Estimation error bounds for denoising by sparse approximation

If a signal is known to have a sparse representation with respect to a given frame, the signal can be estimated from a noise-corrupted observation of the signal by finding the best sparse approximation to the observation. 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 gives bounds to show how inefficiently white Gaussian noise is represented by sparse linear combinations of frame vectors. Combined with knowledge of the approximation efficiency of a given family of frames for a given signal class, this work leads to a better understanding of the merits of frame denoising.