Fast Reconstruction from Random Incoherent Projections

The Compressed Sensing paradigm consists of recovering signals that can be sparsely represented in a given basis from a small set of projections into random vectors; the problem is typically solved using an adapted Basis Pursuit algorithm. We show that the recovery of the signal is equivalent to determining the sparsest representation of the measurement vector using the dictionary obtained by applying the projections to the basis elements, and therefore more efficient algorithms can be used during recovery; we explore the Matching Pursuit and Orthogonal Matching Pursuit algorithms. We also design an algorithm that allows for even faster recovery for piecewise smooth signals: the algorithm exploits the tree structure of the sparse coefficients of the signal in a wavelet representation to select subsets of coefficients to estimate at each iteration. We define a class of signals for which such a Tree Matching Pursuit algorithm performs successful recovery and present variations of the algorithms for different classes of signals. Other applications of TMP include effective denoising, fast approximation in overcomplete wavelet bases, and distributed compression.