Joint Reconstruction and Low-Rank Decomposition for Dynamic Inverse Problems

A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work, we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for three simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.

动态逆问题的一个主要关注点是从外部测量中识别系统潜在的时间行为。在这项工作中，我们考虑这样一种情况：目标可以通过空间和时间基函数的分解来表示，因此可以通过低秩分解有效地表示。然后，我们提出一种基于非负矩阵分解的联合重建和低秩分解方法，以便从高度欠采样的动态测量数据中获取未知量。所提出的框架允许灵活地为空间和时间特征纳入单独的正则化项。对于平稳算子的特殊情况，我们可以有效地利用分解来降低计算复杂度并大幅提高速度。我们针对三个模拟体模对所提出的方法进行了评估，并将所得结果与基于广泛使用的主成分分析的单独的低秩重建及后续分解方法进行了比较。