Signals and images with discontinuities appear in many problems in such diverse areas as biology, medicine, mechanics and electrical engineering. The concrete data are often discrete, indirect and noisy measurements of some quantities describing the signal under consideration. A frequent task is to find the segments of the signal or image which corresponds to finding the discontinuities or jumps in the data. Methods based on minimizing the piecewise constant Mumford–Shah functional—whose discretized version is known as Potts energy—are advantageous in this scenario, in particular, in connection with segmentation. However, due to their non-convexity, minimization of such energies is challenging. In this paper, we propose a new iterative minimization strategy for the multivariate Potts energy dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments.

具有不连续性的信号和图像出现在生物学，医学，力学和电气工程等各个领域的许多问题中。具体数据通常是描述所考虑信号的一些量的离散，间接和噪声测量。常见的任务是找到信号或图像的片段，这些片段对应于发现数据中的不连续性或跳跃。在这种情况下，尤其是在分割方面，基于最小化分段常数Mumford-Shah函数（其离散化版本称为Potts能量）的方法在此情况下是有利的。然而，由于它们的非凸性，使这种能量的最小化具有挑战性。在本文中，我们为多元Potts能量提出了一种新的迭代最小化策略，用于处理间接噪声测量。