Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Projected Gradient Based Color Image Decomposition

  • Conference paper
Scale Space and Variational Methods in Computer Vision (SSVM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5567))

Abstract

This work deals with color image processing, with a focus on color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v = f − u. u contains the geometric information of the original image, while v is made of the oscillating patterns of f, such as textures. We propose a numerical scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vector-valued images. A direct convergence proof of the scheme is provided, and some analysis on color texture modeling is given.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. Rudin, L., Osher, S., Fatemi, E.: Non linear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chan, T., Shen, J.: Image processing and analysis - Variational, PDE, wavelet, and stochastic methods. SIAM Publisher, Philadelphia (2005)

    MATH  Google Scholar 

  3. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  4. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. In: The fifteenth Dean Jacqueline B. Lewis memorial lectures. University Lecture Series, vol. 22. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  5. Mumford, D., Gidas, B.: Stochastic models for generic images. Quarterly of Applied Mathematics LIV(1) (2001)

    Google Scholar 

  6. Vese, L., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing 19(1-3), 553–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H − 1 norm. SIAM Journal on Multiscale Modeling and Simulation 1(3), 349–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aujol, J.F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. Journal of Mathematical Imaging and Vision 22(1), 71–88 (2005)

    Article  MathSciNet  Google Scholar 

  9. Aubert, G., Aujol, J.: Modeling very oscillating signals. Application to image processing. Applied Mathematics and Optimization 51(2), 163–182 (2005)

    Article  MATH  Google Scholar 

  10. Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. International Journal on Computer Vision 63(1), 85–104 (2005)

    Article  Google Scholar 

  11. Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. Journal of Visual Communication and Image Representation 18(3), 240–252 (2007)

    Article  Google Scholar 

  12. Garnett, J., Jones, P., Le, T., Vese, L.: Modeling oscillatory components with the homogeneous spaces BMO  − α and W  − α,p. Pure and Applied Mathematics Quarterly (to appear)

    Google Scholar 

  13. Le, T., Vese, L.: Image decomposition using total variation and div (BMO). Multiscale Modeling and Simulation, SIAM Interdisciplinary Journal 4(2), 390–423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lieu, L., Vese, L.: Image restoration and decomposition via bounded total variation and negative hilbert-sobolev spaces. Applied Mathematics & Optimization 58, 167–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Transactions on Image Processing 5(11), 1582–1586 (1996)

    Article  MATH  Google Scholar 

  16. Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Transactions on Image Processing 7(3), 310–318 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aujol, J.F., Kang, S.H.: Color image decomposition and restoration. Journal of Visual Communication and Image Representation 17(4), 916–928 (2006)

    Article  Google Scholar 

  18. Bresson, X., Chan, T.: Fast minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems and Imaging (IPI) (accepted) (2007)

    Google Scholar 

  19. Blomgren, P., Chan, T.: Color TV: Total variation methods for restoration of vector valued images. IEEE Transactions on Image Processing 7(3), 304–309 (1998)

    Article  Google Scholar 

  20. Vese, L., Osher, S.: Color texture modeling and color image decomposition in a variational-PDE approach. In: Proceedings of the Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2006), pp. 103–110. IEEE, Los Alamitos (2006)

    Chapter  Google Scholar 

  21. Aujol, J., Gilboa, G.: Constrained and SNR-based solutions for TV-Hilbert space image denoising. Journal of Mathematical Imaging and Vision 26(1-2), 217–237 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vogel, C.: Computational Methods for Inverse Problems. Frontiers in Applied Mathematics, vol. 23. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  23. Chan, T., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM Journal on Scientific Computing 20(6), 1964–1977 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chambolle, A.: An algorithm for total variation minimization and its applications. JMIV 20, 89–97 (2004)

    Article  Google Scholar 

  25. Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 136–152. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  26. Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part I: Fast and exact optimization. Journal of Mathematical Imaging and Vision 26(3), 277–291 (2006)

    Article  MathSciNet  Google Scholar 

  27. Weiss, P., Aubert, G., Blanc-Féraud, L.: Efficient schemes for total variation minimization under constraints in image processing. SIAM Journal on Scientific Computing (to appear) (2007)

    Google Scholar 

  28. Aujol, J.: Some algorithms for total variation based image restoration. CMLA Preprint 2008-05 (2008), http://hal.archives-ouvertes.fr/hal-00260494/en/

  29. Zhu, M., Wright, S., Chan, T.: Duality-based algorithms for total variation image restoration, UCLA CAM Report 08-33 (May 2008)

    Google Scholar 

  30. Bermudez, A., Moreno, C.: Duality methods for solving variational inequalities. Comp. and Maths. with Appls. 7(1), 43–58 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. Duval, V., Aujol, J.F., Vese, L.: A projected gradient algorithm for color image decomposition. Technical report, UCLA, CAM Report 08-40 (2008)

    Google Scholar 

  32. Ciarlet, P.G.: Introduction á l’Analyse Numérique Matricielle et á l’Optimisation. Dunod (1998)

    Google Scholar 

  33. Aujol, J., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition - modeling, algorithms, and parameter selection. International Journal of Computer Vision 67(1), 111–136 (2006)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut TELECOM, TELECOM ParisTech, CNRS UMR 5141, France

    Vincent Duval

  2. CMLA, ENS Cachan, CNRS, UniverSud, France

    Jean-François Aujol

  3. Mathematics Department, UCLA, USA

    Luminita Vese

Authors
  1. Vincent Duval
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-François Aujol
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luminita Vese
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Xue-Cheng Tai

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  3. Simula Research Laboratory, M. Linges v 17, Fornebu, P.O.Box 134, N-1325, Lysaker, Norway

    Marius Lysaker

  4. SINTEF ICT, Oslo, Norway

    Knut-Andreas Lie

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duval, V., Aujol, JF., Vese, L. (2009). Projected Gradient Based Color Image Decomposition. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02256-2_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02255-5

  • Online ISBN: 978-3-642-02256-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation