Improving Mutual Coherence with Non-Uniform Discretization of Orthogonal Function for Image Denoising Application
Hani Nozari, Alireza Siamy
DOI: 10.4236/jsip.2011.23025   PDF    HTML     5,670 Downloads   9,112 Views   Citations


This paper presented a novel method on designing redundant dictionary from known orthogonal functions. Usual way of discretization of continuous functions is uniform sampling. Our experiments show that dividing the function definition interval with non-uniform measure makes the redundant dictionary sparser and it is suitable for image denoising via sparse and redundant dictionary. In this case the problem is to find an appropriate measure in order to make each atom of dictionary. It has shown that in sparse approximation context, incoherent dictionary is suitable for sparse approximation method. According to this fact we define some optimization problems to find the best parameter of distribution measure (in our study normal distribution). For better convergence to optimum point we used Genetic Algorithm (GA) with enough diversity on initial population. We show the effect of this type of dictionary design on exact sparse recovery support. Our results also show the advantage of this design method on image denoising task.

Share and Cite:

H. Nozari and A. Siamy, "Improving Mutual Coherence with Non-Uniform Discretization of Orthogonal Function for Image Denoising Application," Journal of Signal and Information Processing, Vol. 2 No. 3, 2011, pp. 184-189. doi: 10.4236/jsip.2011.23025.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. Mallat and Z. Zhang, “Matching Pursuits with Time Frequency Dictionaries,” IEEE Transactions on Signal Processing, Vol. 41, No. 12, 1993, pp. 3397-3415. doi:10.1109/78.258082
[2] G. Davis, “Adaptive Nonlinear Approximations,” Ph.D. Dissertation, New York University, New York, 1994.
[3] T. Blumensath and M. Davies, “Gradient Pursuits,” IEEE Transactions on Signal Processing, Vol. 56, No. 6, 2008, pp. 2370-2382. doi:10.1109/TSP.2007.916124
[4] S. Chen, D. Donoho and M. Saunders, “Atomic Decomposition by Basis Pursuit,” SIAM Journal on Scientific Computing, Vol. 20, No. 1, 1998, pp. 33-61. doi:10.1137/S1064827596304010
[5] M. Elad, “Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing,” Springer, New York, 2010.
[6] R. Rubinstein, A. M. Bruckstein and M. Elad, “Dictionaries for Sparse Representation Modeling,” IEEE Proceedings—Special Issue on Applications of Sparse Representation & Compressive Sensing, Vol. 98, No. 6, April 2010, pp. 1045-1057.
[7] S. Mallat, “A Wavelet Tour of Signal Processing,” 3rd Edition, Academic, New York, 2009.
[8] E. J. Candès and D. L. Donoho, “Curvelets—A Surprisingly Effective Nonadaptive Representation for Objects with Edges,” Vanderbilt University Press, Nashville, 1999.
[9] M. N. Do and M. Vetterli, “The Contourlet Transform: An Efficient Directional Multiresolution Image Representation,” IEEE Transaction on Image Processing, Vol. 14, No. 12, 2005, pp. 2091-2106. doi:10.1109/TIP.2005.859376
[10] E. LePennec and S. Mallat, “Sparse Geometric Image Representations with Bandelets,” IEEE Transaction on Image Processing, Vol. 14, No. 4, 2005, pp. 423-438. doi:10.1109/TIP.2005.843753
[11] K. Engan, S. O. Aase and J. H. Husoy, “Method of Optimal Directions for Frame Design,” IEEE International Conference on Acoustics, Speech, and Signal Processing, Phoenix, 15-19 March 1999, pp. 2443-2446.
[12] M. Aharon, M. Elad and A. M. Bruckstein, “The K-SVD: An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation,” IEEE Transactions on Signal Processing, Vol. 54, No. 11, 2006, pp. 4311-4322. doi:10.1109/TSP.2006.881199
[13] M. Aharon, M. Elad and A. M. Bruckstein, “On the Uniqueness of Overcomplete Dictionaries, and a Practical Way to Retrieve Them,” Journal of Lineaar Algebra and Applications, Vol. 416, No. 1, 2006, pp. 48-67.
[14] T. Strohmer and R. Heath, “Grassmannian Frames with Applications to Coding and Communication,” Applied and Computational Harmonic Analysis, Vol. 14, No. 3, 2003, pp. 257-275. doi:10.1016/S1063-5203(03)00023-X
[15] J. Tropp, I. Dhillon, R. Heath Jr. and T. Strohmer, “Designing Structural Tight Frames via an Alternating Projection Method,” IEEE Transactions on Information Theory, Vol. 51, No. 1, 2005, pp. 188-209. doi:10.1109/TIT.2004.839492
[16] M. Yaghoobi, T. Blumensath and M. Davies, “Dictionary Learning for Sparse Approximations with the Majorization Method,” IEEE Transactions on Signal Processing, Vol. 57, No. 6, 2009, pp. 2178-2191. doi:10.1109/TSP.2009.2016257
[17] J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, “Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM Journal of Optimization, Vol. 9 No. 1, 1998, pp. 112-147. doi:10.1137/S1052623496303470
[18] D. E. Goldberg, “Genetic Algorithms in Search, Optimization and Machine Learning,” Addison Wesley, Reading, 1989.
[19] M. Elad and M. Aharon, “Image Denoising via Sparse and Redundant Representations over Learned Dictionaries,” IEEE Transactions on Image Processing, Vol. 15, No. 12, December 2006, pp. 3736-3745.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.