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

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. This 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 ldquoknownrdquo part of the support. The idea of our solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and whose support contains the smallest number of new additions to the known support. We obtain sufficient conditions for exact reconstruction using modified-CS. These turn out to be much weaker than those needed for CS, particularly when the known part of the support is large compared to the unknown part.