American Institute of Mathematical Sciences

Advanced
November  2015, 9(4): 1171-1191. doi: 10.3934/ipi.2015.9.1171

Iterative choice of the optimal regularization parameter in TV image restoration

Alina Toma 1, , Bruno Sixou 1, and  Françoise Peyrin 1,

1. 

CREATIS, CNRS UMR 5220; INSERM U1044; INSA de Lyon; Université de Lyon 1, Université de Lyon, 69621, Villeurbanne Cedex, France, France, France

Received  June 2014 Revised  April 2015 Published  October 2015

We present iterative methods for choosing the optimal regularization parameter for linear inverse problems with Total Variation regularization. This approach is based on the Morozov discrepancy principle or on a damped version of this principle and on an approximating model function for the data term. The theoretical convergence of the method of choice of the regularization parameter is demonstrated. The choice of the optimal parameter is refined with a Newton method. The efficiency of the method is illustrated on deconvolution and super-resolution experiments on different types of images. Results are provided for different levels of blur, noise and loss of spatial resolution. The damped Morozov discrepancy principle often outerperforms the approaches based on the classical Morozov principle and on the Unbiased Predictive Risk Estimator. Moreover, the proposed methods are fast schemes to select the best parameter for TV regularization.
Keywords: regularization parameter, Morozov principle, total variation regularization, Linear inverse problems, UPRE..
Mathematics Subject Classification: Primary: 65J22, 65J20, 65K10; Secondary: 52A4.
Citation: Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems & Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171
References:
[1]

M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910.  Google Scholar

[2]

M. Afonso, J. Bioucas-Dias and M. Figueiredo, An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems, IEEE Trans. Image Process., 20 (2011), 681-695. doi: 10.1109/TIP.2010.2076294.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000.  Google Scholar

[4]

J. F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hibert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6.  Google Scholar

[5]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Trans. Image Process., 17 (2008), 326-339. doi: 10.1109/TIP.2007.916051.  Google Scholar

[6]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Total Variation Super Resolution Using A Variational Approach, Proc. Int.Conf. Image Process., (2008), 641-644. Google Scholar

[7]

S. Becker, J. Bobin and E. Candes, NESTA: A fast and accurate first-order method for sparse recovery, SIAM Journal on Imaging Sciences, 4 (2011), 1-39. doi: 10.1137/090756855.  Google Scholar

[8]

T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786. doi: 10.1109/TIP.2007.906002.  Google Scholar

[9]

L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math+, 7 (1967), 200-217. Google Scholar

[10]

M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl., 20 (2004), 1411-1421. doi: 10.1088/0266-5611/20/5/005.  Google Scholar

[11]

C. A. Deledalle, S. Vaiter, J. Fadili and G. Peyre, Stein Unbiased Gradient estimator for the risk (SUGAR) for multiple parameter selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487. doi: 10.1137/140968045.  Google Scholar

[12]

Y. C. Eldar, Generalized SURE for exponential families: Application to regularization, IEEE Trans. Signal Process., 57 (2009), 471-481. doi: 10.1109/TSP.2008.2008212.  Google Scholar

[13]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[14]

E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, CAM report, 2009. Google Scholar

[15]

J. M. Fadili, G. Peyre, S. Vaiter, C. Deledalle and J. Salmon, Stable recovery with analysis decomposable priors,, preprint, ().   Google Scholar

[16]

K. Frick, D. A. Lorenz and E. Resmerita, Morozov's principle for the augmented Lagrangian method applied to linear inverse problems, Multiscale Model Simul., 9 (2011), 1528-1548. doi: 10.1137/100812835.  Google Scholar

[17]

G. H. Golub, M. T. Heath and C. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223. doi: 10.2307/1268518.  Google Scholar

[18]

M. Grasmair, Linear convergence rates for Tikhonov regularization with positively homogeneous functionals, Inverse Problems, 27 (2011), 075014, 16pp. doi: 10.1088/0266-5611/27/7/075014.  Google Scholar

[19]

P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problem, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503. doi: 10.1137/0914086.  Google Scholar

[20]

M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smooting splines, Commun. Stat. Simulat., 19 (1990), 433-450. doi: 10.1080/03610919008812864.  Google Scholar

[21]

K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., 35 (1997), 1142-1168. doi: 10.1137/S0363012995281742.  Google Scholar

[22]

K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Probl., 14 (1998), 1247-1264. doi: 10.1088/0266-5611/14/5/010.  Google Scholar

[23]

K. Kunisch, On a class of damped Morozov principles, Computing, 50 (1993), 185-198. doi: 10.1007/BF02243810.  Google Scholar

[24]

H. Liao, F. Li and M. K. Ng, Selection of regularization parameter in total variation restoration, JOSA A, 26 (2009), 2311-2320. doi: 10.1364/JOSAA.26.002311.  Google Scholar

[25]

Y.Lin, B.Wohlberg and H.Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. Google Scholar

[26]

V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.  Google Scholar

[27]

M. K. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM Journal on Scientific Computing, 32 (2010), 2710-2736. doi: 10.1137/090774823.  Google Scholar

[28]

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.  Google Scholar

[29]

S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. A. Fessler, Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672. doi: 10.1109/TIP.2012.2195015.  Google Scholar

[30]

Z. Ren, C. He and Q. Zhang, Fractional order total variation regularization for image super-resolution, Signal Process., 93 (2013), 2408-2421. Google Scholar

[31]

E. Resmerita, Regularization of ill-posed problems in Banach spaces: Convergence rates, Inverse Probl., 21 (2005), 1303-1314. doi: 10.1088/0266-5611/21/4/007.  Google Scholar

[32]

L. I. Rudin , S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. Google Scholar

[33]

M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Barucheland and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Medical Physics, 26 (1999), 2194-2204. Google Scholar

[34]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2008. Google Scholar

[35]

A. N. Tikhonov and V. Y. Arsenin, Solutions to ill-posed problems, Winston-Wiley, New York, 1977.  Google Scholar

[36]

E. Van den Berg E and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912. doi: 10.1137/080714488.  Google Scholar

[37]

S. Vaiter, A. Deledalle, G. Peyre, C. Dossal and J. Fadili, Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451. doi: 10.1016/j.acha.2012.11.006.  Google Scholar

[38]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[39]

C. Vonesch, S. Ramani and M. Unser, Risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, Inverse Problems, 27 (2011), 075014. Google Scholar

[40]

C. Vonesch, S. Ramani and M. Unser, Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, ICIP 2008, 15th IEEE International Conference on Image Processing, (2008), 665-668. Google Scholar

[41]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265.  Google Scholar

[42]

Y. W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781. doi: 10.1109/TIP.2011.2181401.  Google Scholar

[43]

J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM Journal on Imaging Sciences, 2 (2009), 569-592. doi: 10.1137/080730421.  Google Scholar

show all references

References:
[1]

M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910.  Google Scholar

[2]

M. Afonso, J. Bioucas-Dias and M. Figueiredo, An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems, IEEE Trans. Image Process., 20 (2011), 681-695. doi: 10.1109/TIP.2010.2076294.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000.  Google Scholar

[4]

J. F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hibert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6.  Google Scholar

[5]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Trans. Image Process., 17 (2008), 326-339. doi: 10.1109/TIP.2007.916051.  Google Scholar

[6]

S. D. Babacan, R. Molina and A. K. Katsaggelos, Total Variation Super Resolution Using A Variational Approach, Proc. Int.Conf. Image Process., (2008), 641-644. Google Scholar

[7]

S. Becker, J. Bobin and E. Candes, NESTA: A fast and accurate first-order method for sparse recovery, SIAM Journal on Imaging Sciences, 4 (2011), 1-39. doi: 10.1137/090756855.  Google Scholar

[8]

T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786. doi: 10.1109/TIP.2007.906002.  Google Scholar

[9]

L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math+, 7 (1967), 200-217. Google Scholar

[10]

M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl., 20 (2004), 1411-1421. doi: 10.1088/0266-5611/20/5/005.  Google Scholar

[11]

C. A. Deledalle, S. Vaiter, J. Fadili and G. Peyre, Stein Unbiased Gradient estimator for the risk (SUGAR) for multiple parameter selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487. doi: 10.1137/140968045.  Google Scholar

[12]

Y. C. Eldar, Generalized SURE for exponential families: Application to regularization, IEEE Trans. Signal Process., 57 (2009), 471-481. doi: 10.1109/TSP.2008.2008212.  Google Scholar

[13]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[14]

E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, CAM report, 2009. Google Scholar

[15]

J. M. Fadili, G. Peyre, S. Vaiter, C. Deledalle and J. Salmon, Stable recovery with analysis decomposable priors,, preprint, ().   Google Scholar

[16]

K. Frick, D. A. Lorenz and E. Resmerita, Morozov's principle for the augmented Lagrangian method applied to linear inverse problems, Multiscale Model Simul., 9 (2011), 1528-1548. doi: 10.1137/100812835.  Google Scholar

[17]

G. H. Golub, M. T. Heath and C. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223. doi: 10.2307/1268518.  Google Scholar

[18]

M. Grasmair, Linear convergence rates for Tikhonov regularization with positively homogeneous functionals, Inverse Problems, 27 (2011), 075014, 16pp. doi: 10.1088/0266-5611/27/7/075014.  Google Scholar

[19]

P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problem, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503. doi: 10.1137/0914086.  Google Scholar

[20]

M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smooting splines, Commun. Stat. Simulat., 19 (1990), 433-450. doi: 10.1080/03610919008812864.  Google Scholar

[21]

K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., 35 (1997), 1142-1168. doi: 10.1137/S0363012995281742.  Google Scholar

[22]

K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Probl., 14 (1998), 1247-1264. doi: 10.1088/0266-5611/14/5/010.  Google Scholar

[23]

K. Kunisch, On a class of damped Morozov principles, Computing, 50 (1993), 185-198. doi: 10.1007/BF02243810.  Google Scholar

[24]

H. Liao, F. Li and M. K. Ng, Selection of regularization parameter in total variation restoration, JOSA A, 26 (2009), 2311-2320. doi: 10.1364/JOSAA.26.002311.  Google Scholar

[25]

Y.Lin, B.Wohlberg and H.Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. Google Scholar

[26]

V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5280-1.  Google Scholar

[27]

M. K. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM Journal on Scientific Computing, 32 (2010), 2710-2736. doi: 10.1137/090774823.  Google Scholar

[28]

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.  Google Scholar

[29]

S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. A. Fessler, Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672. doi: 10.1109/TIP.2012.2195015.  Google Scholar

[30]

Z. Ren, C. He and Q. Zhang, Fractional order total variation regularization for image super-resolution, Signal Process., 93 (2013), 2408-2421. Google Scholar

[31]

E. Resmerita, Regularization of ill-posed problems in Banach spaces: Convergence rates, Inverse Probl., 21 (2005), 1303-1314. doi: 10.1088/0266-5611/21/4/007.  Google Scholar

[32]

L. I. Rudin , S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. Google Scholar

[33]

M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Barucheland and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Medical Physics, 26 (1999), 2194-2204. Google Scholar

[34]

O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2008. Google Scholar

[35]

A. N. Tikhonov and V. Y. Arsenin, Solutions to ill-posed problems, Winston-Wiley, New York, 1977.  Google Scholar

[36]

E. Van den Berg E and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912. doi: 10.1137/080714488.  Google Scholar

[37]

S. Vaiter, A. Deledalle, G. Peyre, C. Dossal and J. Fadili, Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451. doi: 10.1016/j.acha.2012.11.006.  Google Scholar

[38]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[39]

C. Vonesch, S. Ramani and M. Unser, Risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, Inverse Problems, 27 (2011), 075014. Google Scholar

[40]

C. Vonesch, S. Ramani and M. Unser, Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, ICIP 2008, 15th IEEE International Conference on Image Processing, (2008), 665-668. Google Scholar

[41]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265.  Google Scholar

[42]

Y. W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781. doi: 10.1109/TIP.2011.2181401.  Google Scholar

[43]

J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM Journal on Imaging Sciences, 2 (2009), 569-592. doi: 10.1137/080730421.  Google Scholar

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