Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory
If a signal is known to have a sparse representation with respect to a frame, it can be estimated from a noise-corrupted observation by finding the best sparse approximation to . Removing noise in this manner depends on the frame efficiently representing the signal while it inefficiently represents...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Publishing Corporation
2011
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| Online Access: | http://hdl.handle.net/1721.1/67336 |
| _version_ | 1826213416488927232 |
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| author | Goyal, Vivek K. Fletcher, Alyson K. Rangan, Sundeep Ramchandran, Kannan |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Goyal, Vivek K. Fletcher, Alyson K. Rangan, Sundeep Ramchandran, Kannan |
| author_sort | Goyal, Vivek K. |
| collection | MIT |
| description | If a signal is known to have a sparse representation with respect to a frame, it can be estimated from a noise-corrupted observation by finding the best sparse approximation to . 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. |
| first_indexed | 2024-09-23T15:48:45Z |
| format | Article |
| id | mit-1721.1/67336 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:48:45Z |
| publishDate | 2011 |
| publisher | Hindawi Publishing Corporation |
| record_format | dspace |
| spelling | mit-1721.1/673362022-09-29T16:16:49Z Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory Goyal, Vivek K. Fletcher, Alyson K. Rangan, Sundeep Ramchandran, Kannan Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Research Laboratory of Electronics Goyal, Vivek K. If a signal is known to have a sparse representation with respect to a frame, it can be estimated from a noise-corrupted observation by finding the best sparse approximation to . 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. 2011-11-30T21:53:06Z 2011-11-30T21:53:06Z 2006-03 2005-06 2011-11-10T16:11:31Z Article http://purl.org/eprint/type/JournalArticle 1687-0433 1110-8657 http://hdl.handle.net/1721.1/67336 Fletcher, Alyson K. et al. “Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory.” EURASIP Journal on Advances in Signal Processing 2006 (2006): 1-20. Web. 30 Nov. 2011. © 2006 Alyson K. Fletcher et al. en http://dx.doi.org/10.1155/ASP/2006/26318 EURASIP Journal on Applied Signal Processing et al.; licensee BioMed Central Ltd. application/pdf Hindawi Publishing Corporation |
| spellingShingle | Goyal, Vivek K. Fletcher, Alyson K. Rangan, Sundeep Ramchandran, Kannan Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title | Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title_full | Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title_fullStr | Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title_full_unstemmed | Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title_short | Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory |
| title_sort | denoising by sparse approximation error bounds based on rate distortion theory |
| url | http://hdl.handle.net/1721.1/67336 |
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