### 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.

Original language | English (US) |
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Article number | 26318 |

Journal | Eurasip Journal on Applied Signal Processing |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Hardware and Architecture
- Signal Processing

### Cite this

*Eurasip Journal on Applied Signal Processing*,

*2006*, [26318]. https://doi.org/10.1155/ASP/2006/26318

**Denoising by sparse approximation : Error bounds based on rate-distortion theory.** / Fletcher, Alyson K.; Rangan, Sundeep; Goyal, Vivek K.; Ramchandran, Kannan.

Research output: Contribution to journal › Article

*Eurasip Journal on Applied Signal Processing*, vol. 2006, 26318. https://doi.org/10.1155/ASP/2006/26318

}

TY - JOUR

T1 - Denoising by sparse approximation

T2 - Error bounds based on rate-distortion theory

AU - Fletcher, Alyson K.

AU - Rangan, Sundeep

AU - Goyal, Vivek K.

AU - Ramchandran, Kannan

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33846796161&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846796161&partnerID=8YFLogxK

U2 - 10.1155/ASP/2006/26318

DO - 10.1155/ASP/2006/26318

M3 - Article

VL - 2006

JO - Eurasip Journal on Advances in Signal Processing

JF - Eurasip Journal on Advances in Signal Processing

SN - 1687-6172

M1 - 26318

ER -