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2014, 4(3): 209-225. doi: 10.3934/naco.2014.4.209

Two-step methods for image zooming using duality strategies

Tingting Wu 1, , Yufei Yang 2, and  Huichao Jing 3,

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

Received  November 2013 Revised  July 2014 Published  September 2014

In this paper we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. The second method is based on the TV-Stokes model. By smoothing the tangential vector and imposing the divergence free condition, we propose an image zooming method based on the TV-Stokes model using the dual formulation. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
Keywords: Two-step method, TV-Stokes model, split Bregman iteration, image zooming., duality strategy.
Mathematics Subject Classification: Primary: 65K10, 65C20; Secondary: 94A0.
Citation: Tingting Wu, Yufei Yang, Huichao Jing. Two-step methods for image zooming using duality strategies. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 209-225. doi: 10.3934/naco.2014.4.209
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, UCLA CAM Report 09-31, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TV-Stokes model, UCLA CAM Report 10-30, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213-217. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111-123. Google Scholar

[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.  Google Scholar

[35]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

M. Tao and J. Yang, Alternating direction algorithms for total variation deconvolution in image reconstruction, TR0918, Department of Mathematics, Nanjing University, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

Y. You and M. Kaveh, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, UCLA CAM Report 09-31, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Hahn, C. L. Wu and X. C. Tai, Augmented Lagrangian method for generalized TV-Stokes model, UCLA CAM Report 10-30, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

E. Maeland, On the comparison of interpolation methods, IEEE Trans. Med. Imag., 7 (1988), 213-217. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

E. Polidori and J. L. Dugelay, Zooming using iterated function systems, Fractals, 5 (1997), 111-123. Google Scholar

[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.  Google Scholar

[35]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

M. Tao and J. Yang, Alternating direction algorithms for total variation deconvolution in image reconstruction, TR0918, Department of Mathematics, Nanjing University, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

Y. You and M. Kaveh, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730. Google Scholar

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