稀疏加光滑复合信号反问题的连续域表述
Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals
摘要：
我们提出了一种新的框架，假设是一个混合物的两个添加剂成分，一个稀疏和其他光滑，给定有限数量的线性测量的1D复合信号的重建。我们制定的重建问题作为一个连续域正则化的反问题，多个处罚。我们证明，这些处罚诱导重建的信号，确实采取所需的形式的一个稀疏和光滑的组件的总和。然后，我们离散这个问题使用Riesz基地，这产生了一个离散的问题，可以解决标准算法。我们的离散化是准确的，在这个意义上，我们正在解决的连续域问题的搜索空间指定我们的基地没有任何离散化错误。我们提出了一个完整的算法流水线，并证明其可行性的模拟数据。
英文摘要：
We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a continuous-domain regularized inverse problem with multiple penalties. We prove that these penalties induce reconstructed signals that indeed take the desired form of the sum of a sparse and a smooth component. We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved by standard algorithms. Our discretization is exact in the sense that we are solving the continuous-domain problem over the search space specified by our bases without any discretization error. We propose a complete algorithmic pipeline and demonstrate its feasibility on simulated data.
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