Study on Over-Complete Dictionaries for Sparse Representations of Signals

Abstract:

In recent years there has been a growing interest in the study of sparse representation of signals. The redundancy of over-complete dictionary can make it effectively capture the characteristics of the signals. Using an over-complete dictionary that contains prototype signal-atoms, signals are described as linear combinations of a few of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, Compressed Sensing (CS), and more. Recent activities in this field concentrate mainly on the study of sparse decomposition algorithm and dictionary design algorithm. In this paper, we discuss the advantages of sparse dictionaries, and present the implicit dictionaries for signal sparse presents. The overcomplete dictionaries which combined the different orthonormal transform bases can be used for the compressed sensing. Experimental results demonstrate the effectivity for sparse presents of signals.

• Periodical: Applied Mechanics and Materials (Volumes 157-158)
• Pages: 796-799
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.157-158.796
• Online since: February 2012
• Authors: Guang Chun Gao, Kai Xiong, Li Na Shang, Sheng Ying Zhao, Cui Zhang
• Keywords: Compressed Sensing, Over-Complete Dictionary, Sparse Representation

References

[1] S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev., vol. 43, no. 1, p.129–159, (2001).

DOI: 10.1137/s003614450037906x

[2] D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, ( 2007).

DOI: 10.1109/tit.2011.2173241

[3] A.M. Bruckstein, D. L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev., vol. 51, no. 1, p.34–81, (2009).

DOI: 10.1137/060657704

[4] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., vol. 15, no. 12, p.3736–3745, Dec. (2006).

DOI: 10.1109/tip.2006.881969

[5] G. Davis, S. Mallat, and M. Avellaneda, Adaptive greedy approximations, Constructive Approximation, vol. 13, no. 1, p.57–98, (1997).

DOI: 10.1007/bf02678430

[6] J. Mairal, G. Sapiro, and M. Elad, Learning multiscale sparse representations for image and video restoration, SIAM Multiscale Modeling Simulation, vol. 7, no. 1, p.214–241, (2008).

DOI: 10.1137/070697653

[7] J. Mairal, G. Sapiro, and M. Elad, Learning multiscale sparse representations for image and video restoration, SIAM Multiscale Modeling Simulation, vol. 7, no. 1, p.214–241, (2008).

DOI: 10.1137/070697653

[8] M. Protter and M. Elad, Image sequence denoising via sparse and redundant representations, IEEE Trans. Image Process., vol. 18, no. 1, p.27–35, Jan. (2009).

DOI: 10.1109/tip.2008.2008065

[9] M. D. Carvajalino and G. Sapiro, Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization, IEEE Trans. Image Proc., vol. 18, no. 7, p.1395–1408, Jul. (2009).

DOI: 10.1109/tip.2009.2022459

[10] O. Bryt and M. Elad, Compression of facial images using the K-SVD algorithm, J. Visual Commun. Image Representation, vol. 19, no. 4, p.270–283, (2008).

DOI: 10.1016/j.jvcir.2008.03.001

[11] Ron Rubinstein, Michael Zibulevsky, and Michael Elad, Double Sparsity: Learning Sparse Dictionaries for Sparse Signal Approximation. IEEE Transaction on signal processing, Vol. 58, No. 3, pp.1553-1564, March (2010).

DOI: 10.1109/tsp.2009.2036477

[12] Gabriel Peyr´e, Best Basis Compressed Sensing, IEEE Transaction on signal processing, Vol. 58, No. 1, pp.1-11. Jan . (2010).

[13] Wu-Sheng Lu, Compressed sensing and Sparse Signal processing, University of Victoria, Canada, November (2010).

[14] Orthogonal Matching Pursuit Algorithm for Compressive Sensing, http: /www. eee. hku. hk/~wsha/Freecode/freecode. htm, (2011).