Skip to main content
Springer Nature Link
Log in

Image Denoising Using Euler-Lagrange Equations for Function-Valued Mappings

  • Conference paper
  • First Online:
Image Analysis and Recognition (ICIAR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9730))

Included in the following conference series:

  • 2850 Accesses

Abstract

In this paper, we consider a new method for representing complex images, e.g., hyperspectral images and video sequences, in terms of function-valued mappings (FVMs), also known as Banach-valued functions. At each (pixel) location x, the FVM image u(x) is a function, as opposed to the traditional vector approach. We define the Fourier transform of an FVM as well as Euler-Lagrange conditions for functionals involving FVMs and then show how these results can be used to devise some FVM-based methods of denoising. We consider a very simple functional and present some numerical results.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  2. Bach, F., Jenatton, R., Mairal, J., Obozinski, G.: Convex optimization with sparsity-inducing norms. In: Sra, S., Nowozin, S., Wright, S.J. (eds.) Optimization for Machine Learning, pp. 19–53. MIT Press, Massachusetts (2012)

    Google Scholar 

  3. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. Arch. 2, 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bresson, X., Chan, T.F.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Probl. Imaging 2, 455–484 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buades, A., Coll, B., Morel, J.-M.: A non-local algorithm for image denoising. In: Proceedings of the Conference on Computer Vision and Pattern Recognition, pp. 60–65. IEEE (2005)

    Google Scholar 

  6. Cartan, H., Cartan, H.P.: Differential Calculus. Hermann, Paris (1971)

    MATH  Google Scholar 

  7. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)

    Article  MathSciNet  Google Scholar 

  8. Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)

    Book  MATH  Google Scholar 

  9. Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  10. Grupo de Inteligencia Computacional de la Universidad del País Vasco, Hyperspectral Remote Sensing Scenes. http://www.ehu.eus/ccwintco/index.php

  11. Martín-Herrero, J.: Anisotropic diffusion in the hypercube. IEEE Trans. Geosci. Remote Sens. 45, 1386–1398 (2007)

    Article  Google Scholar 

  12. Kunze, H., La Torre, D., Mendivil, F., Vrscay, E.R.: Fractal-Based Methods in Analysis. Springer Science & Business Media, Berlin (2011)

    MATH  Google Scholar 

  13. La Torre, D., Vrscay, E.R., Ebrahimi, M., Barnsley, M.F.: Measure-valued images associated fractal transforms, and the affine self-similarity of images. SIAM J. Imaging Sci. 2, 470–507 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Michailovich, O., La Torre, D., Vrscay, E.R.: Function-valued mappings, total variation and compressed sensing for diffusion MRI. In: Campilho, A., Kamel, M. (eds.) ICIAR 2012, Part II. LNCS, vol. 7325, pp. 286–295. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  15. Milman, M.: Complex interpolation and geometry of Banach spaces. Annali di Matematica Pura ed Applicata 136, 317–328 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Otero, D.: Function-valued Mappings and SSIM-based Optimization in Imaging, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada (2015)

    Google Scholar 

  17. Peetre, J.: Sur la transformation de Fourier des fonctions à valeurs vectorielles. Rendicoti del Seminario Matematico della Università di Padova 42, 15–26 (1969)

    MathSciNet  MATH  Google Scholar 

  18. Thompson, H.: The Bochner Integral and an Application to Singular Integrals, M.Sc. thesis, Dalhousie University, Halifax, NS, Canada (2014)

    Google Scholar 

  19. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)

    Article  Google Scholar 

  20. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Springer, New York (1990)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant (ERV). Financial support from the Faculty of Mathematics and the Department of Applied Mathematics (DO) is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Daniel Otero, Davide La Torre & Edward R. Vrscay

  2. Department of Economics, Management, and Quantitative Methods, University of Milan, Milan, Italy

    Davide La Torre

  3. Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi, UAE

    Davide La Torre

Authors
  1. Daniel Otero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide La Torre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward R. Vrscay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edward R. Vrscay .

Editor information

Editors and Affiliations

  1. University of Porto, Porto, Portugal

    Aurélio Campilho

  2. Department of Electrical, University of Waterloo, Waterloo, Ontario, Canada

    Fakhri Karray

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Otero, D., La Torre, D., Vrscay, E.R. (2016). Image Denoising Using Euler-Lagrange Equations for Function-Valued Mappings. In: Campilho, A., Karray, F. (eds) Image Analysis and Recognition. ICIAR 2016. Lecture Notes in Computer Science(), vol 9730. Springer, Cham. https://doi.org/10.1007/978-3-319-41501-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-41501-7_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-41500-0

  • Online ISBN: 978-3-319-41501-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics