Mumford–Shah energy functional and Retinex theory. The given image is decomposed into the reflectance component and the illumination component by solving Retinex-based Mumford–Shah model with [Formula: see text] regularizer. Indeed, the existence of the [Formula: see text] regularizer means the nonsmooth term in the model is nonconvex. Thus, it is difficult to solve the proposed model directly. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed nonconvex and nonsmooth model. We apply a novel splitting technique in our algorithm to ensure that all subproblems admit closed-form solutions. Theoretically, we prove that the sequence generated by our proposed algorithm converges to a stationary point under mild conditions. Next, once the reflectance is obtained, the [Formula: see text]-means clustering method is utilized to complete the segmentation. We compare the proposed Retinex-based method with other state-of-the-art segmentation methods under special lighting conditions. Experimental results show that the proposed method has better performance for both gray-scale images and color images efficiently in terms of the quantitative and qualitative results.