SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 248-272, March 2024.
Abstract. The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.

中文翻译：

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使用凸变体 Mumford–Shah 变分模型的贝叶斯推理进行图像分割

SIAM 影像科学杂志，第 17 卷，第 1 期，第 248-272 页，2024 年 3 月。
摘要。Mumford-Shah模型是经典的分割模型，但其目标函数是非凸的。平滑和阈值 (SaT) 方法是 Mumford-Shah 模型的凸变体，它寻求 Mumford-Shah 模型的平滑近似解。SaT方法将分割分为两个阶段：首先，最小化凸能量函数以获得平滑图像；然后，应用阈值技术对平滑图像进行分割。能量函数由三个加权项组成，权重称为正则化参数。选择合适的正则化参数对于获得有效的分割结果至关重要。传统上，正则化参数是通过反复试验来选择的，这是一个非常耗时的过程，并且在实际应用中并不实用。在本文中，我们应用贝叶斯推理方法来推断正则化参数并估计平滑图像。我们从统计角度分析凸变体Mumford-Shah变分模型，然后构建层次贝叶斯模型。使用平均场变分族来近似后验分布。平滑图像的变分密度假设为高斯密度，超参数假设为伽玛变分密度。高斯密度和伽玛密度中的所有参数都被迭代更新。实验结果表明，所提出的方法能够生成高质量的分割结果。尽管所提出的方法包含估计正则化参数的推理步骤，但与以前的方法相比，它需要更少的 CPU 运行时间来获得平滑图像。