Parallel Constraints Disintegration and Approximation Methods for Image Recovery

从概念上讲，图像恢复问题可以简化为约束最小化问题。然而，在实践中，由于恢复问题的复杂性，标准优化算法的有效实现经常遇到严重的困难，这不仅涉及大量的数据和未知数，而且还涉及各种各样的约束。本研究的目标是开发、分析、实现和测试一种通用的凸最小化算法，该算法解决了图像恢复所带来的特定数值困难。其基本原理是将复杂可行集上的最小化问题分解为两个较大半空间的交集上的一系列更简单的最小化问题。约束束被分解成基本组件，在每次迭代中，半空间通过并行激活近似（线性化）约束块来构造。因此，可以以灵活的方式处理各种约束，此外，由于外推松弛，实现了快速收敛。本研究的三个主要目标是研究所提出的约束图像恢复算法，并严格建立其收敛性在非常一般的条件下，调查有关的数值问题，其最佳实施的高性能计算的背景下，通过广泛的数值测试，以证明其灵活性和数值优势，现有的计划，在广泛的应用图像恢复问题。

Conceptually, an image recovery problem can be reduced to a constrained minimization problem. In practice, however, the efficient implementation of standard optimization algorithms often encounters serious difficulties due to the complex nature of recovery problems, which not only involve a sizable amount of data and unknowns, but also a wide variety of constraints. The goal of this research is to develop, analyze, implement, and test a eeneral convex minimization algorithm that addresses the specific numerical difficulties posed by image recovery. The basic principle is to decompose the original problem of minimizing over a complex feasibility set into a sequence of simpler minimizations over the intersection of two larger half-spaces. The bundle of constraints is disintegrated into elementary components and, at every iteration, the half-spaces are constructed by activating in parallel a block of approximated (linearized) constraints. A wide range of constraints can thus be processed in a flexible manner and, moreover, fast convergence is achieved thanks to extrapolated relaxations. The three major objectives of this research are to study the proposed constrained image recovery algorithm and rigorously establish its convergence under very general conditions; to investigate the numerical issues pertaining to its optimal implementation in the context of high performance computing; to demonstrate through extensive numerical testing its flexibility and numerical superiority over existing schemes in a wide range of applied image recovery problems.