Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling.
翻译:反面的问题包括从收集的噪音测量数据中恢复信号,这些问题往往可以作为可行性问题出现;然而,为确保数据扰动方面的准确和稳定的恢复,通常需要增加正规化;人工选择分析性规范化可以产生理想的理论保证,但这类方法由于无法利用大量可用数据,在恢复信号方面效果有限;为此,这项工作以理论上合理的方式结合了数据驱动的正规化和混凝土可行性;这项工作使用基于可行性的固定点网络(F-FPNs)完成。每个F-FPN都定义了非爆炸性操作员的集合,每个操作员都是基于预测的操作员和数据驱动的正规化操作员组成。固定点用于计算这些操作员的固定点,操作员的权重也作了调整,以便固定点能够密切代表现有数据。数字实例表明,与基于电视的标准CT重建恢复方法相比,F-F-FPNs的性能提高,而一个基于算法解动的可比较的神经网络。