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.

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.

We study the problem of sparse reconstruction from noisy undersampled measurements when the following knowledge is available. (1) We are given partial, and partly erroneous, knowledge of the signal's support, denoted by T . (2) We are also given an erroneous estimate of the signal values on T , denoted by (μ) T . In practice, both of these may be available from prior knowledge. Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as T and (μ) T . In this work, we introduce regularized modified-BPDN (reg-mod-BPDN) to solve this problem and obtain computable bounds on its reconstruction error. Reg-mod-BPDN tries to find the signal that is sparsest outside the set T , while being "close enough" to (μ) T on T and while satisfying the data constraint. Corresponding results for modified-BPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided.

Abstract-In this work, we study the problem of reconstructing a sparse signal from a limited number of linear 'incoherent' noisy measurements, when a part of its support is known. The known part of the support may be available from prior knowledge or from the previous time instant (in applications requiring recursive reconstruction of a time sequence of sparse signals, e.g. dynamic MRI). We study a modification of Basis Pursuit Denoising (BPDN) and bound its reconstruction error. A key feature of our work is that the bounds that we obtain are computable. Hence, we are able to use Monte Carlo to study their average behavior as the size of the unknown support increases. We also demonstrate that when the unknown support size is small, modified-BPDN bounds are much tighter than those for BPDN, and hold under much weaker sufficient conditions (require fewer measurements).

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