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.

中文翻译：

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离散Mumford-Shah模型的半线性近邻交替最小化。

Mumford-Shah模型是图像分割的标准模型，由于其困难，已提出了许多近似方法。此功能的主要目的是实现联合图像恢复和轮廓检测。在这项工作中，我们提出了Mumford-Shah泛函的离散对应项的一般表述，适用于不平滑的罚分，符合近交线性最小化（PALM）所要求的假设，并具有收敛性保证。第二个贡献旨在放宽对所涉及功能的某些假设，并推导具有收敛性的新型半线性化近邻交替最小化（SL-PAM）算法。我们将算法的性能与几种不平滑的惩罚进行比较，包括高斯和泊松降噪，图像恢复和RGB颜色降噪。我们将结果与最先进的Mumford-Shah函数凸松弛以及Ambrosio-Tortorelli函数的离散版本进行比较。我们表明，SL-PAM算法比原始的PALM算法更快，并导致竞争性的去噪，恢复和分割结果。