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Twostep 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, SpringerVerlag, New York, 2002. doi: 10.1007/9780387445885. 
[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), 12061215. 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), 805812. doi: 10.1016/S02628856(02)000896. 
[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), 200217. doi: 10.1016/00415553(67)900407. 
[5] 
J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337369. doi: 10.1137/090753504. 
[6] 
Y. Cao, J. X. Yin, Q. Liu and M. H Li, A class of nonlinear parabolichyperbolic equations applied to image restoration, Nonlinear Analysis: Real World Applications, 11 (2010), 253261. 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), 8997. 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), 136152. doi: 10.1007/11585978_10. 
[9] 
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167188. doi: 10.1007/s002110050258. 
[10] 
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primaldual method for total variationbased image restoration, SIAM Journal on Scientific Computing, 20 (1999), 19641977. doi: 10.1137/S1064827596299767. 
[11] 
T. Chan, A. Marquina and P. Mulet, Highorder total variationbased image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503516. doi: 10.1137/S1064827598344169. 
[12] 
T. Chan and J. Shen, Image Processing and AnalysisVariational, 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 tvstokes model with augmented Lagrangian method for image denoising and deconvolution, Journal of Scientific Computing, 51 (2012), 505526. doi: 10.1007/s109150119519x. 
[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), 115130. doi: 10.1007/s1044400890970. 
[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), 8997. doi: 10.1007/s108510080132z. 
[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 TVStokes algorithm for image denoising, In SSVM, LNCS, 5567 (2009), 307318. 
[18] 
E. Esser, Applications of Lagrangianbased alternating direction methods and connections to split Bregman, UCLA CAM Report 0931, 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), 284292. 
[20] 
T. Goldstein and S. Osher, The split Bregman method for L1regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323343. doi: 10.1137/080725891. 
[21] 
J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TVStokes model, UCLA CAM Report 1030, 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), 247260. doi: 10.3934/naco.2013.3.247. 
[23] 
B. S. He and X. M. Yuan, On the O(1/n) convergence rate of DouglasRachford alternating direction method, SIAM J. Num. Anal., 50 (2012), 700709. 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), 367379. doi: 10.1016/j.acha.2009.05.002. 
[25] 
W. Litvinov, T. Rahman and X. C. Tai, A modified TVStokes model for image processing, SIAM J. Sci. Comput., 33 (2011), 15741597. doi: 10.1137/080727506. 
[26] 
M. Lysaker, A. Lundervold and X. C. Tai, Noise removal using fourthorder partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 15791590. 
[27] 
M. Lysaker, S. Osher and X. C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 13451357. doi: 10.1109/TIP.2004.834662. 
[28] 
E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213217. 
[29] 
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variationbased image restoration, Multiscale Model. Simul., 4 (2005), 460489. 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), 117126. 
[31] 
Z. F. Pang and Y. F. Yang, A twostep model for image denoising using a duality strategy and surface fitting, Journal of Computational and Applied Mathematics, 235 (2010), 8290. 
[32] 
J. A. Parker, R. V. Kenyon and D. E. Troxel, Comparison of interpolating methods for image resampling, IEEE Trans. Med. Imag., 2 (1983), 3139. 
[33] 
E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111123. 
[34] 
T. Rahman, X. C. Tai and S. Osher, A TVStokes denoising algorithm, In SSVM, LNCS, 4485 (2007), 473482. doi: 10.1007/9783540728238_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), 259268. doi: 10.1016/01672789(92)90242F. 
[37] 
P. V. Sankar and L. A. Ferrari, Simple algorithms and architecture for Bspline interpolation, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (1988), 271276. doi: 10.1109/34.3889. 
[38] 
X. C. Tai, S. Borok and J. Hahn, Image denoising using TVStokes equation with an orientationmatching minimization, In SSVM, LNCS, 5567 (2009), 490501. 
[39] 
X. C. Tai, S. Osher and R. Holm, Image inpainting using TVStokes equation, In Image Processing Based on Partial Differential Equations, Springer, Heidelberg, (2006), 322. 
[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), 502513. 
[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 fixedpoint iterative algorithm for image restoration using fourthorder PDE model, Appl. Num. Math., 62 (2012), 7990. doi: 10.1016/j.apnum.2011.10.004. 
[43] 
Y. F. Yang, T. T. Wu and Z. F. Pang, Imagezooming 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 l_{1}minimization with applications to compressed sensing, SIAM J. Imaging Sciences, 1 (2008), 143168. doi: 10.1137/070703983. 
[45] 
Y. You and M. Kaveh, Fourthorder partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 17231730. 
show all references
References:
[1] 
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, SpringerVerlag, New York, 2002. doi: 10.1007/9780387445885. 
[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), 12061215. 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), 805812. doi: 10.1016/S02628856(02)000896. 
[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), 200217. doi: 10.1016/00415553(67)900407. 
[5] 
J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337369. doi: 10.1137/090753504. 
[6] 
Y. Cao, J. X. Yin, Q. Liu and M. H Li, A class of nonlinear parabolichyperbolic equations applied to image restoration, Nonlinear Analysis: Real World Applications, 11 (2010), 253261. 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), 8997. 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), 136152. doi: 10.1007/11585978_10. 
[9] 
A. Chambolle and P. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167188. doi: 10.1007/s002110050258. 
[10] 
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primaldual method for total variationbased image restoration, SIAM Journal on Scientific Computing, 20 (1999), 19641977. doi: 10.1137/S1064827596299767. 
[11] 
T. Chan, A. Marquina and P. Mulet, Highorder total variationbased image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503516. doi: 10.1137/S1064827598344169. 
[12] 
T. Chan and J. Shen, Image Processing and AnalysisVariational, 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 tvstokes model with augmented Lagrangian method for image denoising and deconvolution, Journal of Scientific Computing, 51 (2012), 505526. doi: 10.1007/s109150119519x. 
[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), 115130. doi: 10.1007/s1044400890970. 
[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), 8997. doi: 10.1007/s108510080132z. 
[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 TVStokes algorithm for image denoising, In SSVM, LNCS, 5567 (2009), 307318. 
[18] 
E. Esser, Applications of Lagrangianbased alternating direction methods and connections to split Bregman, UCLA CAM Report 0931, 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), 284292. 
[20] 
T. Goldstein and S. Osher, The split Bregman method for L1regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323343. doi: 10.1137/080725891. 
[21] 
J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TVStokes model, UCLA CAM Report 1030, 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), 247260. doi: 10.3934/naco.2013.3.247. 
[23] 
B. S. He and X. M. Yuan, On the O(1/n) convergence rate of DouglasRachford alternating direction method, SIAM J. Num. Anal., 50 (2012), 700709. 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), 367379. doi: 10.1016/j.acha.2009.05.002. 
[25] 
W. Litvinov, T. Rahman and X. C. Tai, A modified TVStokes model for image processing, SIAM J. Sci. Comput., 33 (2011), 15741597. doi: 10.1137/080727506. 
[26] 
M. Lysaker, A. Lundervold and X. C. Tai, Noise removal using fourthorder partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 15791590. 
[27] 
M. Lysaker, S. Osher and X. C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 13451357. doi: 10.1109/TIP.2004.834662. 
[28] 
E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213217. 
[29] 
S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variationbased image restoration, Multiscale Model. Simul., 4 (2005), 460489. 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), 117126. 
[31] 
Z. F. Pang and Y. F. Yang, A twostep model for image denoising using a duality strategy and surface fitting, Journal of Computational and Applied Mathematics, 235 (2010), 8290. 
[32] 
J. A. Parker, R. V. Kenyon and D. E. Troxel, Comparison of interpolating methods for image resampling, IEEE Trans. Med. Imag., 2 (1983), 3139. 
[33] 
E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111123. 
[34] 
T. Rahman, X. C. Tai and S. Osher, A TVStokes denoising algorithm, In SSVM, LNCS, 4485 (2007), 473482. doi: 10.1007/9783540728238_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), 259268. doi: 10.1016/01672789(92)90242F. 
[37] 
P. V. Sankar and L. A. Ferrari, Simple algorithms and architecture for Bspline interpolation, IEEE Transactions on Pattern Analysis Machine Intelligence, 10 (1988), 271276. doi: 10.1109/34.3889. 
[38] 
X. C. Tai, S. Borok and J. Hahn, Image denoising using TVStokes equation with an orientationmatching minimization, In SSVM, LNCS, 5567 (2009), 490501. 
[39] 
X. C. Tai, S. Osher and R. Holm, Image inpainting using TVStokes equation, In Image Processing Based on Partial Differential Equations, Springer, Heidelberg, (2006), 322. 
[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), 502513. 
[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 fixedpoint iterative algorithm for image restoration using fourthorder PDE model, Appl. Num. Math., 62 (2012), 7990. doi: 10.1016/j.apnum.2011.10.004. 
[43] 
Y. F. Yang, T. T. Wu and Z. F. Pang, Imagezooming 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 l_{1}minimization with applications to compressed sensing, SIAM J. Imaging Sciences, 1 (2008), 143168. doi: 10.1137/070703983. 
[45] 
Y. You and M. Kaveh, Fourthorder partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 17231730. 
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