Practical Signal Recovery from Random Projections

Can we recover a signal f 2 R N from a small number of linear measurements? A series of recent papers developed a collection of results showing that it is surprisingly possible to reconstruct certain types of signals accurately from limited measurements. In a nutshell, suppose that f is compressible in the sense that it is well-approximated by a linear combination of M vectors taken from a known basis . Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M logN generic nonadaptive measurements only. The recovery procedure is concrete and consists in solving a simple convex optimization program. In this paper, we show that these ideas are of practical significance. Inspired by theoretical developments, we propose a series of practical recovery procedures and test them on a series of signals and images which are known to be well approximated in wavelet bases. We demonstrate empirically that it is possible to recover an object from about 3M ‐5M projections onto generically chosen vectors with the same accuracy as the ideal M -term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging.