Modified-CS: Modifying compressive sensing for problems with partially known support

Abstract

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.

Iterative hard thresholding for compressed sensing with partially known support

“…Recent works show that modifying the CS framework to include prior knowledge of the support improves the reconstruction results using fewer measurements [6,7]. The a priori information modified CS seeks out a signal that explains the measurements and whose support contains the smallest number of new additions to 0 .…”

“…The a priori information modified CS seeks out a signal that explains the measurements and whose support contains the smallest number of new additions to 0 . Vaswani et al proposed in [6] to modify BP to find an sparse signal assuming uncorrupted measurements. This technique is extended by Jacques in [7] to the case of corrupted measurements and compressible signals.…”

“…Recent works have explored the idea of exploiting prior information about the signal to reduce the number of samples [5][6][7]. Baraniuk et. al exploit a known model for the signal (e.g subspace model or graphical model) and incorporate this model in the reconstruction process achieving accurate reconstructions with fewer samples [5].…”

“…al assume that part of the signal support is known a priori and the problem is recast as finding the unknown support. The remainder of the signal (unknown support) is a sparser signal than the original, thereby requiring fewer samples to yield an accurate reconstruction [6].…”

“…Although the modified CS approach in [6] needs fewer samples to recover a signal, it employs a modified version of basis pursuit (BP) [1] to perform the reconstruction. The computational cost of solving the convex problem posed by BP can be high (or complicated to implement) for large scale problems.…”