Exploiting Prior Knowledge in the Recovery of Non-Sparse Signals from Noisy Random Projections

摘要

The past several years have seen the emergence of the new field of compressive sensing. Given a T sparse signal x ∈ R{sup}K on some basis Ψ = [ψ{sub}1,ψ{sub}2,...,ψ{sub}K], such that x can be approximated by a linear combination of T vectors from Ψ with T K, the theory of compressed sensing [1], [2] shows that x can be recovered from M random projections with high probability when M = CT log K K, where C ≥ 1 is the oversampling factor. The projections are given by y = Φx, where Φ is an M × K random measurement matrix with its rows incoherent with the columns of Ψ. Extensions to the case of noisy projections have appeared recently in the literature (see [3] and references therein). Most of the work in compressed sensing focuses in deterministic signals, and only the sparsity of the signal is exploited in the recovery process. However, in many occasions, additional prior knowledge is available. This paper illustrates that exploiting the source statistics in the recovery process results in significant performance gains, even if the signal is reconstructed in a basis in which it does not admit a sparse representation. Successful recovery will depend on the capability of exploiting all available a priori information in the basis where reconstruction is performed. The proposed framework is similar to joint source-channel coding schemes in digital communications, but applies on the analog domain.