A Discretized Newton Flow for Time-Varying Linear Inverse Problems
The reconstruction of a signal from only a few measurements, deconvolving, or denoising are only a few interesting signal processing applications that can be formulated as linear inverse problems. Commonly, one overcomes the ill-posedness of such problems by finding solutions that match some prior assumptions on the signal best. These are often sparsity assumptions as in the theory of Compressive Sensing. In this paper, we propose a method to track the solutions of linear inverse problems, and consider the two conceptually different approaches based on the synthesis and the analysis signal model. We assume that the corresponding solutions vary smoothly over time. A discretized Newton flow allows to incorporate the time varying information for tracking and predicting the subsequent solution. This prediction requires to solve a linear system of equations, which is in general computationally cheaper than solving a new inverse problem. It may also serve as an additional prior that takes the smooth variation of the solutions into account, or as an initial guess for the preceding reconstruction. We exemplify our approach with the reconstruction of a compressively sampled synthetic video sequence.
KeywordsSparse Representation Compressive Sensing Measurement Matrix Blind Signal Synthesis Model
This work has partially been supported by the Cluster of Excellence CoTeSys – Cognition for Technical Systems, funded by the German Research Foundation (DFG).
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