迭代阈值化与自由间断问题相遇
Iterative thresholding meets free discontinuity problems
AI总结：
本文探索了稀疏恢复中的迭代阈值算法与离散自由间断问题之间的联系，并建立了一种用于求解离散自由间断问题的迭代阈值算法，提供了关于极小化解性质的新见解。
摘要：
自由间断问题描述的是，感兴趣的解由一个函数和一个由该函数的不连续点组成的低维集合定义。因此，解的导数在几乎所有地方都被假设为“小”函数，除了在某些集合上它作为奇异测度集中。例如，这在断裂力学的裂纹检测或某些数字图像分割问题中就是如此。如果为了数值目的离散化这些情况，离散设置下的自由间断问题可以重新表述为寻找一个导数向量，该向量除了少数超过某个阈值的条目外，所有分量都很小。这个问题类似于“稀疏恢复”领域中遇到的问题，其中在绝对值上具有少量主导分量的向量通过最小化相关能量泛函从少数给定的线性测量中恢复。已经设计了几种迭代阈值算法，将梯度型迭代与阈值步骤交织在一起，以恢复此设置中的稀疏解。自然想知道这些算法是否和/或如何用于解决离散自由间断问题。本文探讨了这种联系，并通过为离散自由间断问题建立一种迭代阈值算法，提供了关于其极小化解性质的新见解。
英文摘要：
Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function. Hence, the derivative of the solution is assumed to be a `small' function almost everywhere except on sets where it concentrates as a singular measure. This is the case, for instance, in crack detection from fracture mechanics or in certain digital image segmentation problems. If we discretize such situations for numerical purposes, the free-discontinuity problem in the discrete setting can be re-formulated as that of finding a derivative vector with small components at all but a few entries that exceed a certain threshold. This problem is similar to those encountered in the field of `sparse recovery', where vectors with a small number of dominating components in absolute value are recovered from a few given linear measurements via the minimization of related energy functionals. Several iterative thresholding algorithms that intertwine gradient-type iterations with thresholding steps have been designed to recover sparse solutions in this setting. It is natural to wonder if and/or how such algorithms can be used towards solving discrete free-discontinuity problems. The current paper explores this connection, and, by establishing an iterative thresholding algorithm for discrete free-discontinuity problems, provides new insights on properties of minimizing solutions thereof.
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