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Two-step methods for image zooming using duality strategies
1. | College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China |
2. | Department of Information and Computing Science, Changsha University, Changsha, 410003, China |
3. | College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China |
References:
[1] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-0-387-44588-5. |
[2] |
T. Barbu and V. Barbu, A PDE approach to image restoration problem with observation on a meager domain, Nonlinear Analysis: Real World Applications, 13 (2012), 1206-1215.
doi: 10.1016/j.nonrwa.2011.09.014. |
[3] |
S. Battiato, G. Gallo and F. Stanco, A locally adaptive zooming algorithm for digital images, Image and Vision Computing, 20 (2002), 805-812.
doi: 10.1016/S0262-8856(02)00089-6. |
[4] |
L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.
doi: 10.1016/0041-5553(67)90040-7. |
[5] |
J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[6] |
Y. Cao, J. X. Yin, Q. Liu and M. H Li, A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Analysis: Real World Applications, 11 (2010), 253-261.
doi: 10.1016/j.nonrwa.2008.11.004. |
[7] |
A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imag. Vis., 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011325.36760.1e. |
[8] |
A. Chambolle, Total variation minimization and a class of binary MRF models, In EMMCVPR 05, Lecture Notes in Computer Sciences, 3757 (2005), 136-152.
doi: 10.1007/11585978_10. |
[9] |
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[10] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM Journal on Scientific Computing, 20 (1999), 1964-1977.
doi: 10.1137/S1064827596299767. |
[11] |
T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[12] |
T. Chan and J. Shen, Image Processing and Analysis-Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publisher, Philadelphia, 2005.
doi: 10.1137/1.9780898717877. |
[13] |
D. Q. Chen, L. Z. Cheng and F. Su, A new tv-stokes model with augmented Lagrangian method for image denoising and deconvolution, Journal of Scientific Computing, 51 (2012), 505-526.
doi: 10.1007/s10915-011-9519-x. |
[14] |
H. Z. Chen, J. P. Song and X. C. Tai, A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31 (2009), 115-130.
doi: 10.1007/s10444-008-9097-0. |
[15] |
F. F. Dong, Z. Liu, D. X. Kong and K. F. Liu, An improved LOT model for image restoration, J. Math. Imag. Vis., 34 (2009), 89-97.
doi: 10.1007/s10851-008-0132-z. |
[16] |
C. A. Elo, Image Denoising Algorithms Based on the Dual Formulation of Total Variation, Master Thesis, 2009. Available from: https://bora.uib.no/bitstream/1956/3367/1/Masterthesis_Elo.pdf. |
[17] |
C. A. Elo, A. Malyshev and T. Rahman, A dual formulation of the TV-Stokes algorithm for image denoising, In SSVM, LNCS, 5567 (2009), 307-318. |
[18] |
E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, UCLA CAM Report 09-31, 2009. |
[19] |
R. Gao, J. P. Song and X. C. Tai, Image zooming algorithm based on partial differential equations technique, International Journal of Numerical Analysis and Modeling, 6 (2009), 284-292. |
[20] |
T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[21] |
J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TV-Stokes model, UCLA CAM Report 10-30, 2010. |
[22] |
B. S. He and X. M. Yuan, Linearized alternating direction method with Gaussian back substitution for separable convex programming, Numerical Algebra, Control and Optimization, 3 (2013), 247-260.
doi: 10.3934/naco.2013.3.247. |
[23] |
B. S. He and X. M. Yuan, On the O(1/n) convergence rate of Douglas-Rachford alternating direction method, SIAM J. Num. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[24] |
R. Q. Jia, H. Zhao and W. Zhao, Convergence analysis of the Bregman method for the variational model of image denoising, Appl. Comput. Harmon. Anal., 27 (2009), 367-379.
doi: 10.1016/j.acha.2009.05.002. |
[25] |
W. Litvinov, T. Rahman and X. C. Tai, A modified TV-Stokes model for image processing, SIAM J. Sci. Comput., 33 (2011), 1574-1597.
doi: 10.1137/080727506. |
[26] |
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. |
[27] |
M. Lysaker, S. Osher and X. C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 1345-1357.
doi: 10.1109/TIP.2004.834662. |
[28] |
E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213-217. |
[29] |
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[30] |
Z. F. Pang and Y. F. Yang, A projected gradient algorithm based on the augmented Lagrangian strategy for image restoration and texture extraction, Image and Vision Computing, 29 (2011), 117-126. |
[31] |
Z. F. Pang and Y. F. Yang, A two-step model for image denoising using a duality strategy and surface fitting, Journal of Computational and Applied Mathematics, 235 (2010), 82-90. |
[32] |
J. A. Parker, R. V. Kenyon and D. E. Troxel, Comparison of interpolating methods for image resampling, IEEE Trans. Med. Imag., 2 (1983), 31-39. |
[33] |
E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111-123. |
[34] |
T. Rahman, X. C. Tai and S. Osher, A TV-Stokes denoising algorithm, In SSVM, LNCS, 4485 (2007), 473-482.
doi: 10.1007/978-3-540-72823-8_41. |
[35] |
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1969. |
[36] |
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. |
[37] |
P. V. Sankar and L. A. Ferrari, Simple algorithms and architecture for B-spline interpolation, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (1988), 271-276.
doi: 10.1109/34.3889. |
[38] |
X. C. Tai, S. Borok and J. Hahn, Image denoising using TV-Stokes equation with an orientation-matching minimization, In SSVM, LNCS, 5567 (2009), 490-501. |
[39] |
X. C. Tai, S. Osher and R. Holm, Image inpainting using TV-Stokes equation, In Image Processing Based on Partial Differential Equations, Springer, Heidelberg, (2006), 3-22. |
[40] |
X. C. Tai and C. L. Wu, Augmented Lagrangian method, dual Methods and split Bregman iteration for ROF model, In SSVM, LNCS, 5567 (2009), 502-513. |
[41] |
M. Tao and J. Yang, Alternating direction algorithms for total variation deconvolution in image reconstruction, TR0918, Department of Mathematics, Nanjing University, 2009. |
[42] |
T. T. Wu, Y. F. Yang and Z. F. Pang, A modified fixed-point iterative algorithm for image restoration using fourth-order PDE model, Appl. Num. Math., 62 (2012), 79-90.
doi: 10.1016/j.apnum.2011.10.004. |
[43] |
Y. F. Yang, T. T. Wu and Z. F. Pang, Image-zooming technique based on Bregmanized nonlocal total variation regularization, Optical Engineering, 50 (2011), 097008.
doi: 10.1117/1.3625417. |
[44] |
W. T. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for l1-minimization with applications to compressed sensing, SIAM J. Imaging Sciences, 1 (2008), 143-168.
doi: 10.1137/070703983. |
[45] |
Y. You and M. Kaveh, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730. |
show all references
References:
[1] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-0-387-44588-5. |
[2] |
T. Barbu and V. Barbu, A PDE approach to image restoration problem with observation on a meager domain, Nonlinear Analysis: Real World Applications, 13 (2012), 1206-1215.
doi: 10.1016/j.nonrwa.2011.09.014. |
[3] |
S. Battiato, G. Gallo and F. Stanco, A locally adaptive zooming algorithm for digital images, Image and Vision Computing, 20 (2002), 805-812.
doi: 10.1016/S0262-8856(02)00089-6. |
[4] |
L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.
doi: 10.1016/0041-5553(67)90040-7. |
[5] |
J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369.
doi: 10.1137/090753504. |
[6] |
Y. Cao, J. X. Yin, Q. Liu and M. H Li, A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Analysis: Real World Applications, 11 (2010), 253-261.
doi: 10.1016/j.nonrwa.2008.11.004. |
[7] |
A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imag. Vis., 20 (2004), 89-97.
doi: 10.1023/B:JMIV.0000011325.36760.1e. |
[8] |
A. Chambolle, Total variation minimization and a class of binary MRF models, In EMMCVPR 05, Lecture Notes in Computer Sciences, 3757 (2005), 136-152.
doi: 10.1007/11585978_10. |
[9] |
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.
doi: 10.1007/s002110050258. |
[10] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM Journal on Scientific Computing, 20 (1999), 1964-1977.
doi: 10.1137/S1064827596299767. |
[11] |
T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516.
doi: 10.1137/S1064827598344169. |
[12] |
T. Chan and J. Shen, Image Processing and Analysis-Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publisher, Philadelphia, 2005.
doi: 10.1137/1.9780898717877. |
[13] |
D. Q. Chen, L. Z. Cheng and F. Su, A new tv-stokes model with augmented Lagrangian method for image denoising and deconvolution, Journal of Scientific Computing, 51 (2012), 505-526.
doi: 10.1007/s10915-011-9519-x. |
[14] |
H. Z. Chen, J. P. Song and X. C. Tai, A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31 (2009), 115-130.
doi: 10.1007/s10444-008-9097-0. |
[15] |
F. F. Dong, Z. Liu, D. X. Kong and K. F. Liu, An improved LOT model for image restoration, J. Math. Imag. Vis., 34 (2009), 89-97.
doi: 10.1007/s10851-008-0132-z. |
[16] |
C. A. Elo, Image Denoising Algorithms Based on the Dual Formulation of Total Variation, Master Thesis, 2009. Available from: https://bora.uib.no/bitstream/1956/3367/1/Masterthesis_Elo.pdf. |
[17] |
C. A. Elo, A. Malyshev and T. Rahman, A dual formulation of the TV-Stokes algorithm for image denoising, In SSVM, LNCS, 5567 (2009), 307-318. |
[18] |
E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, UCLA CAM Report 09-31, 2009. |
[19] |
R. Gao, J. P. Song and X. C. Tai, Image zooming algorithm based on partial differential equations technique, International Journal of Numerical Analysis and Modeling, 6 (2009), 284-292. |
[20] |
T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[21] |
J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TV-Stokes model, UCLA CAM Report 10-30, 2010. |
[22] |
B. S. He and X. M. Yuan, Linearized alternating direction method with Gaussian back substitution for separable convex programming, Numerical Algebra, Control and Optimization, 3 (2013), 247-260.
doi: 10.3934/naco.2013.3.247. |
[23] |
B. S. He and X. M. Yuan, On the O(1/n) convergence rate of Douglas-Rachford alternating direction method, SIAM J. Num. Anal., 50 (2012), 700-709.
doi: 10.1137/110836936. |
[24] |
R. Q. Jia, H. Zhao and W. Zhao, Convergence analysis of the Bregman method for the variational model of image denoising, Appl. Comput. Harmon. Anal., 27 (2009), 367-379.
doi: 10.1016/j.acha.2009.05.002. |
[25] |
W. Litvinov, T. Rahman and X. C. Tai, A modified TV-Stokes model for image processing, SIAM J. Sci. Comput., 33 (2011), 1574-1597.
doi: 10.1137/080727506. |
[26] |
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. |
[27] |
M. Lysaker, S. Osher and X. C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 1345-1357.
doi: 10.1109/TIP.2004.834662. |
[28] |
E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213-217. |
[29] |
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.
doi: 10.1137/040605412. |
[30] |
Z. F. Pang and Y. F. Yang, A projected gradient algorithm based on the augmented Lagrangian strategy for image restoration and texture extraction, Image and Vision Computing, 29 (2011), 117-126. |
[31] |
Z. F. Pang and Y. F. Yang, A two-step model for image denoising using a duality strategy and surface fitting, Journal of Computational and Applied Mathematics, 235 (2010), 82-90. |
[32] |
J. A. Parker, R. V. Kenyon and D. E. Troxel, Comparison of interpolating methods for image resampling, IEEE Trans. Med. Imag., 2 (1983), 31-39. |
[33] |
E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111-123. |
[34] |
T. Rahman, X. C. Tai and S. Osher, A TV-Stokes denoising algorithm, In SSVM, LNCS, 4485 (2007), 473-482.
doi: 10.1007/978-3-540-72823-8_41. |
[35] |
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1969. |
[36] |
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. |
[37] |
P. V. Sankar and L. A. Ferrari, Simple algorithms and architecture for B-spline interpolation, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (1988), 271-276.
doi: 10.1109/34.3889. |
[38] |
X. C. Tai, S. Borok and J. Hahn, Image denoising using TV-Stokes equation with an orientation-matching minimization, In SSVM, LNCS, 5567 (2009), 490-501. |
[39] |
X. C. Tai, S. Osher and R. Holm, Image inpainting using TV-Stokes equation, In Image Processing Based on Partial Differential Equations, Springer, Heidelberg, (2006), 3-22. |
[40] |
X. C. Tai and C. L. Wu, Augmented Lagrangian method, dual Methods and split Bregman iteration for ROF model, In SSVM, LNCS, 5567 (2009), 502-513. |
[41] |
M. Tao and J. Yang, Alternating direction algorithms for total variation deconvolution in image reconstruction, TR0918, Department of Mathematics, Nanjing University, 2009. |
[42] |
T. T. Wu, Y. F. Yang and Z. F. Pang, A modified fixed-point iterative algorithm for image restoration using fourth-order PDE model, Appl. Num. Math., 62 (2012), 79-90.
doi: 10.1016/j.apnum.2011.10.004. |
[43] |
Y. F. Yang, T. T. Wu and Z. F. Pang, Image-zooming technique based on Bregmanized nonlocal total variation regularization, Optical Engineering, 50 (2011), 097008.
doi: 10.1117/1.3625417. |
[44] |
W. T. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for l1-minimization with applications to compressed sensing, SIAM J. Imaging Sciences, 1 (2008), 143-168.
doi: 10.1137/070703983. |
[45] |
Y. You and M. Kaveh, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730. |
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