11 March 2005 Signal recovery from random projections
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Abstract
Can we recover a signal f∈RN 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 Ψ. Then not knowing anything in advance about the signal, f can (very nearly) be recovered from about M log N 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 that it is empirically possible to recover an object from about 3M-5M projections onto generically chosen vectors with an accuracy which is as good as that obtained by the ideal M-term wavelet approximation. We briefly discuss possible implications in the areas of data compression and medical imaging.
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Emmanuel J. Candes, Emmanuel J. Candes, Justin K. Romberg, Justin K. Romberg, } "Signal recovery from random projections", Proc. SPIE 5674, Computational Imaging III, (11 March 2005); doi: 10.1117/12.600722; https://doi.org/10.1117/12.600722
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