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Design redundant Chebyshev dictionary with generalized extreme value distribution for sparse approximation and image denoising

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Abstract

This paper illustrates a novel method for designing redundant dictionary from Chebyshev polynomials for sparse coding. Having an overcomplete dictionary in \({{\mathbb R}^{d \times N}:d < N}\) from N, orthogonal functions need to sample d times from orthogonal intervals. It is proved (“Appendix B”) that uniform distribution is not optimal for sampling. Experiments show that using non-uniform measures for dividing orthogonal intervals has some advantages in making incoherent dictionary with a mutual coherence closer to equiangular tight frames, which is appropriate for sparse approximation methods. In this paper, we first describe the dictionary design problem, then modify this design with any kind of distribution, and define an objective function respect to its parameters. Because of the abundant extremums in this objective function, genetic algorithm is used to find the best parameters. Experimental results show that generalized extreme value distribution has better performance among others. This type of dictionary design improves the performance of sparse approximation and image denoising via redundant dictionary. The advantages of this method of designing overcomplete dictionaries are going to be compared with uniform ones in sparse approximation areas.

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Abbreviations

ETF:

Equiangular tight frame

GEVD:

Generalized extreme value distribution

OMP:

Orthogonal matching pursuit

GA:

Genetic algorithm

PSNR:

Peak signal-to-noise ratio

D d × N :

General overcomplete dictionary of size d × N (d < N)

μ (D d × N ):

Mutual coherence of D d × N

T n (t):

Chebyshev polynomials

\({D_{\mathcal C}}\) :

Overcomplete dictionary from T n (t)

\({G_{\mathcal C}}\) :

Gram matrix of \({D_{\mathcal C}}\)

\({{\mathcal G}_G}\) :

Gram matrix of ETFs

μ (G d × N ):

Mutual coherence of ETF

Θ:

Sampling distribution

θ p :

pth parameter of Θ

s u (n):

Uniform sampler

s nu (n):

Non-uniform sampler

D u :

Overcomplete dictionary with uniform distribution of orthogonal function

μ u :

Mutual coherence of uniform overcomplete dictionary

μ nu :

Mutual coherence of non-uniform overcomplete dictionary

μ :

Mean value

σ :

Standard deviations

k :

Shape parameter in GEVD

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  1. P.O. Box 484, Babol, Mazandaran, Iran

    Hani Nozari & Mohammad Reza Karami

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  1. Hani Nozari
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  2. Mohammad Reza Karami
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Correspondence to Hani Nozari.

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Nozari, H., Karami, M.R. Design redundant Chebyshev dictionary with generalized extreme value distribution for sparse approximation and image denoising. SIViP 8, 327–338 (2014). https://doi.org/10.1007/s11760-012-0295-3

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  • DOI: https://doi.org/10.1007/s11760-012-0295-3

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