Sparse approximation, denoising, and large random frames

Alyson K. Fletcher; Sundeep Rangan; Vivek K. Goyal

doi:10.1117/12.615772

17 September 2005 Sparse approximation, denoising, and large random frames

Alyson K. Fletcher, Sundeep Rangan, Vivek K. Goyal

Proceedings Volume 5914, Wavelets XI; 59140M (2005) https://doi.org/10.1117/12.615772
Event: Optics and Photonics 2005, 2005, San Diego, California, United States

Abstract

If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. The ability to remove noise in this manner depends on the frame being designed to efficiently represent the signal while it inefficiently represents the noise. This paper analyzes the mean squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal. Analyses are for dictionaries generated randomly according to a spherically-symmetric distribution. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. In the limit of large dimension, these approximations have simple forms. The asymptotic expressions reveal a critical input signal-to-noise ratio (SNR) for signal recovery.

Citation Download Citation

Alyson K. Fletcher, Sundeep Rangan, and Vivek K. Goyal "Sparse approximation, denoising, and large random frames", Proc. SPIE 5914, Wavelets XI, 59140M (17 September 2005); https://doi.org/10.1117/12.615772

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