SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A1984-A2013, January 2020.
A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of total variation. On the other hand, sparse or noisy data often demand a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori objective function of a suitably chosen Bayesian phase-field approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess a finite perimeter and to have the ability to learn about the true perimeter.

中文翻译：

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调和贝叶斯和周长正则化的二进制反演

SIAM科学计算杂志，第42卷，第4期，第A1984-A2013页，2020年1月。
从观测数据中重建不连续函数的经典算法的中心主题是通过使用总变化量进行周长正则化。另一方面，稀疏或嘈杂的数据通常需要采用概率方法来重建图像，以实现不确定性量化；贝叶斯反演方法本身引入了正则化形式，是执行此操作的自然框架。本文探讨了贝叶斯反演方法与周边正则化之间的联系。在本文中两个链接进行了研究：（i）所述最大适当选择的贝叶斯相场方法的后验的目标函数被示出为密切相关的最小二乘加周长正规化目标;