SIAM Journal on Imaging Sciences, Volume 13, Issue 4, Page 2307-2360, January 2020.
Mumford--Shah models are well-established and powerful variational tools for the regularization of noisy data. In the case of images this includes regularizing both the edge set as well as the image values itself. Thus, these models may be used as a basis for a segmentation pipeline or for smoothing the data. In this paper we consider higher order Mumford--Shah functionals which penalize the deviation from piecewise polynomials instead of piecewise constant functions as first order Mumford--Shah functionals do. Minimizing Mumford--Shah functionals, which are nonsmooth and nonconvex functionals, are NP hard problems. Compared with first order Mumford--Shah functionals, numerically solving higher order models is even more challenging, and in contrast to work on more theoretical aspects there are only very few works dealing with the algorithmic side. In this paper, we propose a new algorithmic framework for second order Mumford--Shah regularization. It is based on a proposed reformulation of higher order Mumford--Shah problems in terms of Taylor jets and a corresponding discretization. Using an ADMM approach, we split the discrete jet-based problem into subproblems which we can solve efficiently, noniteratively, and exactly. We derive numerically stable and fast solvers for these subproblems. In summary, we obtain an efficient overall algorithm. Our method requires a priori knowledge on neither the gray or color levels nor the shape of the discontinuity set of a solution. We demonstrate the applicability of the proposed methods in various numerical experiments. In particular, we quantitatively and qualitatively compare the proposed scheme with the algorithms proposed in the literature.

中文翻译：

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基于泰勒射流公式的二阶Mumford-Shah模型算法

SIAM影像科学杂志，第13卷，第4期，第2307-2360页，2020年1月。
Mumford-Shah模型是完善的功能强大的变异工具，可用于对噪声数据进行正则化。对于图像，这包括对边缘集以及图像值本身进行正则化。因此，这些模型可用作分割管线或平滑数据的基础。在本文中，我们考虑了高阶Mumford-Shah泛函，它惩罚了分段多项式的偏差，而不是像一阶Mumford-Shah泛函那样对分段常数进行了惩罚。最小化Mumford-Shah函数，它们是非光滑和非凸函数，是NP难题。与一阶Mumford-Shah泛函相比，对高阶模型进行数值求解更具挑战性，并且与从事更多理论方面的工作相比，处理算法方面的工作很少。在本文中，我们为二阶Mumford-Shah正则化提出了一种新的算法框架。它基于拟议的泰勒射流对高阶Mumford-Shah问题的重新表述以及相应的离散化。使用ADMM方法，我们将基于离散射流的问题分为多个子问题，这些子问题可以有效，非迭代且准确地解决。我们导出这些子问题的数值稳定和快速求解器。总之，我们获得了一种有效的整体算法。我们的方法既不需要有关灰度或颜色级别的知识，也不需要有关解决方案的不连续集的形状的先验知识。我们证明了所提出的方法在各种数值实验中的适用性。特别是，我们在定量和定性上将提出的方案与文献中提出的算法进行比较。