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

This paper illustrates a novel method for designing redundant dictionary from Chebyshev polynomials for sparse coding. Having an overcomplete dictionary in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}^{d \times N}:d < N}$$\end{document} 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.