### 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.

Original language | English (US) |
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Article number | 59140M |

Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Proceedings of SPIE - The International Society for Optical Engineering |

Volume | 5914 |

DOIs | |

State | Published - Dec 1 2005 |

Event | Wavelets XI - San Diego, CA, United States Duration: Jul 31 2005 → Aug 3 2005 |

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### Keywords

- Dictionary-based representations
- Estimation
- Isotropic random matrices
- Nonlinear approximation
- Stable signal recovery
- Subspace fitting

### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
- Computer Science Applications
- Applied Mathematics
- Electrical and Electronic Engineering

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*,

*5914*, 1-10. [59140M]. https://doi.org/10.1117/12.615772