# American Institute of Mathematical Sciences

November  2013, 7(4): 1409-1432. doi: 10.3934/ipi.2013.7.1409

## Augmented Lagrangian method for a mean curvature based image denoising model

 1 Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487 2 Department of Mathematics, University of Bergen, Bergen 5007, Norway 3 Office of the President, Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowlon, Hong Kong, China

Received  October 2011 Revised  October 2012 Published  November 2013

High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
Citation: Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems and Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409
##### References:
 [1] L. Ambrosio and S. Masnou, A direct variational approach to a problem arising in image reconstruction, Interfaces Free Bound., 5 (2003), 63-81. doi: 10.4171/IFB/72. [2] L. Ambrosio and S. Masnou, On a variational problem arising in image reconstruction, Free Boundary Problems (Trento, 2002), Internat. Ser. Numer. Math., 147, Birkhäuser, Basel, (2004), 17-26. [3] L. Alvarez, F. Guichard, P. L. Lions and J. M. Morel, Axioms and fundamental equations of image-processing, Archive for Rational Mechanics and Analysis, 123 (1993), 199-257. doi: 10.1007/BF00375127. [4] G. Aubert and L. Vese, A variational method in image recovery, SIAM J. Numer. Anal. 34 (1987), 1948-1979. doi: 10.1137/S003614299529230X. [5] A. L. Bertozzi and J. B. Greer, Low curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Comm. Pure Appl. Math., 57 (2004), 764-790. doi: 10.1002/cpa.20019. [6] T. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297. [7] T. Chen, W. Yin, X. S. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition, IEEE. Trans. Pattern Anal. Mach. Intell., 28 (2006), 1519-1524. doi: 10.1109/TPAMI.2006.195. [8] A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [9] T. Chan, S. H. Kang and J. H. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088. [10] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [11] M. P. do Carmo, Differential geometry of curves and surfaces, Translated From the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. viii+503 pp. [12] M. Elsey and S. Esedoglu, Analogue of the total variation denoising model in the context of geometry processing, SIAM J. Multiscale Modeling and Simulation, 7 (2009), 1549-1573. doi: 10.1137/080736612. [13] S. Esedo$\overlineg$lu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904. [14] T. Goldstein and S. Osher, The split bregman method for L1 regularized problems, SIAM J. on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [15] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. doi: 10.1137/S0036141003427373. [16] J. B. Greer, A. L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comp. Phys., 216 (2006), 216-246. doi: 10.1016/j.jcp.2005.11.031. [17] R. Kimmel, R. Malladi and N. Sochen, Image processing via the Beltrami Operator, Proceedings of Asian Conference on Computer Vision, LNCS 1351 (1998), 574-581, Hong Kong. doi: 10.1007/3-540-63930-6_169. [18] R. Kimmel, R. Malladi, and N. Sochen, Images as embedded maps and minimal surfaces: Movies, color, texture and volumetric medical images, International Journal of Computer Vision, 39 (2000), 111-129. [19] P. L. Lions and B. Mercier, Splitting algorithms for the sume of two nonlinear opertors, SIAM J. Numer. Anal., 16 (1979), 964-979. doi: 10.1137/0716071. [20] 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. doi: 10.1109/TIP.2003.819229. [21] 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. [22] S. Masnou, Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76. doi: 10.1109/83.982815. [23] S. Masnou and J. M. Morel, Level lines based disocclusion, Proc. IEEE Int. Conf. on Image Processing, Chicago, IL, (1998), 259-263. doi: 10.1109/ICIP.1998.999016. [24] Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lecture Series, Amer. Math. Soc., 22 (2002). [25] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [26] J. M. Morel and S. Solimini, Variational methods in image segmentation, Birkhauser, Boston, (1995). doi: 10.1007/978-1-4684-0567-5. [27] M. Nitzberg, D. Mumford and T. Shiota, Filtering, segmentation, and depth, Lecture Notes in Computer Science, 662 (1993), Springer Verlag, Berlin. doi: 10.1007/3-540-56484-5. [28] S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, SIAM Multiscale Model., (2003), 349-370. doi: 10.1137/S1540345902416247. [29] P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion, IEEE Trans. on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205. [30] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [31] N. Sochen, R. Kimmel and R. Malladi, A geometrical framework for low level vision, IEEE Trans. on Image Process., 7 (1998), 310-318. doi: 10.1109/83.661181. [32] X. C. Tai, J. Hahn and G. J. Chung, A fast algorithm for Euler's Elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, 4 (2011), 313-344. doi: 10.1137/100803730. [33] T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Transactions on Graphics, 22 (2003), 1012-1033. doi: 10.1145/944020.944024. [34] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillatory patterns im image processing, SINUM., 40 (2003), 2085-2104. doi: 10.1137/S0036142901396715. [35] 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. Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558. [36] W. Yin, T. Chen, X. S. Zhou and A. Chakraborty, Background correction for cDNA microarray image using the TV+L1 model, Bioinformatics, 21 (2005), 2410-2416. doi: 10.1093/bioinformatics/bti341. [37] W. Zhu and T. Chan, A variational model for capturing illusory contours using curvature, J. Math. Imaging Vision, 27 (2007), 29-40. doi: 10.1007/s10851-006-9695-8. [38] W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, 5 (2012), 1-32. doi: 10.1137/110822268. [39] W. Zhu, T. Chan and S. Esedoglu, Segmentation with depth: A level set approach, SIAM J. Sci. Comput., 28 (2006), 1957-1973. doi: 10.1137/050622213.

show all references

##### References:
 [1] L. Ambrosio and S. Masnou, A direct variational approach to a problem arising in image reconstruction, Interfaces Free Bound., 5 (2003), 63-81. doi: 10.4171/IFB/72. [2] L. Ambrosio and S. Masnou, On a variational problem arising in image reconstruction, Free Boundary Problems (Trento, 2002), Internat. Ser. Numer. Math., 147, Birkhäuser, Basel, (2004), 17-26. [3] L. Alvarez, F. Guichard, P. L. Lions and J. M. Morel, Axioms and fundamental equations of image-processing, Archive for Rational Mechanics and Analysis, 123 (1993), 199-257. doi: 10.1007/BF00375127. [4] G. Aubert and L. Vese, A variational method in image recovery, SIAM J. Numer. Anal. 34 (1987), 1948-1979. doi: 10.1137/S003614299529230X. [5] A. L. Bertozzi and J. B. Greer, Low curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Comm. Pure Appl. Math., 57 (2004), 764-790. doi: 10.1002/cpa.20019. [6] T. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837. doi: 10.1137/040604297. [7] T. Chen, W. Yin, X. S. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition, IEEE. Trans. Pattern Anal. Mach. Intell., 28 (2006), 1519-1524. doi: 10.1109/TPAMI.2006.195. [8] A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258. [9] T. Chan, S. H. Kang and J. H. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088. [10] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [11] M. P. do Carmo, Differential geometry of curves and surfaces, Translated From the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. viii+503 pp. [12] M. Elsey and S. Esedoglu, Analogue of the total variation denoising model in the context of geometry processing, SIAM J. Multiscale Modeling and Simulation, 7 (2009), 1549-1573. doi: 10.1137/080736612. [13] S. Esedo$\overlineg$lu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904. [14] T. Goldstein and S. Osher, The split bregman method for L1 regularized problems, SIAM J. on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [15] J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68. doi: 10.1137/S0036141003427373. [16] J. B. Greer, A. L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries, J. Comp. Phys., 216 (2006), 216-246. doi: 10.1016/j.jcp.2005.11.031. [17] R. Kimmel, R. Malladi and N. Sochen, Image processing via the Beltrami Operator, Proceedings of Asian Conference on Computer Vision, LNCS 1351 (1998), 574-581, Hong Kong. doi: 10.1007/3-540-63930-6_169. [18] R. Kimmel, R. Malladi, and N. Sochen, Images as embedded maps and minimal surfaces: Movies, color, texture and volumetric medical images, International Journal of Computer Vision, 39 (2000), 111-129. [19] P. L. Lions and B. Mercier, Splitting algorithms for the sume of two nonlinear opertors, SIAM J. Numer. Anal., 16 (1979), 964-979. doi: 10.1137/0716071. [20] 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. doi: 10.1109/TIP.2003.819229. [21] 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. [22] S. Masnou, Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76. doi: 10.1109/83.982815. [23] S. Masnou and J. M. Morel, Level lines based disocclusion, Proc. IEEE Int. Conf. on Image Processing, Chicago, IL, (1998), 259-263. doi: 10.1109/ICIP.1998.999016. [24] Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lecture Series, Amer. Math. Soc., 22 (2002). [25] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [26] J. M. Morel and S. Solimini, Variational methods in image segmentation, Birkhauser, Boston, (1995). doi: 10.1007/978-1-4684-0567-5. [27] M. Nitzberg, D. Mumford and T. Shiota, Filtering, segmentation, and depth, Lecture Notes in Computer Science, 662 (1993), Springer Verlag, Berlin. doi: 10.1007/3-540-56484-5. [28] S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, SIAM Multiscale Model., (2003), 349-370. doi: 10.1137/S1540345902416247. [29] P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion, IEEE Trans. on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205. [30] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [31] N. Sochen, R. Kimmel and R. Malladi, A geometrical framework for low level vision, IEEE Trans. on Image Process., 7 (1998), 310-318. doi: 10.1109/83.661181. [32] X. C. Tai, J. Hahn and G. J. Chung, A fast algorithm for Euler's Elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, 4 (2011), 313-344. doi: 10.1137/100803730. [33] T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Transactions on Graphics, 22 (2003), 1012-1033. doi: 10.1145/944020.944024. [34] L. Vese and S. Osher, Modeling textures with total variation minimization and oscillatory patterns im image processing, SINUM., 40 (2003), 2085-2104. doi: 10.1137/S0036142901396715. [35] 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. Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558. [36] W. Yin, T. Chen, X. S. Zhou and A. Chakraborty, Background correction for cDNA microarray image using the TV+L1 model, Bioinformatics, 21 (2005), 2410-2416. doi: 10.1093/bioinformatics/bti341. [37] W. Zhu and T. Chan, A variational model for capturing illusory contours using curvature, J. Math. Imaging Vision, 27 (2007), 29-40. doi: 10.1007/s10851-006-9695-8. [38] W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, 5 (2012), 1-32. doi: 10.1137/110822268. [39] W. Zhu, T. Chan and S. Esedoglu, Segmentation with depth: A level set approach, SIAM J. Sci. Comput., 28 (2006), 1957-1973. doi: 10.1137/050622213.
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