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

Xue-Cheng Tai[1]

  • [1] Department of Mathematics University of Bergen, Bergen, Norway

Actes des rencontres du CIRM (2013)

  • Volume: 3, Issue: 1, page 17-30
  • ISSN: 2105-0597

Abstract

top
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.

How to cite

top

Tai, Xue-Cheng. "Fast numerical schemes related to curvature minimization: a brief and elementary review." Actes des rencontres du CIRM 3.1 (2013): 17-30. <http://eudml.org/doc/275347>.

@article{Tai2013,
abstract = {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.},
affiliation = {Department of Mathematics University of Bergen, Bergen, Norway},
author = {Tai, Xue-Cheng},
journal = {Actes des rencontres du CIRM},
keywords = {variaitonal models; curvature minimization; Augmented Lagrangian methods},
language = {eng},
month = {11},
number = {1},
pages = {17-30},
publisher = {CIRM},
title = {Fast numerical schemes related to curvature minimization: a brief and elementary review},
url = {http://eudml.org/doc/275347},
volume = {3},
year = {2013},
}

TY - JOUR
AU - Tai, Xue-Cheng
TI - Fast numerical schemes related to curvature minimization: a brief and elementary review
JO - Actes des rencontres du CIRM
DA - 2013/11//
PB - CIRM
VL - 3
IS - 1
SP - 17
EP - 30
AB - 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.
LA - eng
KW - variaitonal models; curvature minimization; Augmented Lagrangian methods
UR - http://eudml.org/doc/275347
ER -

References

top
  1. Luigi Ambrosio, Simon Masnou, A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries 5 (2003), 63-82 Zbl1029.49037MR1959769
  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), 1-18 Zbl1235.68244MR2775242
  3. Kristian Bredies, Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty, SFB Report 6 (2012) 
  4. Kristian Bredies, Thomas Pock, Benedikt Wirth, Convex relaxation of a class of vertex penalizing functionals, Journal of mathematical imaging and vision 47 (2013), 278-302 Zbl1293.49063MR3097120
  5. Carlos Brito-Loeza, Ke Chen, On high-order denoising models and fast algorithms for vector-valued images, Image Processing, IEEE Transactions on 19 (2010), 1518-1527 Zbl1205.68474MR2778518
  6. Jeff Calder, A Mansouri, Anthony Yezzi, Image sharpening via Sobolev gradient flows, SIAM Journal on Imaging Sciences 3 (2010), 981-1014 Zbl1202.94019MR2763705
  7. Antonin Chambolle, Pierre-Louis Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik 76 (1997), 167-188 Zbl0874.68299MR1440119
  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 6 (2013) Zbl1289.68196MR3011641
  9. T. Chan, L.A. Vese, Active contours without edges, IEEE Trans Image Proc. 10 (2001), 266-277 Zbl1039.68779
  10. Tony F. Chan, Selim Esedoglu, Mila Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math. 66 (2006), 1632-1648 (electronic) Zbl1117.94002MR2246072
  11. Tony F Chan, Sung Ha Kang, Jianhong Shen, Euler’s elastica and curvature-based inpainting, SIAM Journal on Applied Mathematics (2002), 564-592 Zbl1028.68185MR1951951
  12. Eduardo Cuesta, Mokhtar Kirane, Salman A Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing 92 (2012), 553-563 
  13. Stephan Didas, Joachim Weickert, Bernhard Burgeth, Properties of higher order nonlinear diffusion filtering, Journal of mathematical imaging and vision 35 (2009), 208-226 Zbl1171.68788MR2533935
  14. Yuping Duan, Weimin Huang, A fixed-point augmented Lagrangian method for total variation minimization problems, Journal of Visual Communication and Image Representation 24 (2013), 1168-1181 
  15. R. Glowinski, P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, 9 (1989), Society for Industrial Mathematics Zbl0698.73001MR1060954
  16. T. Goldstein, S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences 2 (2009), 323-343 Zbl1177.65088MR2496060
  17. John B Greer, Andrea L Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM Journal on Mathematical Analysis 36 (2004), 38-68 Zbl1082.35080MR2083852
  18. Patrick Guidotti, Kate Longo, Two enhanced fourth order diffusion models for image denoising, Journal of Mathematical Imaging and Vision 40 (2011), 188-198 Zbl1255.68235MR2782126
  19. Langhua Hu, Duan Chen, Guo-Wei Wei, High-order fractional partial differential equation transform for molecular surface construction, Molecular based mathematical biology 1 (2013), 1-25 Zbl1277.35338
  20. P Jidesh, Santhosh George, Fourth–order variational model with local–constraints for denoising images with textures, International Journal of Computational Vision and Robotics 2 (2011), 330-340 
  21. Michael Kass, Andrew Witkin, Demetri Terzopoulos, Snakes: Active contour models, International journal of computer vision 1 (1988), 321-331 Zbl0646.68105
  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 39 (2000), 111-129 Zbl1060.68656
  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 15 (2006), 1171-1181 Zbl1286.94018
  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 12 (2003), 1579-1590 Zbl1286.94020
  25. Kent Andre Mardal, Xue-Cheng Tai, Ragnar Winther, A robust finite element method for Darcy-Stokes flow, SIAM Journal on Numerical Analysis (2003), 1605-1631 Zbl1037.65120MR1950614
  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 15 (2013) Zbl1273.35153MR3073448
  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) Zbl1302.35208MR3158470
  28. D Mumford, M Nitzberg, T Shiota, Filtering, Segmentation and Depth, Lecture Notes in Computer Science 662 (1993) Zbl0801.68171MR1226232
  29. D. Mumford, J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math, 42 42 (1989), 577-685 Zbl0691.49036MR997568
  30. Ehsan Nadernejad, Søren Forchhammer, Wavelet-based image enhancement using fourth order PDE, Intelligent Signal Processing (WISP), 2011 IEEE 7th International Symposium on (2011), 1-6 
  31. Carola-Bibiane Papafitsorosand Schönlieb, A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision 48 (2014), 308-338 
  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), 52-62 
  33. Guy Rosman, Yu Wang, Xue-Cheng Tai, Ron Kimmel, Alfred M Bruckstein, Fast regularization of matrix-valued images, Computer Vision–ECCV 2012 (2012), 173-186, Springer Zbl1292.65072
  34. L. Rudin, S. Osher, E. Fatemi, Nonlinear Total Variation based noise removal algorithms, Physica D 60 (1992), 259-268 Zbl0780.49028
  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), 17-23 
  36. Carola-bibiane Schönlieb, Andrea Bertozzi, Unconditionally stable schemes for higher order inpainting, Communications in Mathematical Sciences 9 (2011), 413-457 Zbl1216.94016MR2815678
  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 4 (2011) Zbl1215.68262MR2792415
  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), 502-513 Zbl1206.90245
  39. Wekipedia, Co-area formula (http://en.wikipedia.org/wiki/Coarea_formula), (2013) 
  40. Wikipedia, Total variation (http://en.wikipedia.org/wiki/Total_variation), (2014) 
  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 3 (2010), 300-339 Zbl1206.90245MR2679430
  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 9 (2012) Zbl1264.35288MR2926494
  43. W. Yin, S. Osher, D. Goldfarb, J. Darbon, Bregman iterative algorithms for ∖ ell_1-minimization with applications to compressed sensing, SIAM Journal on Imaging Sciences 1 (2008), 143-168 Zbl1203.90153MR2475828
  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), 2217-2224 
  45. J. Yuan, E. Bae, X.C. Tai, Y. Boykov, A continuous max-flow approach to potts model, Computer Vision–ECCV 2010 (2010), 379-392 
  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), 1-29 Zbl1290.49060MR3164146
  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) Zbl1286.94027MR3119514
  48. Weili Zeng, Xiaobo Lu, Xianghua Tan, Non-linear fourth-order telegraph-diffusion equation for noise removal, IET Image Processing 7 (2013), 335-342 MR3113012
  49. W. Zhu, T. Chan, Image denoising using mean curvature of image surface, SIAM Journal on Imaging Sciences 5 (2012), 1-32 Zbl1258.94021MR2902655
  50. W. Zhu, T. Chan, Selim Esedoglu, Segmentation with depth: A level set approach, SIAM Journal on Scientific Computing 28 (2006), 1957-1973 Zbl05194935MR2272196
  51. Wei Zhu, Tony Chan, A variational model for capturing illusory contours using curvature, Journal of Mathematical Imaging and Vision 27 (2007), 29-40 MR2297995
  52. Wei Zhu, Xue-Cheng Tai, Tony Chan, Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging 7 (2012), 1075-1097 Zbl1311.94015MR3180687
  53. Wei Zhu, Xue-Cheng Tai, Tony Chan, Augmented Lagrangian Method for A Mean Curvature Based Image Denoising Model, Inverse Problems and Imaging 7 (2013), 1409-1432 Zbl1311.94015MR3180687
  54. Wei Zhu, Xue-Cheng Tai, Tony Chan, Image Segmentation Using Euler’s Elastica as the Regularization, Journal of Scientific Computing 57 (2013), 414-438 Zbl1282.65037MR3108757

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.