# American Institute of Mathematical Sciences

November  2011, 5(4): 815-841. doi: 10.3934/ipi.2011.5.815

## A comparison of dictionary based approaches to inpainting and denoising with an emphasis to independent component analysis learned dictionaries

 1 Division of Laser and Atomic Research and Development, Rudjer Bošković Institute, Bijenička cesta 54, P.O. Box 180, 10002, Zagreb, Croatia, Croatia

Received  February 2010 Revised  July 2011 Published  November 2011

The first contribution of this paper is the comparison of learned dictionary based approaches to inpainting and denoising of images in natural scenes, where emphasis is given on the use of complete and overcomplete dictionary learned by independent component analysis. The second contribution of the paper relates to the formulation of a problem of denoising an image corrupted by a salt and pepper type of noise (this problem is equivalent to estimating saturated pixel values), as a noiseless inpainting problem, whereupon noise corrupted pixels are treated as missing pixels. A maximum a posteriori (MAP) approach to image denoising is not applicable in such a case due to the fact that variance of the impulsive noise is infinite and the MAP based estimation relies on solving an optimization problem with an inequality constraint that depends on the variance of the additive noise. Through extensive comparative performance analysis of the inpainting task, it is demonstrated that ICA-learned basis outperforms K-SVD and morphological component analysis approaches in terms of visual quality. It yielded similar performance as a field of experts method but with more than two orders of magnitude lower computational complexity. On the same problems, Fourier and wavelet bases as representatives of fixed bases, exhibited the poorest performance. It is also demonstrated that noiseless inpainting-based approach to image denoising (estimation of the saturated pixel values) greatly outperforms denoising based on two-dimensional myriad filtering that is a theoretically optimal solution for this class of additive impulsive noise.
Citation: Marko Filipović, Ivica Kopriva. A comparison of dictionary based approaches to inpainting and denoising with an emphasis to independent component analysis learned dictionaries. Inverse Problems & Imaging, 2011, 5 (4) : 815-841. doi: 10.3934/ipi.2011.5.815
##### References:
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Google Scholar [47] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity,, IEEE Trans. Image Process., 13 (2004), 600. doi: 10.1109/TIP.2003.819861. Google Scholar [48] X. Zhang and D. H. Brainard, Estimation of saturated pixel values in digital color imaging,, J. Opt. Soc. Amer. A, 21 (2004), 2301. doi: 10.1364/JOSAA.21.002301. Google Scholar [49] L. Zhang, A. Cichocki and S.-i. Amari, Self-adaptive blind source separation based on activation function adaptation,, IEEE Trans. Neural Net., 15 (2004), 233. doi: 10.1109/TNN.2004.824420. Google Scholar

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##### References:
 [1] M. Aharon, M. Elad and A. M. Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,, IEEE Trans. Signal Process., 54 (2006), 4311. doi: 10.1109/TSP.2006.881199. Google Scholar [2] G. R. Arce, "Nonlinear Signal Processing - A Statistical Approach,", John Wiley & Sons, (2005). Google Scholar [3] A. J. Bell and T. J. Sejnowski, An information-maximization approach to blind separation and blind deconvolution,, Neural Comput., 7 (1995), 1129. doi: 10.1162/neco.1995.7.6.1129. Google Scholar [4] A. J. Bell and T. J. Sejnowski, The 'independent components' of natural scenes are edge filters,, Vision Research, 37 (1997), 3327. doi: 10.1016/S0042-6989(97)00121-1. Google Scholar [5] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, in, (2000), 417. doi: 10.1145/344779.344972. Google Scholar [6] M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting,, IEEE Trans. Image Process., 12 (2003), 882. doi: 10.1109/TIP.2003.815261. Google Scholar [7] M. Bethge, Factorial coding of natural images: how effective are linear models in removing higher-order dependencies?,, J. Opt. Soc. Amer. A, 23 (2006), 1253. doi: 10.1364/JOSAA.23.001253. Google Scholar [8] T. Blumensath and M. E. Davies, Normalised iterative hard thresholding; guaranteed stability and performance,, IEEE J. Sel. Top. Signal Process., 4 (2010), 298. doi: 10.1109/JSTSP.2010.2042411. Google Scholar [9] P. Bofill and M. Zibulevsky, Underdetermined blind source separation using sparse representations,, Signal Proc., 81 (2001), 2353. doi: 10.1016/S0165-1684(01)00120-7. Google Scholar [10] A. M. Bruckstein, M. Elad and D. L. Donoho, From sparse solutions of systems of equations to sparse modeling of signals and images,, SIAM Rev., 51 (2009), 34. doi: 10.1137/060657704. Google Scholar [11] E. Candès, M. B. Wakin and S. Boyd, Enhancing sparsity by reweighted ℓ$_1$ minimization,, J. of Fourier Anal. Appl., 14 (2008), 877. doi: 10.1007/s00041-008-9045-x. Google Scholar [12] R. Chartrand, Exact reconstructions of sparse signals via nonconvex minimization,, IEEE Signal Process. Lett., 14 (2007), 707. doi: 10.1109/LSP.2007.898300. Google Scholar [13] S. Choi, A. Cichocki and S.-i. Amari, Flexible independent component analysis,, J. VLSI Signal Process. Sys., 26 (2000), 25. doi: 10.1023/A:1008135131269. Google Scholar [14] M. Elad, J.-L. Starck, P. Querre and D. L. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA),, Appl. Comput. Harmon. Anal., 19 (2005), 340. doi: 10.1016/j.acha.2005.03.005. Google Scholar [15] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries,, IEEE Trans. Image Proc., 15 (2006), 3736. doi: 10.1109/TIP.2006.881969. Google Scholar [16] K. Engan, S. O. Aase and J. H. Husoy, Multi-frame compression: Theory and design,, Signal Process., 80 (2000), 2121. doi: 10.1016/S0165-1684(00)00072-4. Google Scholar [17] K. Engan, K. Skretting and J. H. Husoy, Family of iterative LS-based dictionary learning algorithms, ILS-DLA, for sparse signal representation,, Dig. Signal Process., 17 (2007), 32. doi: 10.1016/j.dsp.2006.02.002. Google Scholar [18] D. Erdogmus, K. E. Hild II, Y. N. Rao and J. C. Principe, Minimax mutual information approach for independent component analysis,, Neural Comput., 16 (2004), 1235. doi: 10.1162/089976604773717595. Google Scholar [19] S. Foucart and M.-J. Lai, Sparsest solution of underdetermined linear systems via ℓ$_q$ minimization for $0 < q \leq 1$,, Appl. Comput. Harmon. Anal., 26 (2009), 395. doi: 10.1016/j.acha.2008.09.001. Google Scholar [20] P. Georgiev, F. Theis and A. Cichocki, Sparse component analysis and blind source separation of underdetermined mixtures,, IEEE Trans. Neural Netw., 16 (2005), 992. doi: 10.1109/TNN.2005.849840. Google Scholar [21] J. G. Gonzalez and G. R. Arce, Statistically-efficient filtering in impulsive environments: Weighted myriad filters,, EURASIP J. Appl. Signal Process., 1 (2002), 4. doi: 10.1155/S1110865702000483. Google Scholar [22] I. F. Gorodnitsky and B. D. Rao, Sparse signal reconstruction from limited data using FOCUSS: a re-weighted norm minimization algorithm,, IEEE Trans. Signal Process., 45 (1997), 600. doi: 10.1109/78.558475. Google Scholar [23] O. G. Guleryuz, Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising: Part I - Theory,, IEEE Trans. Image Process., 15 (2006), 539. doi: 10.1109/TIP.2005.863057. Google Scholar [24] O. G. Guleryuz, Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising: Part II - Adaptive algorithms,, IEEE Trans. Image Process., 15 (2006), 555. doi: 10.1109/TIP.2005.863055. Google Scholar [25] A. Hyvärinen, R. Cristescu and E. Oja, A fast algorithm for estimating overcomplete ICA bases for image windows,, in, (1999), 894. doi: 10.1109/IJCNN.1999.831071. Google Scholar [26] A. Hyvärinen, J. Karhunen and E. Oja, "Independent Component Analysis,", John Wiley & Sons, (2001). Google Scholar [27] A. Hyvärinen and E. Oja, A fast fixed-point algorithm for independent component analysis,, Neural Comput., 9 (1997), 1483. doi: 10.1162/neco.1997.9.7.1483. Google Scholar [28] J. Kovačević and A. Chebira, Life beyond bases: The advent of frames (Part I),, IEEE Signal Process. Mag., 25 (2007), 86. doi: 10.1109/MSP.2007.4286567. Google Scholar [29] J. Kovačević and A. Chebira, Life beyond bases: The advent of frames (Part II),, IEEE Signal Process. Mag., 25 (2007), 115. doi: 10.1109/MSP.2007.904809. Google Scholar [30] K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee and T. J. Sejnowski, Dictionary learning algorithms for sparse representation,, Neural Comput., 15 (2003), 349. doi: 10.1162/089976603762552951. Google Scholar [31] M. S. Lewicki and B. A. Olshausen, Probabilistic framework for the adaptation and comparison of image codes,, J. Opt. Soc. Amer. A, 16 (1999), 1587. doi: 10.1364/JOSAA.16.001587. Google Scholar [32] M. S. Lewicki and T. J. Sejnowski, Learning overcomplete representations,, Neural Comput., 12 (2000), 337. doi: 10.1162/089976600300015826. Google Scholar [33] L. Ma and Y. Zhang, Bayesian estimation of overcomplete independent feature subspaces for natural images,, in, (2007), 746. Google Scholar [34] J. Mairal, G. Sapiro and M. Elad, Sparse representation for color image restoration,, IEEE Trans. Image Process., 17 (2008), 53. doi: 10.1109/TIP.2007.911828. Google Scholar [35] J. Mairal, F. Bach, J. Ponce and G. Sapiro, Online learning for matrix factorization and sparse coding,, J. Mach. Learn. Res., 11 (2010), 19. Google Scholar [36] H. Mansour, R. Saab, P. Nasiopoulos and R. Ward, Color image desaturation using sparse reconstruction,, in, (2010), 778. doi: 10.1109/ICASSP.2010.5494984. Google Scholar [37] H. Mohimani, M. Babaie-Zadeh and C. Jutten, A fast approach for overcomplete sparse decomposition based on smoothed ℓ$_0$ norm,, IEEE Trans. Signal Process., 57 (2009), 289. doi: 10.1109/TSP.2008.2007606. Google Scholar [38] B. A. Olshausen and D. J. Field, Emergence of simple-cell receptive field properties by learning a sparse code for natural images,, Nature, 381 (1996), 607. doi: 10.1038/381607a0. Google Scholar [39] D. T. Pham and P. Garat, Blind separation of mixtures of independent sources through a quasimaximum likelihood approach,, IEEE Trans. Signal Process., 45 (1997), 1712. doi: 10.1109/78.599941. Google Scholar [40] S. Roth and M. J. Black, Fields of experts,, Int. J. Computer Vision, 82 (2009), 205. doi: 10.1007/s11263-008-0197-6. Google Scholar [41] U. Schmidt, Q. Gao and S. Roth, A generative perspective on Markov random fields in low-level vision,, in, (2010), 1751. doi: 10.1109/CVPR.2010.5539844. Google Scholar [42] I. W. Selesnick, R. V. Slyke and O. G. Guleryuz, Pixel recovery via ℓ$_1$ minimization in wavelet domain,, in, (2004), 1819. doi: 10.1109/ICIP.2004.1421429. Google Scholar [43] J. Shen, Inpainting and the fundamental problem of image processing,, SIAM News, 36 (2003). Google Scholar [44] K. Skretting, J. H. Husoy and S. O. Aase, General design algorithm for sparse frame expansions,, Signal Process., 86 (2006), 117. doi: 10.1016/j.sigpro.2005.04.013. Google Scholar [45] J. Tropp and S. J. Wright, Computational methods for sparse solution of linear inverse problems,, Proc. of the IEEE, 98 (2010), 948. doi: 10.1109/JPROC.2010.2044010. Google Scholar [46] Z. Wang and A. Bovik, Mean squared error: Love it or leave it? A new look at signal fidelity measures,, IEEE Signal Process. Mag., 26 (2009), 98. doi: 10.1109/MSP.2008.930649. Google Scholar [47] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity,, IEEE Trans. Image Process., 13 (2004), 600. doi: 10.1109/TIP.2003.819861. Google Scholar [48] X. Zhang and D. H. Brainard, Estimation of saturated pixel values in digital color imaging,, J. Opt. Soc. Amer. A, 21 (2004), 2301. doi: 10.1364/JOSAA.21.002301. Google Scholar [49] L. Zhang, A. Cichocki and S.-i. Amari, Self-adaptive blind source separation based on activation function adaptation,, IEEE Trans. Neural Net., 15 (2004), 233. doi: 10.1109/TNN.2004.824420. Google Scholar
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