Volumetric Nonlinear Anisotropic Diffusion on GPUs

  • Andreas Schwarzkopf
  • Thomas Kalbe
  • Chandrajit Bajaj
  • Arjan Kuijper
  • Michael Goesele
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)


We present an efficient implementation of volumetric nonlinear anisotropic image diffusion on modern programmable graphics processing units (GPUs). We avoid the computational bottleneck of a time consuming eigenvalue decomposition in ℝ3. Instead, we use a projection of the Hessian matrix along the surface normal onto the tangent plane of the local isodensity surface and solve for the remaining two tangent space eigenvectors. We derive closed formulas to achieve this resulting in efficient GPU code. We show that our most complex volumetric nonlinear anisotropic diffusion gains a speed up of more than 600 compared to a CPU solution.


Tangent Space Diffusion Tensor Tangent Plane Anisotropic Diffusion Closed Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Koenderink, J.J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Lindeberg, T.: Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  3. 3.
    Weickert, J.: A review of nonlinear diffusion filtering. In: ter Haar Romeny, B.M., Florack, L.M.J., Viergever, M.A. (eds.) Scale-Space 1997. LNCS, vol. 1252, pp. 1–28. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  4. 4.
    Lindeberg, T.: Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. Journal of Mathematical Imaging and Vision, 1–46 (2010)Google Scholar
  5. 5.
    Kuijper, A.: Geometrical PDEs based on second order derivatives of gauge coordinates in image processing. Image and Vision Computing 27(8), 1023–1034 (2009)CrossRefGoogle Scholar
  6. 6.
    Weickert, J.: Coherence enhancing diffusion filtering. International Journal of Computer Vision 31, 111–127 (1999)CrossRefGoogle Scholar
  7. 7.
    Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface smoothing via anisotropic diffusion of normals. In: Proc. VIS 2002, pp. 125–132 (2002)Google Scholar
  8. 8.
    Bajaj, C.L., Xu, G.: Adaptive surfaces fairing by geometric diffusion. In: Symp. CAGD, pp. 731–737 (2001)Google Scholar
  9. 9.
    Bajaj, C.L., Xu, G.: Anisotropic diffusion of surfaces and functions on surfaces. ACM Trans. Graph. 22(1), 4–32 (2003)CrossRefGoogle Scholar
  10. 10.
    Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. Journal of Computational Physics 227(1), 492–512 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Agelas, L., Masson, R.: Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes. Comptes Rendus Mathematique 346(17-18), 1007–1012 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Zhang, X., Chen, W., Qian, L., Ye, H.: Affine invariant non-linear anisotropic diffusion smoothing strategy for vector-valued images. Imaging Science Journal 58(3), 119–124 (2010)CrossRefGoogle Scholar
  13. 13.
    Hadwiger, M., Sigg, C., Scharsach, H., Bühler, K., Gross, M.H.: Real-time ray-casting and advanced shading of discrete isosurfaces. Comp. Graph. Forum 24(3), 303–312 (2005)CrossRefGoogle Scholar
  14. 14.
    Tabik, S., Garzon, E., Garcia, I., Fernandez, J.: Implementation of anisotropic nonlinear diffusion for filtering 3D images in structural biology on SMP clusters. In: Proc. Int. Conf. Parallel Computing: Current & Future Issues of High-End Computing, ParCo., vol. 33, pp. 727–734 (2005)Google Scholar
  15. 15.
    Interactive 3D seismic fault detection on the graphics hardware. In: Proc. Volume Graphics (2006)Google Scholar
  16. 16.
    Zhao, Y.: Lattice Boltzman based PDE solver on the GPU. The Visual Computer 24(5), 323–333 (2008)CrossRefGoogle Scholar
  17. 17.
    Beyer, J., Langer, C., Fritz, L., Hadwiger, M., Wolfsberger, S., Bühler, K.: Interactive diffusion-based smoothing and segmentation of volumetric datasets on graphics hardware. Methods Inf. Med. 46(3), 270–274Google Scholar
  18. 18.
    Bajaj, C.L., Wu, Q., Xu, G.: Level set based volumetric anisotropic diffusion for 3D image denoising. ICES TR03-10, UTexas, Austin USA (2003)Google Scholar
  19. 19.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  20. 20.
    Weickert, J.: Anisotropic Diffusion in Image Processing. B.G. Teubne, Stuttgart (1998)zbMATHGoogle Scholar
  21. 21.
    Kindlmann, G., Whitaker, R., Tasdizen, T., Mller, T.: Curvature-based transfer functions for direct volume rendering: Methods and applications. In: Proc. IEEE Vis., pp. 513–520 (2003)Google Scholar
  22. 22.
    Sigg, C.: Representation and Rendering of Implicit Surfaces. PhD thesis, ETH Zurich (2006)Google Scholar
  23. 23.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. IEEE Int. Conf. on Computer Vision 1998, pp. 839–846 (1998)Google Scholar
  24. 24.
    Durand, F., Dorsey, J.: Fast bilateral filtering for the display of high-dynamic-range images. ACM Trans. Graph. 21(3), 257–266 (2002)CrossRefGoogle Scholar
  25. 25.
    Paris, S., Kornprobst, P., Tumblin, J., Durand, F.: A gentle introduction to bilateral filtering and its applications. In: SIGGRAPH Course (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Schwarzkopf
    • 1
  • Thomas Kalbe
    • 1
  • Chandrajit Bajaj
    • 2
  • Arjan Kuijper
    • 1
    • 3
  • Michael Goesele
    • 1
  1. 1.Technische Universität DarmstadtGermany
  2. 2.ICES-CVC University of Texas at AustinUSA
  3. 3.Fraunhofer IGDDarmstadtGermany

Personalised recommendations