On Modified l_1-Minimization Problems in Compressed Sensing
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Sparse signal modeling has received much attention recently because of its application in medical imaging, group testing and radar technology, among others. Compressed sensing, a recently coined term, has showed us, both in theory and practice, that various
signals of interest which are sparse or approximately sparse can be efficiently recovered by using far fewer samples than suggested by Shannon sampling theorem.
Sparsity is the only prior information about an unknown signal assumed in traditional compressed sensing techniques. But in many applications, other kinds of prior information are also available, such as partial knowledge of the support, tree structure
of signal and clustering of large coefficients around a small set of coefficients.
In this thesis, we consider compressed sensing problems with prior information on the support of the signal, together with sparsity. We modify regular l_1 -minimization problems considered in compressed sensing, using this extra information. We call these
modified l_1 -minimization problems.
We show that partial knowledge of the support helps us to weaken sufficient conditions for the recovery of sparse signals using modified ` 1 minimization problems. In case
of deterministic compressed sensing, we show that a sharp condition for sparse recovery
can be improved using modified ` 1 minimization problems. We also derive algebraic necessary and sufficient condition for modified basis pursuit problem and use an open source algorithm known as l_1 -homotopy algorithm to perform some numerical experiments and compare the performance of modified Basis Pursuit Denoising with the regular Basis Pursuit Denoising.