Practical Signal Recovery from Random Projections
نویسندگان
چکیده
Can we recover a signal f ∈ R 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 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. EDICS: 2-REST, 2-RECO, 2-NLIN E. C. is partially supported by a National Science Foundation grant DMS 01-40698 (FRG) and by an Alfred P. Sloan Fellowship. J. R. is supported by National Science Foundation grants DMS 01-40698 and ITR ACI-0204932. E. C. and J. R. are with the Department of Applied and Computational Mathematics, Caltech, MC 217-50, Pasadena, CA 91125. Email: emmanuel,[email protected]. Phone: (626) 395-2291 (EC), (626) 395-5760 (JR). Fax: (626) 578-0124. The material in this paper was presented at IS&T/SPIE’s 17th Annual Symposium on Electronic Imaging. January 25, 2005 DRAFT
منابع مشابه
Sparse signal recovery using sparse random projections
Sparse signal recovery using sparse random projections
متن کاملCompressed Sensing with Very Sparse Gaussian Random Projections
We study the use of very sparse random projections [12, 11] for compressed sensing (sparse signal recovery) when the nonzero coordinates of signals can be either positive or negative. In our setting, the entries of a Gaussian design matrix are randomly sparsified so that only a very small fraction of entries are nonzero. Our proposed decoding algorithm is simple and efficient in that the major ...
متن کاملScalable Inference and Recovery from Compressive Measurements
Despite the apparent need for adaptive, nonlinear techniques for dimensionality reduction, random linear projections have proven to be extremely effective at capturing signal structure in cases where the signal obeys a low-dimensional model. Similarly, random projections are a useful tool for solving problems where the ultimate question of interest about the data requires a small amount of info...
متن کاملAn analysis of blocks sampling strategies in compressed sensing
Compressed sensing is a theory which guarantees the exact recovery of sparse signals from a small number of linear projections. The sampling schemes suggested by current compressed sensing theories are often of little practical relevance since they cannot be implemented on real acquisition systems. In this paper, we study a new random sampling approach that consists in projecting the signal ove...
متن کاملRecovery of Clustered Sparse Signals from Compressive Measurements
We introduce a new signal model, called (K,C)-sparse, to capture K-sparse signals in N dimensions whose nonzero coefficients are contained within at most C clusters, with C < K ≪ N . In contrast to the existing work in the sparse approximation and compressive sensing literature on block sparsity, no prior knowledge of the locations and sizes of the clusters is assumed. We prove that O (K + C lo...
متن کامل