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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 |
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, 1976 ().
|
[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, 1976 ().
|
[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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