Abstract
If a signal x is known to have a sparse representation with respect to a frame, it can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. Removing noise in this manner depends on the frame efficiently representing the signal while it inefficiently represents the noise. The mean-squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal are analyzed. First an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary is given. Further analyses are for dictionaries generated randomly according to a spherically-symmetric distribution and signals expressible with single dictionary elements. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. Asymptotic expressions reveal a critical input signal-to-noise ratio for signal recovery.
Highlights
Estimating a signal from a noise-corrupted observation of the signal is a recurring task in science and engineering
Before addressing the denoising performance of sparse approximation, we give an approximation result for Gaussian signals. This result is a lower bound on the mean-squared error (MSE) when sparsely approximating a Gaussian signal; it is the basis for an upper bound on the MSE for denoising when the signal-to-noise ratio (SNR) is low
This paper has addressed properties of denoising by sparse approximation that are geometric in that the signal model is membership in a specified union of subspaces, without a probability density on that set
Summary
Estimating a signal from a noise-corrupted observation of the signal is a recurring task in science and engineering. This paper explores the limits of estimation performance in the case where the only a priori structure on the signal x ∈ RN is that it has known sparsity K with respect to a given set of vectors Φ = {φi}Mi=1 ⊂ RN. This is a restrictive model, even if there is some approximation error in (1). This paper presents progress in explaining the value of a sparsity model for signal denoising as a function of (N, M, K)
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