Skip to main content
SpringerLink
Log in

Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D

  • Conference paper
  • First Online:
Scale Space Methods in Computer Vision (Scale-Space 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2695))

Included in the following conference series:

Abstract

Recently, several different approaches for digital inpainting have been proposed in the literature. We give a review and introduce a novel approach based on the complex Ginzburg-Landau equation. The use of this equation is motivated by some of its remarkable analytical properties. While common inpainting technology is especially designed for restorations of two dimensional image data, the Ginzburg-Landau equation can straight forwardly be applied to restore higher dimensional data, which has applications in frame interpolation, improving sparsely sampled volumetric data and to fill in fragmentary surfaces. The latter application is of importance in architectural heritage preservation. We discuss a stable and efficient scheme for the numerical solution of the Ginzburg-Landau equation and present some numerical experiments. We compare the performance of our algorithm with other well established methods for inpainting.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Lückenfüller und Farbmischer, Bildkorrekturverfahren: beim Menschen gelernt, c’t-Magazin für Computer Technik, 24/2002, Heise Zeitschriften Verlag, Hannover, p.190

    Google Scholar 

  2. L. Joyeux, O. Buisson, B. Besserer, S. Boukir, Detection and removal of line scratches in motion picture films, Proceedings of CVPR’99, IEEE Int. Conf. on Computer Vision and Pattern Recognition, Fort Collins, Colorado, USA, June 1999

    Google Scholar 

  3. M. Ramasubramanian, S. Pattanaik, D. Greenberg, A Perceptually Based Physical Error Metric for Realistic Image Synthesis, Proceedings of SIGGRAPH 99. In Computer Graphics Proceedings, Annual Conference Series, 1999, ACM SIGGRAPH, p.73–82

    Google Scholar 

  4. V. Caselles, B. Coll, J. Morel, A Kanizsa programme, Progress in Nonlinear Differential Equations and their Applications, Vol. 25, p.35–55, 1996

    MathSciNet  Google Scholar 

  5. A.A. Efros and T.K. Leung, Texture synthesis by non-parametric sampling, Proceedings of the Seventh International Conference on Computer Vision, Corfu, Greece, 1999

    Google Scholar 

  6. H. Igehy, L. Pereira, Image Replacement through Texture Synthesis, Proceedings of the 1997 IEEE International Conference on Image Processing

    Google Scholar 

  7. M. Unser, Sampling-50 years after Shannon, Proceedings of the IEEE, vol. 88, no. 4, p.569–587, April 2000

    Article  Google Scholar 

  8. P. Thenaz, T. Blu, M. Unser, Handbook of Medical Imaging, Processing and Analysis, I.N. Bankman, Ed., Academic Press, San Diego CA, USA, p.393–420, 2000

    Chapter  Google Scholar 

  9. S. Masnou, Disocclusion: A Variational Approach Using Level Lines, IEEE Transactions on Signal Processing, 11(2), February 2002, p.68–76

    MathSciNet  Google Scholar 

  10. S. Masnou, J.-M. Morel, Level Lines based Disocclusion, Proceedings of the 1998 IEEE International Conference on Image Processing, p.259–263

    Google Scholar 

  11. C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, J. Verdera, Filling-In by Joint Interpolation of Vector Fields and Gray Levels, IEEE Transactions on Signal Processing, 10(8), August 2001, p.1200–1211

    MATH  MathSciNet  Google Scholar 

  12. M. Bertalmio, G. Sapiro, C. Ballester, V. Caselles, Image inpainting, Computer Graphics, SIGGRAPH 2000, July 2000

    Google Scholar 

  13. M. Bertalmio, A. Bertozzi, G. Sapiro, Navier-Stokes, Fluid Dynamics, and Image and Video Inpainting, IEEE CVPR 2001, Hawaii, USA, December 2001

    Google Scholar 

  14. T. Chan, J. Shen, Mathematical Models for Local Nontexture Inpaintings, SIAM Journal of Applied Mathematics, 62(3), 2002, p.1019–1043

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Chan, J. Shen, Non-Texture Inpainting by Curvature-Driven Diffusions (CDD), Journal of Visual Communication and Image Representation, 12(4), 2001, p.436–449

    Article  Google Scholar 

  16. T. Chan, S. Kang, J. Shen, Euler’s Elastica and Curvature Based Inpainting, SIAM Journal of Applied Mathematics, 63(2), pp. 564–592, 2002

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Esedoglu, J. Shen, Digital Inpainting Based on the Mumford-Shah-Euler Image Model, European Journal of Applied Mathematics, 13, pp. 353–370, 2002

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Oliveira, B. Bowen, R. McKenna, Y. Chang, Fast Digital Inpainting, Proceedings of the International Conference on Visualization, Imaging and Image Processing (VIIP 2001), Marbella, Spain, pp. 261–266

    Google Scholar 

  19. L. Landau, V. Ginzburg, On the Theory of Superconductivity, Journal of Experimental and Theoretical Physics (USSR), 20 (1950), p.1064

    Google Scholar 

  20. M. Ipsen, P. Sorensen, Finite Wavelength Instabilities in a Slow Mode Coupled Complex Ginzburg-Landau Equation, Physical Review Letters, Vol. 84/11, p.2389, 2000

    Article  Google Scholar 

  21. M. van Hecke, E. de Wit, W. van Saarloos, Coherent and Incoherent Drifting Pulse Dynamics in a Complex Ginzburg-Landau Equation, Physical Review Letters, Vol. 75/21, p.3830, 1995

    Article  Google Scholar 

  22. T. Bohr, G. Huber, E. Ott, The structure of spiral-domain patterns and shocks in the 2D complex Ginzburg-Landau equation, Physica D, Vol. 106, p.95–112, 1997

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Ambrosio, N. Dancer, Calculus of Variations and Partial Differential Equations, Springer Verlag

    Google Scholar 

  24. F. Bethuel, H. Brezis, F. Hélein, Ginzburg-Landau Vortices. In “Progress in Nonlinear Differential Equations and Their Applications”, 13 (1994), Birkhäuser

    Google Scholar 

  25. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes, Cambridge University Press

    Google Scholar 

  26. J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, Vol. 22

    Google Scholar 

  27. L. Chen, J. Shen, Applications of semi-implicit Fourier-spectral methods to phase field equations, Computer Physics Communications, Vol. 108, p.147–158, 1998

    Article  MATH  Google Scholar 

  28. R. H. Nochetto, M. Paolini, C. Verdi, A Dynamic Mesh Algorithm for Curvature Dependent Evolving Interfaces, Journal of Computational Physics, 123, 1996, p.296–310

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Weickert, Anisotropic Diffusion in Image Processing, B.G.Teubner, Stuttgart, 1998

    MATH  Google Scholar 

  30. J. Davis, S. Marschner, M. Garr, M. Levoy, Filling holes in complex surfaces using volumetric diffusion, to appear in First International Symposium on 3D Data Processing, Visualization, and Transmission Padua, Italy, June 19–21, 2002

    Google Scholar 

  31. J. Taylor, Demonstration of Canadian 3D Technology for Heritage Recording in China, Proceedings of the Italy-Canada 2001 Workshop on 3D Digital Imaging and Modeling Applications, Padova, Italy, April 3–4, 2001

    Google Scholar 

  32. M. Bertalmio, L. Vese, G. Sapiro, S. Osher, Simultaneous Structure and Texture Image Inpainting, submitted

    Google Scholar 

  33. http://www.people.fas.harvard.edu/~hchong/Spring2002/cs276r/

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Innsbruck, Technikerstr. 25, A-6020, Innsbruck, Austria

    Harald Grossauer & Otmar Scherzer

Authors
  1. Harald Grossauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Otmar Scherzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Imaging Sciences, 5th Floor Thomas Guy House, Guy’s Campus, London, SE1 9RT, UK

    Lewis D. Griffin

  2. IT University of Copenhagen, Glentevej 67-69, Copenhagen NV, Denmark

    Martin Lillholm

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grossauer, H., Scherzer, O. (2003). Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D. In: Griffin, L.D., Lillholm, M. (eds) Scale Space Methods in Computer Vision. Scale-Space 2003. Lecture Notes in Computer Science, vol 2695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44935-3_16

Download citation

  • DOI: https://doi.org/10.1007/3-540-44935-3_16

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40368-5

  • Online ISBN: 978-3-540-44935-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation