IEEE Trans Image Process. 2020;29(1):2176-2189. doi: 10.1109/TIP.2019.2944561. Epub 2019 Oct 7.
The Mumford-Shah model is a standard model in image segmentation, and due to its difficulty, many approximations have been proposed. The major interest of this functional is to enable joint image restoration and contour detection. In this work, we propose a general formulation of the discrete counterpart of the Mumford-Shah functional, adapted to nonsmooth penalizations, fitting the assumptions required by the Proximal Alternating Linearized Minimization (PALM), with convergence guarantees. A second contribution aims to relax some assumptions on the involved functionals and derive a novel Semi-Linearized Proximal Alternated Minimization (SL-PAM) algorithm, with proved convergence. We compare the performances of the algorithm with several nonsmooth penalizations, for Gaussian and Poisson denoising, image restoration and RGB-color denoising. We compare the results with state-of-the-art convex relaxations of the Mumford-Shah functional, and a discrete version of the Ambrosio-Tortorelli functional. We show that the SL-PAM algorithm is faster than the original PALM algorithm, and leads to competitive denoising, restoration and segmentation results.
Mumford-Shah 模型是图像分割的标准模型，由于其难度，已经提出了许多近似方法。该函数的主要兴趣在于能够联合进行图像恢复和轮廓检测。在这项工作中，我们提出了 Mumford-Shah 函数的离散对应物的一般形式，适用于非平滑惩罚，符合 Proximal Alternating Linearized Minimization（PALM）所需的假设，并保证收敛性。第二个贡献旨在放宽对所涉及函数的一些假设，并推导出一种新的半线性化 Proximal Alternated Minimization（SL-PAM）算法，具有已证明的收敛性。我们将算法的性能与几种非平滑惩罚方法进行了比较，用于高斯和泊松去噪、图像恢复和 RGB 颜色去噪。我们将结果与 Mumford-Shah 函数的最先进的凸松弛方法以及 Ambrosio-Tortorelli 函数的离散版本进行了比较。我们表明，SL-PAM 算法比原始 PALM 算法更快，并导致具有竞争力的去噪、恢复和分割结果。
