Journal of Mathematical Imaging and Vision

, Volume 41, Issue 1–2, pp 59–85 | Cite as

Heat Equations on Vector Bundles—Application to Color Image Regularization

  • Thomas BatardEmail author


We use the framework of heat equations associated to generalized Laplacians on vector bundles over Riemannian manifolds in order to regularize color images. We show that most methods devoted to image regularization may be considered in this framework. From a geometric viewpoint, they differ by the metric of the base manifold and the connection of the vector bundle involved. By the regularization operator we propose in this paper, the diffusion process is completely determined by the geometry of the vector bundle. More precisely, the metric of the base manifold determines the anisotropy of the diffusion through the computation of geodesic distances whereas the connection determines the data regularized by the diffusion process through the computation of the parallel transport maps. This regularization operator generalizes the ones based on short-time Beltrami kernel and oriented Gaussian kernels. Then we construct particular connections and metrics involving color information such as luminance and chrominance in order to perform new kinds of regularization.


Heat equation Generalized Laplacian Differential geometry of vector bundle Riemannian geometry Regularization Color image 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, Berlin (2006) zbMATHGoogle Scholar
  2. 2.
    Batard, T.: Clifford bundles: a common framework for images. Vector fields and orthonormal frame fields regularization. SIAM J. Imaging Sci. 3(3), 670–701 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Batard, T., Berthier, M.: Heat kernels of generalized Laplacians: applications to color images smoothing. In: Proceedings of IEEE International Conference on Image Processing ICIP (2009) Google Scholar
  4. 4.
    Batard, T., Saint-Jean, C., Berthier, M.: A Metric Approach to nD Images Edge Detection with Clifford Algebras. J. Math. Imaging Vis. 33(3), 293–312 (2009) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Batard, T., Berthier, M., Saint-Jean, C.: Clifford-Fourier transform for color image processing. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science. Springer, London (2010) Google Scholar
  6. 6.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels of Dirac Operators. Springer, Heidelberg (2004) Google Scholar
  7. 7.
    Chan, T., Shen, J.: Image Processing and Analysis: Variational, Pde, Wavelet, and Stochastic Methods. Society for Industrial and Applied Mathematic (2005) zbMATHGoogle Scholar
  8. 8.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology, vols. I–III. Academic Press, New York (1972) (1973 and 1976) zbMATHGoogle Scholar
  9. 9.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, London (1978) zbMATHGoogle Scholar
  10. 10.
    Husemoller, D.: Fibre Bundles, 3rd edn. Springer, New York (1994) Google Scholar
  11. 11.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proceedings of 8th IEEE International Conference on Computer Vision ICCV, pp. 416–423 (2001) CrossRefGoogle Scholar
  12. 12.
    Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2002) Google Scholar
  13. 13.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2006) zbMATHGoogle Scholar
  14. 14.
    Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. Image Process. 7(3), 310–318 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Spira, A., Kimmel, R., Sochen, N.: A Short-time Beltrami kernel for smoothing images and manifolds. IEEE Trans. Image Process. 16, 1628–1636 (2007) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tschumperlé, D.: Fast anisotropic smoothing of multi-valued images using curvature-preserving PDE. Int. J. Comput. Vis. 68, 65–82 (2006) CrossRefGoogle Scholar
  17. 17.
    Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDE’s: a common framework for different applications. IEEE Trans. Pattern Anal. Mach. Intell. 27, 506–517 (2005) CrossRefGoogle Scholar
  18. 18.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998) zbMATHGoogle Scholar
  19. 19.
    Weickert, J.: Coherence-enhancing diffusion of colour images. Image Vis. Comput. 17, 199–210 (1999) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Laboratoire Mathématiques, Image et ApplicationsUniversité de La RochelleLa Rochelle CedexFrance

Personalised recommendations