### 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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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Editors | M. Papadakis, A.F. Laine, M.A. Unser |

Pages | 1-10 |

Number of pages | 10 |

Volume | 5914 |

DOIs | |

State | Published - 2005 |

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

### Other

Other | Wavelets XI |
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Country | United States |

City | San Diego, CA |

Period | 7/31/05 → 8/3/05 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 5914, pp. 1-10). [59140M] https://doi.org/10.1117/12.615772

**Sparse approximation, denoising, and large random frames.** / Fletcher, Alyson K.; Rangan, Sundeep; Goyal, Vivek K.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 5914, 59140M, pp. 1-10, Wavelets XI, San Diego, CA, United States, 7/31/05. https://doi.org/10.1117/12.615772

}

TY - GEN

T1 - Sparse approximation, denoising, and large random frames

AU - Fletcher, Alyson K.

AU - Rangan, Sundeep

AU - Goyal, Vivek K.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

KW - Dictionary-based representations

KW - Estimation

KW - Isotropic random matrices

KW - Nonlinear approximation

KW - Stable signal recovery

KW - Subspace fitting

UR - http://www.scopus.com/inward/record.url?scp=30844457729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30844457729&partnerID=8YFLogxK

U2 - 10.1117/12.615772

DO - 10.1117/12.615772

M3 - Conference contribution

AN - SCOPUS:30844457729

VL - 5914

SP - 1

EP - 10

BT - Proceedings of SPIE - The International Society for Optical Engineering

A2 - Papadakis, M.

A2 - Laine, A.F.

A2 - Unser, M.A.

ER -