  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.
