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

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

Published online:

DOI: 10.5802/acirm.52

Classification: 00X99

Keywords: variaitonal models, curvature minimization, Augmented Lagrangian methods

Author's affiliations:

DOI: 10.5802/acirm.52

Classification: 00X99

Keywords: variaitonal models, curvature minimization, Augmented Lagrangian methods

Author's affiliations:

Xue-Cheng Tai ^{1}
^{1} Department of Mathematics University of Bergen, Bergen, Norway

@article{ACIRM_2013__3_1_17_0, author = {Xue-Cheng Tai}, title = {Fast numerical schemes related to curvature minimization: a brief and elementary review}, journal = {Actes des rencontres du CIRM}, pages = {17--30}, publisher = {CIRM}, volume = {3}, number = {1}, year = {2013}, doi = {10.5802/acirm.52}, zbl = {06938600}, language = {en}, url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.52/} }

TY - JOUR TI - Fast numerical schemes related to curvature minimization: a brief and elementary review JO - Actes des rencontres du CIRM PY - 2013 DA - 2013/// SP - 17 EP - 30 VL - 3 IS - 1 PB - CIRM UR - https://acirm.centre-mersenne.org/articles/10.5802/acirm.52/ UR - https://zbmath.org/?q=an%3A06938600 UR - https://doi.org/10.5802/acirm.52 DO - 10.5802/acirm.52 LA - en ID - ACIRM_2013__3_1_17_0 ER -

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

[1] Luigi Ambrosio; Simon Masnou A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries, Volume 5 (2003) no. 1, pp. 63-82 | Article | MR: 1959769 | Zbl: 1029.49037

[2] Egil Bae; Jing Yuan; Xue-Cheng Tai Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach, International Journal of Computer Vision (2010), pp. 1-18 | Article | Zbl: 1235.68244

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

[4] Kristian Bredies; Thomas Pock; Benedikt Wirth Convex relaxation of a class of vertex penalizing functionals, Journal of mathematical imaging and vision, Volume 47 (2013) no. 3, pp. 278-302 | Article | MR: 3097120 | Zbl: 1293.49063

[5] Carlos Brito-Loeza; Ke Chen 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 | Article | MR: 2778518 | Zbl: 1371.94060

[6] Jeff Calder; A Mansouri; Anthony Yezzi Image sharpening via Sobolev gradient flows, SIAM Journal on Imaging Sciences, Volume 3 (2010) no. 4, pp. 981-1014 | Article | MR: 2763705 | Zbl: 1202.94019

[7] Antonin Chambolle; Pierre-Louis Lions Image recovery via total variation minimization and related problems, Numerische Mathematik, Volume 76 (1997) no. 2, pp. 167-188 | Article | MR: 1440119 | Zbl: 0874.68299

[8] RH Chan; A Lanza; S Morigi; F Sgallari An Adaptive Strategy for the Restoration of Textured Images using Fractional Order Regularization., Numerical Mathematics: Theory, Methods & Applications, Volume 6 (2013) no. 1 | MR: 3011641 | Zbl: 1289.68196

[9] T. Chan; L.A. Vese Active contours without edges, IEEE Trans Image Proc., Volume 10 (2001), pp. 266-277 | Article | Zbl: 1039.68779

[10] Tony F. Chan; Selim Esedoglu; Mila Nikolova Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., Volume 66 (2006) no. 5, p. 1632-1648 (electronic) | Article | MR: 2246072 | Zbl: 1117.94002

[11] Tony F Chan; Sung Ha Kang; Jianhong Shen Euler’s elastica and curvature-based inpainting, SIAM Journal on Applied Mathematics (2002), pp. 564-592 | MR: 1951951 | Zbl: 1028.68185

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

[13] Stephan Didas; Joachim Weickert; Bernhard Burgeth Properties of higher order nonlinear diffusion filtering, Journal of mathematical imaging and vision, Volume 35 (2009) no. 3, pp. 208-226 | Article | MR: 2533935

[14] Yuping Duan; Weimin Huang 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 | Article

[15] R. Glowinski; P. Le Tallec Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, 9, Society for Industrial Mathematics, 1989 | MR: 1060954 | Zbl: 0698.73001

[16] T. Goldstein; S. Osher The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, Volume 2 (2009) no. 2, pp. 323-343 | Article | MR: 2496060 | Zbl: 1177.65088

[17] John B Greer; Andrea L Bertozzi Traveling wave solutions of fourth order PDEs for image processing, SIAM Journal on Mathematical Analysis, Volume 36 (2004) no. 1, pp. 38-68 | Article | MR: 2083852 | Zbl: 1082.35080

[18] Patrick Guidotti; Kate Longo Two enhanced fourth order diffusion models for image denoising, Journal of Mathematical Imaging and Vision, Volume 40 (2011) no. 2, pp. 188-198 | Article | MR: 2782126 | Zbl: 1255.68235

[19] Langhua Hu; Duan Chen; Guo-Wei Wei High-order fractional partial differential equation transform for molecular surface construction, Molecular based mathematical biology, Volume 1 (2013), pp. 1-25 | Zbl: 1277.35338

[20] P Jidesh; Santhosh George 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 | Article

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

[22] Ron Kimmel; Ravi Malladi; Nir Sochen 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 | Article | Zbl: 1060.68656

[23] Johan Lie; Marius Lysaker; Xue Tai 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: 1286.94018

[24] M. Lysaker; A. Lundervold; X.C. Tai 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 | Article | Zbl: 1286.94020

[25] Kent Andre Mardal; Xue-Cheng Tai; Ragnar Winther A robust finite element method for Darcy-Stokes flow, SIAM Journal on Numerical Analysis (2003), pp. 1605-1631 | Zbl: 1037.65120

[26] Lihua Min; Xiaoping Yang; Changfeng Gui 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: 3073448 | Zbl: 1273.35153

[27] Lihua Min; Xiaoping Yang; Dong Ye Well-posedness for a fourth order nonlinear equation related to image processing, Nonlinear Analysis: Real World Applications (2013) | Zbl: 1302.35208

[28] D Mumford; M Nitzberg; T Shiota Filtering, Segmentation and Depth, Lecture Notes in Computer Science, Volume 662 (1993) | MR: 1226232 | Zbl: 0801.68171

[29] D. Mumford; J. Shah Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math, 42, Volume 42 (1989), pp. 577-685 | Article | MR: 997568 | Zbl: 0691.49036

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

[31] Carola-Bibiane Papafitsorosand Schönlieb 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: 3152107

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

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

[34] L. Rudin; S. Osher; E. Fatemi Nonlinear Total Variation based noise removal algorithms, Physica D, Volume 60 (1992), pp. 259-268 | Article | MR: 3363401 | Zbl: 0780.49028

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

[36] Carola-bibiane Schönlieb; Andrea Bertozzi Unconditionally stable schemes for higher order inpainting, Communications in Mathematical Sciences, Volume 9 (2011) no. 2, pp. 413-457 | MR: 2815678 | Zbl: 1216.94016

[37] X.C. Tai; J. Hahn; G.J. Chung A Fast Algorithm for Euler’s Elastica Model Using Augmented Lagrangian Method, SIAM Journal on Imaging Sciences, Volume 4 (2011), 313 pages | MR: 2792415

[38] Xue-Cheng Tai; Chunlin Wu 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: 1371.94362

[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] C. Wu; X.C. Tai 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 | Article | MR: 2679430 | Zbl: 1206.90245

[42] Fenlin Yang; Ke Chen; Bo Yu 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: 2926494 | Zbl: 1264.35288

[43] W. Yin; S. Osher; D. Goldfarb; J. Darbon Bregman iterative algorithms for $\setminus $ell_1-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences, Volume 1 (2008) no. 1, pp. 143-168 | Article | Zbl: 1203.90153

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

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

[46] Jing Yuan; Egil Bae; Xue-Cheng Tai; Yuri Boykov A spatially continuous max-flow and min-cut framework for binary labeling problems, Numerische Mathematik (2013), pp. 1-29 | Zbl: 1290.49060

[47] Jing Yuan; Juan Shi; Xue-Cheng Tai A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation, East Asian Journal on Applied Mathematics (to appear) | Zbl: 1286.94027

[48] Weili Zeng; Xiaobo Lu; Xianghua Tan Non-linear fourth-order telegraph-diffusion equation for noise removal, IET Image Processing, Volume 7 (2013) no. 4, pp. 335-342 | Article | MR: 3113012

[49] W. Zhu; T. Chan Image denoising using mean curvature of image surface, SIAM Journal on Imaging Sciences, Volume 5 (2012) no. 1, pp. 1-32 | Article | MR: 2902655 | Zbl: 1258.94021

[50] W. Zhu; T. Chan; Selim Esedoglu Segmentation with depth: A level set approach, SIAM Journal on Scientific Computing, Volume 28 (2006) no. 5, pp. 1957-1973 | Article | MR: 2272196 | Zbl: 1344.68266

[51] Wei Zhu; Tony Chan A variational model for capturing illusory contours using curvature, Journal of Mathematical Imaging and Vision, Volume 27 (2007) no. 1, pp. 29-40 | Article | MR: 2297995

[52] Wei Zhu; Xue-Cheng Tai; Tony Chan Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging, Volume 7 (2012) no. 3, pp. 1075-1097 | MR: 3180687 | Zbl: 1311.94015

[53] Wei Zhu; Xue-Cheng Tai; Tony Chan Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging, Volume 7 (2013) no. 4, pp. 1409-1432 | Article | MR: 3180687 | Zbl: 1311.94015

[54] Wei Zhu; Xue-Cheng Tai; Tony Chan Image Segmentation Using Euler’s Elastica as the Regularization, Journal of Scientific Computing, Volume 57 (2013) no. 2, pp. 414-438 | Article | MR: 3108757 | Zbl: 1282.65037

*Cited by Sources: *