Augmented Lagrangian method for total variation restoration with non-quadratic fidelity

Chunlin Wu; Juyong Zhang; Xue-Cheng Tai

doi:10.3934/ipi.2011.5.237

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February 2011, 5(1): 237-261. doi: 10.3934/ipi.2011.5.237

Augmented Lagrangian method for total variation restoration with non-quadratic fidelity

Chunlin Wu ^1, , Juyong Zhang ^2, and Xue-Cheng Tai ^3,

1.	Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore
2.	Division of Computer Communications, School of Computer Engineering, Nanyang Technological University, Singapore
3.	University of Bergen, University of Bergen Bergen, Norway

Received December 2009 Revised September 2010 Published February 2011

Abstract
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Recently augmented Lagrangian method has been successfully applied to image restoration. We extend the method to total variation (TV) restoration models with non-quadratic fidelities. We will first introduce the method and present an iterative algorithm for TV restoration with a quite general fidelity. In each iteration, three sub-problems need to be solved, two of which can be very efficiently solved via Fast Fourier Transform (FFT) implementation or closed form solution. In general the third sub-problem need iterative solvers. We then apply our method to TV restoration with $L^1$ and Kullback-Leibler (KL) fidelities, two common and important data terms for deblurring images corrupted by impulsive noise and Poisson noise, respectively. For these typical fidelities, we show that the third sub-problem also has closed form solution and thus can be efficiently solved. In addition, convergence analysis of these algorithms are given. Numerical experiments demonstrate the efficiency of our method.

Keywords: impulsive noise, Poisson noise, Augmented Lagrangian method, TV-KL, convergence., TV-$L^1$, total variation.

Mathematics Subject Classification: 80M30, 80M50, 68U1.

Citation: Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237

References:

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[8]	E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. Google Scholar
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References:

[1]	S. Alliney, Digital filters as absolute norm regularizers, IEEE Trans. Signal Process., 40 (1992), 1548-1562. doi: 10.1109/78.139258. Google Scholar
[2]	P. Besbeas, I. D. Fies and T. Sapatinas, A comparative simulation study of wavelet shrinkage estimators for Poisson counts, International Statistical Review, 72 (2004), 209-237. doi: 10.1111/j.1751-5823.2004.tb00234.x. Google Scholar
[3]	P. Blomgren and T. F. Chan, Color TV: Total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7 (1998), 304-309. doi: 10.1109/83.661180. Google Scholar
[4]	A. Bovik, "Handbook of Image and Video Processing," Academic Press, 2000. Google Scholar
[5]	X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Problems and Imaging, 2 (2008), 455-484. doi: 10.3934/ipi.2008.2.455. Google Scholar
[6]	C. Brune, A. Sawatzky and M. Burger, Bregman-EM-TV methods with application to optical nanoscopy, LNCS, 5567 (2009), 235-246. Google Scholar
[7]	A. Caboussat, R. Glowinski and V. Pons, An augmented Lagrangian approach to the numerical solution of a non-smooth eigenvalue problem, J. Numer. Math., 17 (2009), 3-26. doi: 10.1515/JNUM.2009.002. Google Scholar
[8]	E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. Google Scholar
[9]	J. L. Carter, "Dual Methods for Total Variation - Based Image Restoration," Ph.D thesis, UCLA, 2001. Google Scholar
[10]	A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. Google Scholar
[11]	A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88. Google Scholar
[12]	R. Chan, C. W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detector and detail-preserving regularization, IEEE Trans. Image Process., 14 (2005), 1479-1485. doi: 10.1109/TIP.2005.852196. Google Scholar
[13]	R. H. Chan and K. Chen, Multilevel algorithms for a Poisson noise removal model with total variation regularization, Int. J. Comput. Math., 84 (2007), 1183-1198. doi: 10.1080/00207160701450390. Google Scholar
[14]	T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977. doi: 10.1137/S1064827596299767. Google Scholar
[15]	T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. Google Scholar
[16]	T. F. Chan, S. H. Kang and J. H. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Visual Commun. Image Repres., 12 (2001), 422-435. doi: 10.1006/jvci.2001.0491. Google Scholar
[17]	T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297. Google Scholar
[18]	S. Chen, D. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), pp. 33-61. doi: 10.1137/S1064827596304010. Google Scholar
[19]	T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 48 (2001), 784-789. Google Scholar
[20]	Y. Dong, M. Hintermüller and M. Neri, An efficient primal-dual method for $L^1$TV image restoration, SIAM J. Imaging Sciences, 2 (2009), 1168-1189. doi: 10.1137/090758490. Google Scholar
[21]	D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. Google Scholar
[22]	I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," SIAM, 1999. Google Scholar
[23]	H. L. Eng and K. K. Ma, Noise adaptive soft-switching median filter, IEEE Trans. Image Process., 10 (2001), 242-251. doi: 10.1109/83.902289. Google Scholar
[24]	E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman,, UCLA CAM Report, (): 09. Google Scholar
[25]	M. A. T. Figueiredo and J. M. Bioucas-Dias, Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization, in "IEEE Workshop on Statistical Signal Processing," Cardiff, (2009), 733-736. doi: 10.1109/SSP.2009.5278459. Google Scholar
[26]	H. Y. Fu, M. K. Ng, M. Nikolova and J. L. Barlow, Efficient minimization methods of mixed l2-l1 and l1-l1 norms for image restoration, SIAM J. Sci. Comput., 27 (2006), 1881-1902. doi: 10.1137/040615079. Google Scholar
[27]	R. Glowinski and P. Le Tallec, "Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics," SIAM, Philadelphia, 1989. Google Scholar
[28]	T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM J. Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. Google Scholar
[29]	M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory and Appl., 4 (1969), 303-320. doi: 10.1007/BF00927673. Google Scholar
[30]	W. Hinterberger and O. Scherzer, Variational methods on the space of functions of bounded Hessian for convexification and denoising, Computing, 76 (2006), 109-133. doi: 10.1007/s00607-005-0119-1. Google Scholar
[31]	M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2002), 865-888. doi: 10.1137/S1052623401383558. Google Scholar
[32]	Y. Huang, M. Ng and Y. Wen, A fast total variation minimization method for image restoration, SIAM Multi. Model. Simul., 7 (2008), 774-795. doi: 10.1137/070703533. Google Scholar
[33]	H. Hwang and R. A. Haddad, Adaptive median filters: New algorithms and results, IEEE Trans. Image Process., 4 (1995), 499-502. doi: 10.1109/83.370679. Google Scholar
[34]	C. Kervrann and A. Trubuil, An adaptive window approach for poisson noise reduction and structure preserving in confocal microscopy, in "Proc. International Symposium on Biomedical Imaging (ISBI'04)," Arlington, (2004), 788-791. Google Scholar
[35]	E. Kolaczyk, Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds, Statist. Sinica, 9 (1999), 119-135. Google Scholar
[36]	T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y. Google Scholar
[37]	Y. Li and S. Osher, A new median formula with applications to PDE based denoising, Commun. Math. Sci., 7 (2009), 741-753. Google Scholar
[38]	M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590. doi: 10.1109/TIP.2003.819229. Google Scholar
[39]	M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second order functional, Int'l J. Computer Vision, 2005. Google Scholar
[40]	P. Mrazek, J. Weickert and A. Bruhn, On robust estimations and smoothing with spatial and tonal kernels, in "Geometric Properties from Incomplete Data," Springer, (2006), 335-352. doi: 10.1007/1-4020-3858-8_18. Google Scholar
[41]	P. E. Ng and K. K. Ma, A switching median filter with boundary discriminative noise detection for extremely corrupted images, IEEE Trans. Image Process., 15 (2006), 1506-1516. doi: 10.1109/TIP.2005.871129. Google Scholar
[42]	M. Nikolova, Minimizers of cost-functions involving non-smooth data fidelity terms, SIAM J. Num. Anal., 40 (2002), 965-994. doi: 10.1137/S0036142901389165. Google Scholar
[43]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar
[44]	V. Y. Panin, G. L. Zeng and G. T. Gullberg, Total variation regulated EM algorithm [SPECT reconstruction], IEEE Trans. Nucl. Sci., 46 (1999), 2202-2210. doi: 10.1109/23.819305. Google Scholar
[45]	G. Pok, J. C. Liu and A. S. Nair, Selective removal of impulse noise based on homogeneity level information, IEEE Trans. Image Process., 12 (2003), 85-92. doi: 10.1109/TIP.2002.804278. Google Scholar
[46]	M. J. D. Powell, A method for nonlinear constraints in minimization problems, in "Optimization"(ed. R. Fletcher), Academic Press, (1972), 283-298. Google Scholar
[47]	R. T. Rockafellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Mathematical Programming, 5 (1973), 354-373. doi: 10.1007/BF01580138. Google Scholar
[48]	L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[49]	G. Sapiro and D. L. Ringach, Anisotropic diffusion of multivalued images with applications to color filtering, IEEE Trans. Image Process., 5 (1996), 1582-1586. doi: 10.1109/83.541429. Google Scholar
[50]	O. Scherer, Denoising with higher order derivatives of bounded variation and an application to parameter estimation, Computing, 60 (1998), 1-27. doi: 10.1007/BF02684327. Google Scholar
[51]	S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, LNCS, 5567 (2009), 464-476. Google Scholar
[52]	S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Visual Commun. Image Repres., accepted (2009). Google Scholar
[53]	L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Medical Imaging, 1 (1982), 113-122. doi: 10.1109/TMI.1982.4307558. Google Scholar
[54]	X. C. Tai and C. L. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, LNCS, 5567 (2009), 502-513. Google Scholar
[55]	K. Timmermann and R. Novak, Multiscale modeling and estimation of Poisson processes with applications to photon-limited imaging, IEEE Trans. Inf. Theor., 45 (1999), 846-852. doi: 10.1109/18.761328. Google Scholar
[56]	Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265. Google Scholar
[57]	C. L. Wu and X. C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558. Google Scholar
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Augmented Lagrangian method for total variation restoration with non-quadratic fidelity

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Augmented Lagrangian method for total variation restoration with non-quadratic fidelity

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