Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support

TL;DR

This paper introduces Modified-CS, a new approach for reconstructing sparse signals when part of the support is known but may contain errors, improving conditions for exact recovery over traditional compressive sensing.

Contribution

The paper proposes Modified-CS and its extension RegModCS, which leverage partial support knowledge to enhance sparse signal reconstruction, with weaker conditions than standard CS.

Findings

Modified-CS achieves exact reconstruction under weaker conditions.

RegModCS incorporates prior signal estimates for improved accuracy.

Simulation results demonstrate superior performance on sparse and compressible signals.

Abstract

We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The ``known" part of the support, denoted T, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the ``known" part. The idea of our proposed solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T. We obtain sufficient conditions for exact reconstruction using modified-CS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called Regularized Modified-CS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown.