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, 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. 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. Index Terms—Compressive sensing, modified-CS, partially known support, prior knowledge, sparse reconstruction.

We study recovery conditions of weighted 1 minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50 % of the (partial) support information is accurate, then weighted 1 minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard 1 minimization. Moreover, weighted 1 minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals. Index Terms Compressed sensing, weighted 1 minimization, adaptive recovery. I.

Abstract — In this work, we show the “stability ” of two of our recently proposed algorithms, LS-CS-residual (LS-CS) and the noisy version of modified-CS, designed for recursive reconstruction of sparse signal sequences from noisy measurements. By “stability ” we mean that the number of misses from the current support estimate and the number of extras in it remain bounded by a time-invariant value at all times. The concept is meaningful only if the bound is small compared to the current signal support size. A direct corollary is that the reconstruction errors are also bounded by a time-invariant value. I.