Fast numerical schemes related to curvature minimization: a brief and elementary review

Tai, Xue-Cheng

doi:10.5802/acirm.52

Tai, Xue-Cheng ¹

¹ Department of Mathematics University of Bergen, Bergen, Norway

Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 17-30.

Résumé

We will treat variational models that use Euler’s elastica and related higher order derivatives as regularizers. These models normally lead to higher order partial differential equations with complicated nonlinearities. It is difficult to solve these equations numerically. Recently, some fast numerical techniques have been proposed that can solve these equations with very good numerical speed. We will try to explain the essential ideas of these numerical techniques and point to some central implementation details for these algorithms.

Publié le : 2014-11-13

Zbl

DOI : 10.5802/acirm.52

Classification : 00X99
Mots clés : variaitonal models, curvature minimization, Augmented Lagrangian methods

Affiliations des auteurs :

Tai, Xue-Cheng ¹

¹ Department of Mathematics University of Bergen, Bergen, Norway

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Tai, Xue-Cheng. Fast numerical schemes related to curvature minimization: a brief and elementary review. Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 17-30. doi : 10.5802/acirm.52. http://www.numdam.org/articles/10.5802/acirm.52/

Bibliographie
Cité par

[1] Ambrosio, Luigi; Masnou, Simon A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries, Volume 5 (2003) no. 1, pp. 63-82 | DOI | MR | Zbl

[2] Bae, Egil; Yuan, Jing; Tai, Xue-Cheng Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach, International Journal of Computer Vision (2010), pp. 1-18 | DOI | Zbl

[3] Bredies, Kristian Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty, SFB Report, Volume 6 (2012)

[4] Bredies, Kristian; Pock, Thomas; Wirth, Benedikt Convex relaxation of a class of vertex penalizing functionals, Journal of mathematical imaging and vision, Volume 47 (2013) no. 3, pp. 278-302 | DOI | MR | Zbl

[5] Brito-Loeza, Carlos; Chen, Ke On high-order denoising models and fast algorithms for vector-valued images, Image Processing, IEEE Transactions on, Volume 19 (2010) no. 6, pp. 1518-1527 | DOI | MR | Zbl

[6] Calder, Jeff; Mansouri, A; Yezzi, Anthony Image sharpening via Sobolev gradient flows, SIAM Journal on Imaging Sciences, Volume 3 (2010) no. 4, pp. 981-1014 | DOI | MR | Zbl

[7] Chambolle, Antonin; Lions, Pierre-Louis Image recovery via total variation minimization and related problems, Numerische Mathematik, Volume 76 (1997) no. 2, pp. 167-188 | DOI | MR | Zbl

[8] Chan, RH; Lanza, A; Morigi, S; Sgallari, F An Adaptive Strategy for the Restoration of Textured Images using Fractional Order Regularization., Numerical Mathematics: Theory, Methods & Applications, Volume 6 (2013) no. 1 | MR | Zbl

[9] Chan, T.; Vese, L.A. Active contours without edges, IEEE Trans Image Proc., Volume 10 (2001), pp. 266-277 | DOI | Zbl

[10] Chan, Tony F.; Esedoglu, Selim; Nikolova, Mila Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., Volume 66 (2006) no. 5, p. 1632-1648 (electronic) | DOI | MR | Zbl

[11] Chan, Tony F; Kang, Sung Ha; Shen, Jianhong Euler’s elastica and curvature-based inpainting, SIAM Journal on Applied Mathematics (2002), pp. 564-592 | MR | Zbl

[12] Cuesta, Eduardo; Kirane, Mokhtar; Malik, Salman A Image structure preserving denoising using generalized fractional time integrals, Signal Processing, Volume 92 (2012) no. 2, pp. 553-563 | DOI

[13] Didas, Stephan; Weickert, Joachim; Burgeth, Bernhard Properties of higher order nonlinear diffusion filtering, Journal of mathematical imaging and vision, Volume 35 (2009) no. 3, pp. 208-226 | DOI | MR

[14] Duan, Yuping; Huang, Weimin A fixed-point augmented Lagrangian method for total variation minimization problems, Journal of Visual Communication and Image Representation, Volume 24 (2013) no. 7, pp. 1168-1181 | DOI

[15] Glowinski, R.; Le Tallec, P. Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, 9, Society for Industrial Mathematics, 1989 | MR | Zbl

[16] Goldstein, T.; Osher, S. The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, Volume 2 (2009) no. 2, pp. 323-343 | DOI | MR | Zbl

[17] Greer, John B; Bertozzi, Andrea L Traveling wave solutions of fourth order PDEs for image processing, SIAM Journal on Mathematical Analysis, Volume 36 (2004) no. 1, pp. 38-68 | DOI | MR | Zbl

[18] Guidotti, Patrick; Longo, Kate Two enhanced fourth order diffusion models for image denoising, Journal of Mathematical Imaging and Vision, Volume 40 (2011) no. 2, pp. 188-198 | DOI | MR | Zbl

[19] Hu, Langhua; Chen, Duan; Wei, Guo-Wei High-order fractional partial differential equation transform for molecular surface construction, Molecular based mathematical biology, Volume 1 (2013), pp. 1-25 | Zbl

[20] Jidesh, P; George, Santhosh Fourth–order variational model with local–constraints for denoising images with textures, International Journal of Computational Vision and Robotics, Volume 2 (2011) no. 4, pp. 330-340 | DOI

[21] Kass, Michael; Witkin, Andrew; Terzopoulos, Demetri Snakes: Active contour models, International journal of computer vision, Volume 1 (1988) no. 4, pp. 321-331 | DOI

[22] Kimmel, Ron; Malladi, Ravi; Sochen, Nir Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images, International Journal of Computer Vision, Volume 39 (2000) no. 2, pp. 111-129 | DOI | Zbl

[23] Lie, Johan; Lysaker, Marius; Tai, Xue A Binary Level Set Model and some Applications to Mumford-Shah Image Segmentation, IEEE Transactions on Image Processing, Volume 15 (2006) no. 5, pp. 1171-1181 | Zbl

[24] Lysaker, M.; Lundervold, A.; Tai, X.C. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, Image Processing, IEEE Transactions on, Volume 12 (2003) no. 12, pp. 1579-1590 | DOI | Zbl

[25] Mardal, Kent Andre; Tai, Xue-Cheng; Winther, Ragnar A robust finite element method for Darcy-Stokes flow, SIAM Journal on Numerical Analysis (2003), pp. 1605-1631 | Zbl

[26] Min, Lihua; Yang, Xiaoping; Gui, Changfeng ENTROPY ESTIMATES AND LARGE-TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER NONLINEAR DEGENERATE EQUATION, Communications in Contemporary Mathematics, Volume 15 (2013) no. 04 | MR | Zbl

[27] Min, Lihua; Yang, Xiaoping; Ye, Dong Well-posedness for a fourth order nonlinear equation related to image processing, Nonlinear Analysis: Real World Applications (2013) | Zbl

[28] Mumford, D; Nitzberg, M; Shiota, T Filtering, Segmentation and Depth, Lecture Notes in Computer Science, Volume 662 (1993) | MR | Zbl

[29] Mumford, D.; Shah, J. Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math, 42, Volume 42 (1989), pp. 577-685 | DOI | MR | Zbl

[30] Nadernejad, Ehsan; Forchhammer, Søren Wavelet-based image enhancement using fourth order PDE, Intelligent Signal Processing (WISP), 2011 IEEE 7th International Symposium on, IEEE (2011), pp. 1-6

[31] Papafitsorosand Schönlieb, Carola-Bibiane A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, Volume 48 (2014) no. 2, pp. 308-338 | MR

[32] Rosman, Guy; Bronstein, Alex M; Bronstein, Michael M; Tai, Xue-Cheng; Kimmel, Ron Group-Valued regularization for analysis of articulated motion, Computer Vision–ECCV 2012. Workshops and Demonstrations, Springer (2012), pp. 52-62 | DOI

[33] Rosman, Guy; Wang, Yu; Tai, Xue-Cheng; Kimmel, Ron; Bruckstein, Alfred M Fast regularization of matrix-valued images, Computer Vision–ECCV 2012, Springer, 2012, pp. 173-186 | DOI

[34] Rudin, L.; Osher, S.; Fatemi, E. Nonlinear Total Variation based noise removal algorithms, Physica D, Volume 60 (1992), pp. 259-268 | DOI | MR | Zbl

[35] Schoenemann, T.; Kahl, F.; Cremers, D. Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation, Computer Vision, 2009 IEEE 12th International Conference on, IEEE (2009), pp. 17-23 | DOI

[36] Schönlieb, Carola-bibiane; Bertozzi, Andrea Unconditionally stable schemes for higher order inpainting, Communications in Mathematical Sciences, Volume 9 (2011) no. 2, pp. 413-457 | MR | Zbl

[37] Tai, X.C.; Hahn, J.; Chung, G.J. A Fast Algorithm for Euler’s Elastica Model Using Augmented Lagrangian Method, SIAM Journal on Imaging Sciences, Volume 4 (2011), 313 pages | MR

[38] Tai, Xue-Cheng; Wu, Chunlin Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, Scale Space and Variational Methods in Computer Vision (2009), pp. 502-513 | Zbl

[39] Wekipedia Co-area formula (http://en.wikipedia.org/wiki/Coarea_formula), 2013 http://en.wikipedia.org/wiki/Coarea_formula

[40] Wikipedia Total variation (http://en.wikipedia.org/wiki/Total_variation), 2014 http://en.wikipedia.org/wiki/Total_variation

[41] Wu, C.; Tai, X.C. Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, Volume 3 (2010) no. 3, pp. 300-339 | DOI | MR | Zbl

[42] Yang, Fenlin; Chen, Ke; Yu, Bo EFFICIENT HOMOTOPY SOLUTION AND A CONVEX COMBINATION OF ROF AND LLT MODELS FOR IMAGE RESTORATION., International Journal of Numerical Analysis & Modeling, Volume 9 (2012) no. 4 | MR | Zbl

[43] Yin, W.; Osher, S.; Goldfarb, D.; Darbon, J. Bregman iterative algorithms for $∖$ ell_1-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, Volume 1 (2008) no. 1, pp. 143-168 | DOI | Zbl

[44] Yuan, J.; Bae, E.; Tai, X.C. A study on continuous max-flow and min-cut approaches, Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, IEEE (2010), pp. 2217-2224 | DOI

[45] Yuan, J.; Bae, E.; Tai, X.C.; Boykov, Y. A continuous max-flow approach to potts model, Computer Vision–ECCV 2010 (2010), pp. 379-392 | DOI

[46] Yuan, Jing; Bae, Egil; Tai, Xue-Cheng; Boykov, Yuri A spatially continuous max-flow and min-cut framework for binary labeling problems, Numerische Mathematik (2013), pp. 1-29 | Zbl

[47] Yuan, Jing; Shi, Juan; Tai, Xue-Cheng A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation, East Asian Journal on Applied Mathematics (to appear) | Zbl

[48] Zeng, Weili; Lu, Xiaobo; Tan, Xianghua Non-linear fourth-order telegraph-diffusion equation for noise removal, IET Image Processing, Volume 7 (2013) no. 4, pp. 335-342 | DOI | MR

[49] Zhu, W.; Chan, T. Image denoising using mean curvature of image surface, SIAM Journal on Imaging Sciences, Volume 5 (2012) no. 1, pp. 1-32 | DOI | MR | Zbl

[50] Zhu, W.; Chan, T.; Esedoglu, Selim Segmentation with depth: A level set approach, SIAM Journal on Scientific Computing, Volume 28 (2006) no. 5, pp. 1957-1973 | DOI | MR | Zbl

[51] Zhu, Wei; Chan, Tony A variational model for capturing illusory contours using curvature, Journal of Mathematical Imaging and Vision, Volume 27 (2007) no. 1, pp. 29-40 | DOI | MR

[52] Zhu, Wei; Tai, Xue-Cheng; Chan, Tony Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging, Volume 7 (2012) no. 3, pp. 1075-1097 | MR | Zbl

[53] Zhu, Wei; Tai, Xue-Cheng; Chan, Tony Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging, Volume 7 (2013) no. 4, pp. 1409-1432 | DOI | MR | Zbl

[54] Zhu, Wei; Tai, Xue-Cheng; Chan, Tony Image Segmentation Using Euler’s Elastica as the Regularization, Journal of Scientific Computing, Volume 57 (2013) no. 2, pp. 414-438 | DOI | MR | Zbl

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