American Institute of Mathematical Sciences

Advanced
August  2013, 7(3): 1031-1050. doi: 10.3934/ipi.2013.7.1031

Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm

Xiaojing Ye 1, and  Haomin Zhou 1,

1. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30308, United States

Received  May 2012 Revised  November 2012 Published  September 2013

The primal-dual hybrid gradient (PDHG) algorithm has been successfully applied to a number of total variation (TV) based image reconstruction problems for fast numerical solutions. We show that PDHG can also effectively solve the computational problem of image inpainting in wavelet domain, where high quality images are to be recovered from incomplete wavelet coefficients due to lossy data transmission. In particular, as the original PDHG algorithm requires the orthogonality of encoding operators for optimal performance, we propose an approximated PDHG algorithm to tackle the non-orthogonality of Daubechies 7-9 wavelet which is widely used in practice. We show that this approximated version essentially alters the gradient descent direction in the original PDHG algorithm, but eliminates its orthogonality restriction and retains low computation complexity. Moreover, we prove that the sequences generated by the approximated PDHG algorithm always converge monotonically to an exact solution of the TV based image reconstruction problem starting from any initial guess. We demonstrate that the approximated PDHG algorithm also works on more general image reconstruction problems with total variation regularizations, and analyze the condition on the step sizes that guarantees the convergence.
Keywords: approximation, Total variation, wavelet, primal-dual., inpainting.
Mathematics Subject Classification: Primary: 65K15, 49K35; Secondary: 90C2.
Citation: Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031
References:
[1]

C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro and J. Verdera, Filling-in by joint interpolation of vector fields and gray levels,, Image Processing, 10 (2001), 1200.  doi: 10.1109/83.935036.  Google Scholar

[2]

J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA J. Numer. Anal., 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[3]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, In, (2000), 417.  doi: 10.1145/344779.344972.  Google Scholar

[4]

M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting,, IEEE Trans. Image Process., 12 (2003), 882.   Google Scholar

[5]

D. Bertsekas, "Parallel and Distributed Computation,", Prentice Hall, (1989).   Google Scholar

[6]

M. Burger, L. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images,, SIAM J. Imag. Sci., 2 (2009), 1129.  doi: 10.1137/080728548.  Google Scholar

[7]

J.-F. Cai, R. Chan and Z. Shen, A framelet-based image inpainting algorithm,, Appl. Comput. Harmon. Anal., 24 (2008), 131.  doi: 10.1016/j.acha.2007.10.002.  Google Scholar

[8]

J.-F. Cai, H. Ji, F. Shang and Z. Shen, Inpainting for compressed images,, Appl. Comput. Harmon. Anal., 29 (2010), 368.  doi: 10.1016/j.acha.2010.01.005.  Google Scholar

[9]

A. Chambolle, An algorithm for total variation minimization and applications,, Special issue on mathematics and image analysis. J. Math. Imaging Vis., 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[10]

A. Chambolle, V. Caselles, M. Novaga, D. Cremers and T. Pock, "An Introduction to Total Variation for Image Analysis,", Technical Report, (2009).   Google Scholar

[11]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, J. Math. Imaging Vis., 40 (2011), 120.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[12]

R. Chan, Y. Wen and A. Yip, A fast optimization transfer algorithm for image inpainting in wavelet domains,, IEEE Trans. Image Process., 18 (2009), 1467.  doi: 10.1109/TIP.2009.2019806.  Google Scholar

[13]

R. Chan, Y. Wen and A. Yip, A primal-dual method for total variation-based wavelet domain inpainting,, IEEE Trans. Image Process, 21 (2012), 106.  doi: 10.1109/TIP.2011.2159983.  Google Scholar

[14]

R. H. Chan, J. Yang and X. Yuan, Alternating direction method for image inpainting in wavelet domains,, SIAM J. Imag. Sci., 4 (2012), 807.  doi: 10.1137/100807247.  Google Scholar

[15]

T. Chan, S. Kang and J. Shen, Euler's elastica and curvature-based inpainting,, SIAM J. Appl. Math., 63 (2002), 564.  doi: 10.1137/S0036139901390088.  Google Scholar

[16]

T. Chan, S. Osher and J. Shen, The digital TV filter and nonlinear denoising,, IEEE Trans. Image Process., 10 (2001), 231.  doi: 10.1109/83.902288.  Google Scholar

[17]

T. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM J. Appl. Math., 62 (2002), 1019.  doi: 10.1137/S0036139900368844.  Google Scholar

[18]

T. Chan, J. Shen and H. Zhou, Total variation wavelet inpainting,, J. Math. Imaging Vis., 25 (2006), 107.  doi: 10.1007/s10851-006-5257-3.  Google Scholar

[19]

T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variationbased image restoration,, SIAM J. Optim., 20 (1999), 1964.  doi: 10.1137/S1064827596299767.  Google Scholar

[20]

Y. Chen, W. W. Hager, F. Huang, D. T. Phan, X. Ye and W. Yin, Fast algorithms for image reconstruction with application to partially parallel MR imaging,, SIAM J. Imag. Sci., 5 (2012), 90.  doi: 10.1137/100792688.  Google Scholar

[21]

J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,, Mathematical Programming, 55 (1992), 293.  doi: 10.1007/BF01581204.  Google Scholar

[22]

A. Efros and T. Leung, Texture synthesis by non-parametric sampling,, In, 2 (1999), 1033.  doi: 10.1109/ICCV.1999.790383.  Google Scholar

[23]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar

[24]

E. Esser, X. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for tv minimization,, SIAM J. Imag. Sci., 3 (2010), 1015.  doi: 10.1137/09076934X.  Google Scholar

[25]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,, Comput. Math. Appl., 2 (1976), 17.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar

[26]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité d'une classe de problèmes de dirichlet nonlinéaires,, RAIRO Analyse Numérique, 9 (1975), 41.   Google Scholar

[27]

T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar

[28]

B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective,, SIAM J. Imag. Sci., 5 (2011), 119.  doi: 10.1137/100814494.  Google Scholar

[29]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives,, Rev. Francaise Inform. Rech. Oper. Ser. R-3, 4 (1970), 154.   Google Scholar

[30]

S. Masnou and J.-M. Morel, Level lines based discocclusion,, In, (1998).  doi: 10.1109/ICIP.1998.999016.  Google Scholar

[31]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Model. Simul., 4 (2005), 460.  doi: 10.1137/040605412.  Google Scholar

[32]

R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control, 14 (1976), 877.  doi: 10.1137/0314056.  Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Non-linear total variation noise removal algorithm,, Physics D., 60 (1992), 259.   Google Scholar

[34]

C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising,, SIAM J. Sci. Comput., 17 (1996), 227.  doi: 10.1137/0917016.  Google Scholar

[35]

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

[36]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,, SIAM J. Imag. Sci., 3 (2010), 300.  doi: 10.1137/090767558.  Google Scholar

[37]

J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data,, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288.   Google Scholar

[38]

X. Ye, Y. Chen and F. Huang, Computational acceleration for MR image reconstruction in partially parallel imaging,, IEEE Trans. Med. Imag., 30 (2011), 1055.   Google Scholar

[39]

X. Ye, Y. Chen, W. Lin and F. Huang, Fast MR image reconstruction for partially parallel imaging with arbitrary k-space trajectories,, IEEE Trans. Med. Imag., 30 (2011), 575.   Google Scholar

[40]

X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,, SIAM J. Imag. Sci., 3 (2010), 253.  doi: 10.1137/090746379.  Google Scholar

[41]

X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration,, J. Sci. Comput., 46 (2011), 20.  doi: 10.1007/s10915-010-9408-8.  Google Scholar

[42]

X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation,, Inverse Probl. Imag., 4 (2010), 191.  doi: 10.3934/ipi.2010.4.191.  Google Scholar

[43]

M. Zhu and T. Chan, "An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration,", Technical Report 08-34, (2008), 08.   Google Scholar

show all references

References:
[1]

C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro and J. Verdera, Filling-in by joint interpolation of vector fields and gray levels,, Image Processing, 10 (2001), 1200.  doi: 10.1109/83.935036.  Google Scholar

[2]

J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA J. Numer. Anal., 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[3]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, In, (2000), 417.  doi: 10.1145/344779.344972.  Google Scholar

[4]

M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting,, IEEE Trans. Image Process., 12 (2003), 882.   Google Scholar

[5]

D. Bertsekas, "Parallel and Distributed Computation,", Prentice Hall, (1989).   Google Scholar

[6]

M. Burger, L. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images,, SIAM J. Imag. Sci., 2 (2009), 1129.  doi: 10.1137/080728548.  Google Scholar

[7]

J.-F. Cai, R. Chan and Z. Shen, A framelet-based image inpainting algorithm,, Appl. Comput. Harmon. Anal., 24 (2008), 131.  doi: 10.1016/j.acha.2007.10.002.  Google Scholar

[8]

J.-F. Cai, H. Ji, F. Shang and Z. Shen, Inpainting for compressed images,, Appl. Comput. Harmon. Anal., 29 (2010), 368.  doi: 10.1016/j.acha.2010.01.005.  Google Scholar

[9]

A. Chambolle, An algorithm for total variation minimization and applications,, Special issue on mathematics and image analysis. J. Math. Imaging Vis., 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[10]

A. Chambolle, V. Caselles, M. Novaga, D. Cremers and T. Pock, "An Introduction to Total Variation for Image Analysis,", Technical Report, (2009).   Google Scholar

[11]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, J. Math. Imaging Vis., 40 (2011), 120.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[12]

R. Chan, Y. Wen and A. Yip, A fast optimization transfer algorithm for image inpainting in wavelet domains,, IEEE Trans. Image Process., 18 (2009), 1467.  doi: 10.1109/TIP.2009.2019806.  Google Scholar

[13]

R. Chan, Y. Wen and A. Yip, A primal-dual method for total variation-based wavelet domain inpainting,, IEEE Trans. Image Process, 21 (2012), 106.  doi: 10.1109/TIP.2011.2159983.  Google Scholar

[14]

R. H. Chan, J. Yang and X. Yuan, Alternating direction method for image inpainting in wavelet domains,, SIAM J. Imag. Sci., 4 (2012), 807.  doi: 10.1137/100807247.  Google Scholar

[15]

T. Chan, S. Kang and J. Shen, Euler's elastica and curvature-based inpainting,, SIAM J. Appl. Math., 63 (2002), 564.  doi: 10.1137/S0036139901390088.  Google Scholar

[16]

T. Chan, S. Osher and J. Shen, The digital TV filter and nonlinear denoising,, IEEE Trans. Image Process., 10 (2001), 231.  doi: 10.1109/83.902288.  Google Scholar

[17]

T. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM J. Appl. Math., 62 (2002), 1019.  doi: 10.1137/S0036139900368844.  Google Scholar

[18]

T. Chan, J. Shen and H. Zhou, Total variation wavelet inpainting,, J. Math. Imaging Vis., 25 (2006), 107.  doi: 10.1007/s10851-006-5257-3.  Google Scholar

[19]

T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variationbased image restoration,, SIAM J. Optim., 20 (1999), 1964.  doi: 10.1137/S1064827596299767.  Google Scholar

[20]

Y. Chen, W. W. Hager, F. Huang, D. T. Phan, X. Ye and W. Yin, Fast algorithms for image reconstruction with application to partially parallel MR imaging,, SIAM J. Imag. Sci., 5 (2012), 90.  doi: 10.1137/100792688.  Google Scholar

[21]

J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,, Mathematical Programming, 55 (1992), 293.  doi: 10.1007/BF01581204.  Google Scholar

[22]

A. Efros and T. Leung, Texture synthesis by non-parametric sampling,, In, 2 (1999), 1033.  doi: 10.1109/ICCV.1999.790383.  Google Scholar

[23]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar

[24]

E. Esser, X. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for tv minimization,, SIAM J. Imag. Sci., 3 (2010), 1015.  doi: 10.1137/09076934X.  Google Scholar

[25]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,, Comput. Math. Appl., 2 (1976), 17.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar

[26]

R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité d'une classe de problèmes de dirichlet nonlinéaires,, RAIRO Analyse Numérique, 9 (1975), 41.   Google Scholar

[27]

T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar

[28]

B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective,, SIAM J. Imag. Sci., 5 (2011), 119.  doi: 10.1137/100814494.  Google Scholar

[29]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives,, Rev. Francaise Inform. Rech. Oper. Ser. R-3, 4 (1970), 154.   Google Scholar

[30]

S. Masnou and J.-M. Morel, Level lines based discocclusion,, In, (1998).  doi: 10.1109/ICIP.1998.999016.  Google Scholar

[31]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Model. Simul., 4 (2005), 460.  doi: 10.1137/040605412.  Google Scholar

[32]

R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control, 14 (1976), 877.  doi: 10.1137/0314056.  Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Non-linear total variation noise removal algorithm,, Physics D., 60 (1992), 259.   Google Scholar

[34]

C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising,, SIAM J. Sci. Comput., 17 (1996), 227.  doi: 10.1137/0917016.  Google Scholar

[35]

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

[36]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,, SIAM J. Imag. Sci., 3 (2010), 300.  doi: 10.1137/090767558.  Google Scholar

[37]

J. Yang, Y. Zhang and W. Yin, A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data,, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288.   Google Scholar

[38]

X. Ye, Y. Chen and F. Huang, Computational acceleration for MR image reconstruction in partially parallel imaging,, IEEE Trans. Med. Imag., 30 (2011), 1055.   Google Scholar

[39]

X. Ye, Y. Chen, W. Lin and F. Huang, Fast MR image reconstruction for partially parallel imaging with arbitrary k-space trajectories,, IEEE Trans. Med. Imag., 30 (2011), 575.   Google Scholar

[40]

X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,, SIAM J. Imag. Sci., 3 (2010), 253.  doi: 10.1137/090746379.  Google Scholar

[41]

X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration,, J. Sci. Comput., 46 (2011), 20.  doi: 10.1007/s10915-010-9408-8.  Google Scholar

[42]

X. Zhang and T. Chan, Wavelet inpainting by nonlocal total variation,, Inverse Probl. Imag., 4 (2010), 191.  doi: 10.3934/ipi.2010.4.191.  Google Scholar

[43]

M. Zhu and T. Chan, "An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration,", Technical Report 08-34, (2008), 08.   Google Scholar

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