Skip to main content
SpringerLink
Log in

New Possibilities in Image Diffusion and Sharpening via High-Order Sobolev Gradient Flows

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

We study the use of high-order Sobolev gradients for PDE-based image smoothing and sharpening, extending our previous work on this problem. In particular, we study the gradient descent equation on the heat equation energy functional obtained by modifying the usual metric on the space of images, which is the L 2 metric, to a weighted H k Sobolev metric. We present existence and uniqueness results which show that the Sobolev diffusion PDE are well-posed both in the forward and backward direction. Furthermore, we perform a Fourier analysis on the scale space generated by the Sobolev PDE and show that as the order of the Sobolev metric tends to infinity, the Sobolev gradients converge to a Gaussian smoothed L 2 gradient. We then present experimental results which exploit the theoretical stability results by applying the various Sobolev gradient flows in the backward direction for image sharpening effects. Furthermore, we show that as the Sobolev order is increased, the sharpening effects become more global in nature and more immune to noise.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29, 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences. Springer, Berlin (2006)

    MATH  Google Scholar 

  3. Calder, J., Mansouri, A., Yezzi, A.: Image diffusion and sharpening via Sobolev gradient flows. SIAM J. Imaging Sci. 3(4), 981–1014 (2010). doi:10.1137/090771260

    Article  MATH  Google Scholar 

  4. Catté, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chambolle, A.: Partial differential equations and image processing. In: IEEE International Conference on Image Processing, vol. 1, pp. 16–20. (1994)

    Google Scholar 

  6. Charpiat, G., Maurel, P., Pons, J.P., Keriven, R., Faugeras, O.: Generalized gradients: Priors on minimization flows. Int. J. Comput. Vis. 73, 325–344 (2007)

    Article  Google Scholar 

  7. Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P.: Image compression with anisotropic diffusion. J. Math. Imaging Vis. 31(2–3), 255–269 (2008). doi:10.1007/s10851-008-0087-0

    Google Scholar 

  8. Lieu, L., Vese, L.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58, 167–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12, 1579–1590 (2003)

    Article  Google Scholar 

  10. Neuberger, J.: Sobolev Gradients and Differential Equations, 2nd edn. Lecture Notes in Mathematics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  11. Osher, S., Rudin, L.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27(4), 919–940 (1990). doi:10.1137/0727053

    Article  MATH  Google Scholar 

  12. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  13. Richardson, W.J.: High-order Sobolev preconditioning. Nonlinear Anal. 63(5–7), e1779–e1787 (2005). doi:10.1016/j.na.2005.02.072 Invited Talks from the Fourth World Congress of Nonlinear Analysts (WCNA 2004)

    Article  Google Scholar 

  14. Richardson, W.J.: Sobolev gradient preconditioning for image-processing PDEs. Commun. Numer. Methods Eng. 24, 493–504 (2008)

    Article  MATH  Google Scholar 

  15. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  16. Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Sobolev active contours. Int. J. Comput. Vis. 73, 109–120 (2005)

    Google Scholar 

  17. Sundaramoorthi, G., Jackson, J., Yezzi, A.: Tracking with Sobolev active contours. In: Proceedings Computer Vision and Pattern Recognition, pp. 674–680 (2006)

    Google Scholar 

  18. Sundaramoorthi, G., Yezzi, A., Mennucci, A., Sapiro, G.: New possibilities with Sobolev active contours. In: Scale Space Variational Methods, pp. 153–164. Springer, Berlin (2007)

    Chapter  Google Scholar 

  19. Witkin, A.: Scale-space filtering. In: International Joint Conference on Artificial Intelligence, Karlsruhe, West Germany, pp. 1019–1021 (1983)

    Google Scholar 

  20. Witkin, A.: Scale-space filtering: A new approach to multi-scale description. In: IEEE International Conference on Acoustics, Speech, and Signal Processing. vol. 9, pp. 159–153 (1984)

    Google Scholar 

  21. You, Y., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9, 1723–1730 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, 99 University Avenue, Kingston, ON, Canada, K7L 3N6

    J. Calder & A. Mansouri

  2. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    A. Yezzi

Authors
  1. J. Calder
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Mansouri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Yezzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Calder.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calder, J., Mansouri, A. & Yezzi, A. New Possibilities in Image Diffusion and Sharpening via High-Order Sobolev Gradient Flows. J Math Imaging Vis 40, 248–258 (2011). https://doi.org/10.1007/s10851-010-0250-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-010-0250-2

Keywords

Search

Navigation