The Markov random field (MRF) is one of the most widely used models in image processing, constituting a prior model for addressing problems such as image segmentation, object detection, and reconstruction. What is not often appreciated is that the MRF owes its origin to the physics of solids, making it an ideal prior model for processing microscopic observations of materials. While both fields know of their respective interpretations of the MRF, each knows very little about the other’s version of it. Hence, both fields have “blind spots,” where some concepts readily appreciated by one field are completely obscured from the other. With this in mind, the objectives of this article are to 1) develop a synoptic view of the MRF, the related Gibbs distribution, and the Hammersley–Clifford theorem that links them, in such a way that signal processing and materials readers will see them from the same perspective; and 2) explain physics-based regularization using the MRF and describe how it can provide insight into the performance of MRF-based segmentation methods. While the MRF has already been used in many machine learning contexts, we will use a simpler, more transparent method to illustrate the fundamental behavior of the MRF, with the understanding that this behavior should be inherent in more complex learning approaches.

中文翻译：

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材料应用中的马尔可夫随机场：信号处理和材料阅读器的概要视图

马尔可夫随机场 (MRF) 是图像处理中使用最广泛的模型之一，构成了解决图像分割、对象检测和重建等问题的先验模型。不常被理解的是，MRF 起源于固体物理学，使其成为处理材料微观观察的理想先验模型。虽然这两个领域都知道他们各自对 MRF 的解释，但每个领域都对对方的版本知之甚少。因此，这两个领域都有“盲点”，即一个领域容易理解的一些概念与另一个领域完全不同。考虑到这一点，本文的目标是 1) 对 MRF、相关的 Gibbs 分布以及将它们联系起来的 Hammersley-Clifford 定理形成一个概要视图，以这样的方式，信号处理和材料的读者将从相同的角度看待它们；2) 使用 MRF 解释基于物理的正则化，并描述它如何深入了解基于 MRF 的分割方法的性能。虽然 MRF 已经在许多机器学习环境中使用，但我们将使用一种更简单、更透明的方法来说明 MRF 的基本行为，并理解这种行为应该是更复杂的学习方法所固有的。