Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity

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November 2013, 7(4): 1331-1366. doi: 10.3934/ipi.2013.7.1331

Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity

Marc C. Robini ^1, , Yuemin Zhu ^1, and Jianhua Luo ^2,

1.	CREATIS (CNRS research unit UMR5220 and INSERM research unit U1044), INSA-Lyon, 69621 Villeurbanne Cedex, France, France
2.	College of Life Science and Technology, Shanghai Jiao Tong University, Shanghai 200240, China

Received March 2012 Revised March 2013 Published November 2013

Abstract
Related Papers

The standard approach for image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. State-of-the-art methods to correct these undesirable effects either have weak convergence guarantees or are limited to specific situations; furthermore, most of them use a quadratic data-fidelity term. In this paper, we propose a simple alternative regularization model to improve contour regularity and to reduce the staircase effect-our model incorporates the smoothness of the edge field in an implicit way by adding a simple penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data-fidelity term is not quadratic and the cost function is not convex. Our approach either extends or supplements existing methods and offers strong convergence guarantees. Numerical experiments show that it outperforms first-order total variation regularization as well as state-of-the-art second-order regularization techniques.

Keywords: linear inverse problems, Image restoration, half-quadratic algorithms, non-convex optimization., edge-preserving regularization.

Mathematics Subject Classification: 94A08, 49N45, 68U10, 65K10, 49M2.

Citation: Marc C. Robini, Yuemin Zhu, Jianhua Luo. Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity. Inverse Problems and Imaging, 2013, 7 (4) : 1331-1366. doi: 10.3934/ipi.2013.7.1331

References:

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[2]	M. Banham and A. Katsaggelos, Digital image restoration, IEEE Signal Processing Mag., 14 (1997), 24-41. doi: 10.1109/79.581363.
[3]	L. Bedini, L. Benvenuti, E. Salerno and A. Tonazzini, A mixed-annealing algorithm for edge preserving image reconstruction using a limited number of projections, Signal Process., 32 (1993), 397-408. doi: 10.1016/0165-1684(93)90009-Y.
[4]	M. Belge, M. Kilmer and E. Miller, Wavelet domain image restoration with adaptive edge-preserving regularization, IEEE Trans. Image Process., 9 (2000), 597-608. doi: 10.1109/83.841937.
[5]	M. Belge, M. Kilmer and E. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse Problems, 18 (2002), 1161-1183. doi: 10.1088/0266-5611/18/4/314.
[6]	L. Blanc-Féraud, S. Teboul, G. Aubert and M. Barlaud, Nonlinear regularization using constrained edges in image reconstruction, in Proc. IEEE Int. Conf. Image Processing, 2, Lausanne, Switzerland, (1996), 449-452. doi: 10.1109/ICIP.1996.560882.
[7]	J.-F. Cai, R. Chan and M. Nikolova, Fast two-phase image deblurring under impulse noise, J. Math. Imaging Vis., 36 (2010), 46-53. doi: 10.1007/s10851-009-0169-7.
[8]	R. Chan, Y. Dong and M. Hintermüller, An efficient two-phase $L^1$-TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), 1731-1739. doi: 10.1109/TIP.2010.2045148.
[9]	T. Chan, S. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.
[10]	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.
[11]	P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Trans. Image Process., 6 (1997), 298-311. doi: 10.1109/83.551699.
[12]	H. Choi and R. Baraniuk, Wavelet statistical models and Besov spaces, in Proc. SPIE, Wavelet Applications in Signal and Image Processing VII, vol. 3813, 1999, 489-501. doi: 10.1007/978-0-387-21579-2_2.
[13]	E. Chouzenoux, J. Idier and S. Moussaoui, A majorize-minimize strategy for subspace optimization applied to image restoration, IEEE Trans. Image Process, 20 (2011), 1517-1528. doi: 10.1109/TIP.2010.2103083.
[14]	A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45 (1992), 485-560. doi: 10.1002/cpa.3160450502.
[15]	I. Daubechies, M. Defrise and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.
[16]	A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography, IEEE Trans. Image Process, 7 (1998), 204-221. doi: 10.1109/83.660997.
[17]	G. Demoment, Image reconstruction and restoration: Overview of common estimation structures and problems, IEEE Trans. Acoust. Speech Signal Process, 37 (1989), 2024-2036. doi: 10.1109/29.45551.
[18]	R. DeVore, Nonlinear approximation, Acta Numer., 7 (1998), 51-150. doi: 10.1017/S0962492900002816.
[19]	D. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., 56 (1996), 1181-1198. doi: 10.1137/S003613999427560X.
[20]	Y. Dong, M. Hintermüller and M. Neri, An efficient primal-dual method for $L^1$-TV image restoration, SIAM J. Imaging Sci., 2 (2009), 1168-1189. doi: 10.1137/090758490.
[21]	H. Farid and E. Simoncelli, Differentiation of discrete multidimensional signals, IEEE Trans. Image Process., 13 (2004), 496-508. doi: 10.1109/TIP.2004.823819.
[22]	D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinuities, IEEE Trans. Pattern Anal. Machine Intell., 14 (1992), 367-383. doi: 10.1109/34.120331.
[23]	S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596.
[24]	S. Geman and D. McClure, Bayesian image analysis: an application to single photon emission tomography, in Proc. Stat. Comput. Section: Annual meeting of the Amer. Stat. Assoc., Las Vegas, Nevada, 1985, 12-18.
[25]	P. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Trans. Med. Imag., 9 (1990), 84-93. doi: 10.1109/42.52985.
[26]	T. Hebert and R. Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag., 8 (1989), 194-202. doi: 10.1109/42.24868.
[27]	T. Hou, S. Wang and H. Qin, Image deconvolution with multi-stage convex relaxation and its perceptual evaluation, IEEE Trans. Image Process., 20 (2011), 1057-7149. doi: 10.1109/TIP.2011.2150236.
[28]	Y. Hu and M. Jacob, Higher degree total variation (HDTV) regularization for image recovery, IEEE Trans. Image Process., 21 (2012), 2559-2571. doi: 10.1109/TIP.2012.2183143.
[29]	J. Idier, Convex half-quadratic criteria and interacting auxiliary variables for image restoration, IEEE Trans. Image Process., 10 (2001), 1001-1009. doi: 10.1109/83.931094.
[30]	K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization, SIAM J. Sci. Comput., 33 (2011), 1415-1438. doi: 10.1137/100790756.
[31]	P. Kohli, A. Shekhovtsov, C. Rother, V. Kolmogorov and P. Torr, On partial optimality in multi-label MRFs, in Proc. $25^{th}$ Int. Conf. Machine Learning, Helsinki, Finland, 2008, 480-487. doi: 10.1145/1390156.1390217.
[32]	S. Lefkimmiatis, A. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications, IEEE Trans. Image Process., 21 (2012), 983-995. doi: 10.1109/TIP.2011.2168232.
[33]	S. Li, On discontinuity-adaptive smoothness priors in computer vision, IEEE Trans. Pattern Anal. Machine Intell., 17 (1995), 576-586. doi: 10.1109/34.387504.
[34]	Y.-R. Li, L. Shen, D.-Q. Dai and B. Suter, Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise, IEEE Trans. Image Process., 20 (2011), 1822-1837. doi: 10.1109/TIP.2010.2103950.
[35]	M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis., 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7.
[36]	J. Ma and G. Plonka, The curvelet transform, IEEE Signal Processing Mag., 27 (2010), 118-133. doi: 10.1109/MSP.2009.935453.
[37]	S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674-693. doi: 10.1109/34.192463.
[38]	J. Marroquin, S. Mitter and T. Poggio, Probabilistic solution of ill-posed problems in computational vision, J. Amer. Statist. Assoc., 82 (1987), 76-89. doi: 10.1080/01621459.1987.10478393.
[39]	R. Meyer, Sufficient conditions for the convergence of monotonic mathematical programming algorithms, J. Comput. System Sci., 12 (1976), 108-121. doi: 10.1016/S0022-0000(76)80021-9.
[40]	P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors, IEEE Trans. Inform. Theory, 45 (1999), 909-919. doi: 10.1109/18.761332.
[41]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c.
[42]	M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares, Multiscale Model. Simul., 4 (2005), 960-991. doi: 10.1137/040619582.
[43]	M. Nikolova, M. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Trans. Image Process., 19 (2010), 3073-3088. doi: 10.1109/TIP.2010.2052275.
[44]	S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: a black-box optimization of regularization parameters for general denoising algorithms, IEEE Trans. Image Process., 17 (2008), 1540-1554. doi: 10.1109/TIP.2008.2001404.
[45]	M. C. Robini and I. Magnin, Optimization by stochastic continuation, SIAM J. Imaging Sci., 3 (2010), 1096-1121. doi: 10.1137/090756181.
[46]	M. C. Robini, T. Rastello and I. Magnin, Simulated annealing, acceleration techniques and image restoration, IEEE Trans. Image Process., 8 (1999), 1374-1387. doi: 10.1109/83.791963.
[47]	P. Rodríguez and B. Wohlberg, Efficient minimization method for a generalized total variation functional, IEEE Trans. Image Process., 18 (2009), 322-332. doi: 10.1109/TIP.2008.2008420.
[48]	C. Rother, V. Kolmogorov, V. Lempitsky and M. Szummer, Optimizing binary MRFs via extended roof duality, in Proc. IEEE Conf. Comp. Vis. Patt. Recognition, Minneapolis, MN, 2007, 1-8. doi: 10.1109/CVPR.2007.383203.
[49]	X.-C. Tai, J. Hahn and G. Chung, A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4 (2011), 313-344. doi: 10.1137/100803730.
[50]	M. Unser, Ten good reasons for using spline wavelets, in Proc. SPIE, Wavelet Applications in Signal and Image Processing V, vol. 3169, 1997, 422-431. doi: 10.1117/12.292801.
[51]	L. Wang, T.-T. Wong, P. Heng and J. Cheng, Template-matching approach to edge detection of volume data, in Proc. Int. Workshop on Medical Imaging and Augmented Reality, Shatin, Hong Kong, 2001, 286-291.
[52]	T. Zhang, Analysis of multi-stage convex relaxation for sparse regularization, J. Mach. Learn. Res., 11 (2010), 1081-1107.
[53]	W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32. doi: 10.1137/110822268.
[54]	X. Zhu and P. Milanfar, Automatic parameter selection for denoising algorithms using a no-reference measure of image content, IEEE Trans. Image Process., 19 (2010), 3116-3132. doi: 10.1109/TIP.2010.2052820.

show all references

References:

[1]	E. Bae, J. Shi and X.-C. Tai, Graph cuts for curvature based image denoising, IEEE Trans. Image Process., 20 (2011), 1199-1210. doi: 10.1109/TIP.2010.2090533.
[2]	M. Banham and A. Katsaggelos, Digital image restoration, IEEE Signal Processing Mag., 14 (1997), 24-41. doi: 10.1109/79.581363.
[3]	L. Bedini, L. Benvenuti, E. Salerno and A. Tonazzini, A mixed-annealing algorithm for edge preserving image reconstruction using a limited number of projections, Signal Process., 32 (1993), 397-408. doi: 10.1016/0165-1684(93)90009-Y.
[4]	M. Belge, M. Kilmer and E. Miller, Wavelet domain image restoration with adaptive edge-preserving regularization, IEEE Trans. Image Process., 9 (2000), 597-608. doi: 10.1109/83.841937.
[5]	M. Belge, M. Kilmer and E. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse Problems, 18 (2002), 1161-1183. doi: 10.1088/0266-5611/18/4/314.
[6]	L. Blanc-Féraud, S. Teboul, G. Aubert and M. Barlaud, Nonlinear regularization using constrained edges in image reconstruction, in Proc. IEEE Int. Conf. Image Processing, 2, Lausanne, Switzerland, (1996), 449-452. doi: 10.1109/ICIP.1996.560882.
[7]	J.-F. Cai, R. Chan and M. Nikolova, Fast two-phase image deblurring under impulse noise, J. Math. Imaging Vis., 36 (2010), 46-53. doi: 10.1007/s10851-009-0169-7.
[8]	R. Chan, Y. Dong and M. Hintermüller, An efficient two-phase $L^1$-TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), 1731-1739. doi: 10.1109/TIP.2010.2045148.
[9]	T. Chan, S. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.
[10]	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.
[11]	P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Trans. Image Process., 6 (1997), 298-311. doi: 10.1109/83.551699.
[12]	H. Choi and R. Baraniuk, Wavelet statistical models and Besov spaces, in Proc. SPIE, Wavelet Applications in Signal and Image Processing VII, vol. 3813, 1999, 489-501. doi: 10.1007/978-0-387-21579-2_2.
[13]	E. Chouzenoux, J. Idier and S. Moussaoui, A majorize-minimize strategy for subspace optimization applied to image restoration, IEEE Trans. Image Process, 20 (2011), 1517-1528. doi: 10.1109/TIP.2010.2103083.
[14]	A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45 (1992), 485-560. doi: 10.1002/cpa.3160450502.
[15]	I. Daubechies, M. Defrise and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.
[16]	A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography, IEEE Trans. Image Process, 7 (1998), 204-221. doi: 10.1109/83.660997.
[17]	G. Demoment, Image reconstruction and restoration: Overview of common estimation structures and problems, IEEE Trans. Acoust. Speech Signal Process, 37 (1989), 2024-2036. doi: 10.1109/29.45551.
[18]	R. DeVore, Nonlinear approximation, Acta Numer., 7 (1998), 51-150. doi: 10.1017/S0962492900002816.
[19]	D. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., 56 (1996), 1181-1198. doi: 10.1137/S003613999427560X.
[20]	Y. Dong, M. Hintermüller and M. Neri, An efficient primal-dual method for $L^1$-TV image restoration, SIAM J. Imaging Sci., 2 (2009), 1168-1189. doi: 10.1137/090758490.
[21]	H. Farid and E. Simoncelli, Differentiation of discrete multidimensional signals, IEEE Trans. Image Process., 13 (2004), 496-508. doi: 10.1109/TIP.2004.823819.
[22]	D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinuities, IEEE Trans. Pattern Anal. Machine Intell., 14 (1992), 367-383. doi: 10.1109/34.120331.
[23]	S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596.
[24]	S. Geman and D. McClure, Bayesian image analysis: an application to single photon emission tomography, in Proc. Stat. Comput. Section: Annual meeting of the Amer. Stat. Assoc., Las Vegas, Nevada, 1985, 12-18.
[25]	P. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Trans. Med. Imag., 9 (1990), 84-93. doi: 10.1109/42.52985.
[26]	T. Hebert and R. Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag., 8 (1989), 194-202. doi: 10.1109/42.24868.
[27]	T. Hou, S. Wang and H. Qin, Image deconvolution with multi-stage convex relaxation and its perceptual evaluation, IEEE Trans. Image Process., 20 (2011), 1057-7149. doi: 10.1109/TIP.2011.2150236.
[28]	Y. Hu and M. Jacob, Higher degree total variation (HDTV) regularization for image recovery, IEEE Trans. Image Process., 21 (2012), 2559-2571. doi: 10.1109/TIP.2012.2183143.
[29]	J. Idier, Convex half-quadratic criteria and interacting auxiliary variables for image restoration, IEEE Trans. Image Process., 10 (2001), 1001-1009. doi: 10.1109/83.931094.
[30]	K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization, SIAM J. Sci. Comput., 33 (2011), 1415-1438. doi: 10.1137/100790756.
[31]	P. Kohli, A. Shekhovtsov, C. Rother, V. Kolmogorov and P. Torr, On partial optimality in multi-label MRFs, in Proc. $25^{th}$ Int. Conf. Machine Learning, Helsinki, Finland, 2008, 480-487. doi: 10.1145/1390156.1390217.
[32]	S. Lefkimmiatis, A. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications, IEEE Trans. Image Process., 21 (2012), 983-995. doi: 10.1109/TIP.2011.2168232.
[33]	S. Li, On discontinuity-adaptive smoothness priors in computer vision, IEEE Trans. Pattern Anal. Machine Intell., 17 (1995), 576-586. doi: 10.1109/34.387504.
[34]	Y.-R. Li, L. Shen, D.-Q. Dai and B. Suter, Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise, IEEE Trans. Image Process., 20 (2011), 1822-1837. doi: 10.1109/TIP.2010.2103950.
[35]	M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis., 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7.
[36]	J. Ma and G. Plonka, The curvelet transform, IEEE Signal Processing Mag., 27 (2010), 118-133. doi: 10.1109/MSP.2009.935453.
[37]	S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674-693. doi: 10.1109/34.192463.
[38]	J. Marroquin, S. Mitter and T. Poggio, Probabilistic solution of ill-posed problems in computational vision, J. Amer. Statist. Assoc., 82 (1987), 76-89. doi: 10.1080/01621459.1987.10478393.
[39]	R. Meyer, Sufficient conditions for the convergence of monotonic mathematical programming algorithms, J. Comput. System Sci., 12 (1976), 108-121. doi: 10.1016/S0022-0000(76)80021-9.
[40]	P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors, IEEE Trans. Inform. Theory, 45 (1999), 909-919. doi: 10.1109/18.761332.
[41]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120. doi: 10.1023/B:JMIV.0000011920.58935.9c.
[42]	M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares, Multiscale Model. Simul., 4 (2005), 960-991. doi: 10.1137/040619582.
[43]	M. Nikolova, M. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Trans. Image Process., 19 (2010), 3073-3088. doi: 10.1109/TIP.2010.2052275.
[44]	S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: a black-box optimization of regularization parameters for general denoising algorithms, IEEE Trans. Image Process., 17 (2008), 1540-1554. doi: 10.1109/TIP.2008.2001404.
[45]	M. C. Robini and I. Magnin, Optimization by stochastic continuation, SIAM J. Imaging Sci., 3 (2010), 1096-1121. doi: 10.1137/090756181.
[46]	M. C. Robini, T. Rastello and I. Magnin, Simulated annealing, acceleration techniques and image restoration, IEEE Trans. Image Process., 8 (1999), 1374-1387. doi: 10.1109/83.791963.
[47]	P. Rodríguez and B. Wohlberg, Efficient minimization method for a generalized total variation functional, IEEE Trans. Image Process., 18 (2009), 322-332. doi: 10.1109/TIP.2008.2008420.
[48]	C. Rother, V. Kolmogorov, V. Lempitsky and M. Szummer, Optimizing binary MRFs via extended roof duality, in Proc. IEEE Conf. Comp. Vis. Patt. Recognition, Minneapolis, MN, 2007, 1-8. doi: 10.1109/CVPR.2007.383203.
[49]	X.-C. Tai, J. Hahn and G. Chung, A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4 (2011), 313-344. doi: 10.1137/100803730.
[50]	M. Unser, Ten good reasons for using spline wavelets, in Proc. SPIE, Wavelet Applications in Signal and Image Processing V, vol. 3169, 1997, 422-431. doi: 10.1117/12.292801.
[51]	L. Wang, T.-T. Wong, P. Heng and J. Cheng, Template-matching approach to edge detection of volume data, in Proc. Int. Workshop on Medical Imaging and Augmented Reality, Shatin, Hong Kong, 2001, 286-291.
[52]	T. Zhang, Analysis of multi-stage convex relaxation for sparse regularization, J. Mach. Learn. Res., 11 (2010), 1081-1107.
[53]	W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32. doi: 10.1137/110822268.
[54]	X. Zhu and P. Milanfar, Automatic parameter selection for denoising algorithms using a no-reference measure of image content, IEEE Trans. Image Process., 19 (2010), 3116-3132. doi: 10.1109/TIP.2010.2052820.

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