Fast Reconstruction of Piecewise Smooth Signals from Incoherent Projections

The Compressed Sensing framework aims to recover a sparse signal from a small set of projections onto random vectors; the problem reduces to searching for a sparse approximation of this measurement vector. Conventional solutions involve linear programming or greedy algorithms and can be computationally expensive. These techniques are generic, however, and assume no structure in the signal aside from sparsity. In this paper, we design an algorithm that enables fast recovery of piecewise smooth signals, sparse signals that have a distinct “connected tree” structure in the wavelet domain. Our Tree Matching Pursuit (TMP) algorithm significantly reduces the search space of the traditional Matching Pursuit greedy algorithm, resulting in a substantial decrease in computational complexity for recovering piecewise smooth signals. An additional advantage of TMP is that it performs an implicit regularization to combat noise in the reconstruction. TMP also applies to the more general case of “incoherent” measurement vectors.