Diffusion-Like Reconstruction Schemes from Linear Data Models

  • Hanno Scharr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4174)


In this paper we extend anisotropic diffusion with a diffusion tensor to be applicable to data that is well modeled by linear models. We focus on its variational theory, and investigate simple discretizations and their performance on synthetic data fulfilling the underlying linear models. To this end, we first show that standard anisotropic diffusion with a diffusion tensor is directly linked to a data model describing single orientations. In the case of spatio-temporal data this model is the well known brightness constancy constraint equation often used to estimate optical flow. Using this observation, we construct extended anisotropic diffusion schemes that are based on more general linear models. These schemes can be thought of as higher order anisotropic diffusion. As an example we construct schemes for noise reduction in the case of two orientations in 2d images. By comparison to the denoising result via standard single orientation anisotropic diffusion, we demonstrate the better suited behavior of the novel schemes for double orientation data.


Anisotropic Diffusion Structure Tensor Orientation Estimation Reconstruction Scheme Single Orientation 
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.


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  1. 1.
    Acton, S.T.: Multigrid anisotropic diffusion. Trans. Im. Proc. 7, 280–291 (1998)CrossRefGoogle Scholar
  2. 2.
    Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. International Journal of Computer Vision 12(1), 43–77 (1994)CrossRefGoogle Scholar
  3. 3.
    Bigün, J., Granlund, G.H.: Optimal orientation detection of linear symmetry. In: ICCV, London, UK, pp. 433–438 (1987)Google Scholar
  4. 4.
    Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. IEEE Transactions on Image Processing 7(3), 412–432 (1998)CrossRefGoogle Scholar
  5. 5.
    Felsberg, M., Forssén, P.-E., Scharr, H.: Channel smoothing: Efficient robust smoothing of low-level signal features. PAMI 28(2) (2006)Google Scholar
  6. 6.
    Fleet, D.J., Black, M.J., Yacoob, Y., Jepson, A.D.: Design and use of linear models for image motion analysis. Int. J. Computer Vision 36(3), 171–193 (2000)CrossRefGoogle Scholar
  7. 7.
    Fröhlich, J., Weickert, J.: Image processing using a wavelet algorithm for nonlinear diffusion. Technical Report 104, Laboratory of Technomathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany (1994)Google Scholar
  8. 8.
    Haußecker, H., Fleet, D.J.: Computing optical flow with physical models of brightness variation. PAMI 23(6), 661–673 (2001)Google Scholar
  9. 9.
    Jähne, B.: Spatio-Temporal Image Processing. LNCS, vol. 751. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  10. 10.
    Mota, C., Stuke, I., Barth, E.: Analytic solutions for multiple motions. In: Proc. ICIP, pp. 917–920 (2001)Google Scholar
  11. 11.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. XLII, 577–685 (1989)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Nestares, O., Fleet, D.J., Heeger, D.: Likelihood functions and confidence bounds for total-least-squares problems. In: CVPR 2000, vol. 1 (2000)Google Scholar
  13. 13.
    Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.): Scale-Space 1999. LNCS, vol. 1682. Springer, Heidelberg (1999)Google Scholar
  14. 14.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  15. 15.
    Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P.: Image denoising using scale mixtures of gaussians in the wavelet domain. IEEE TIP 12(11) (2003)Google Scholar
  16. 16.
    Preußer, T., Rumpf, M.: An Adaptive Finite Element Method for Large Scale Image Processing. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space 1999. LNCS, vol. 1682, pp. 223–234. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Roth, S., Black, M.J.: Fields of experts: A framework for learning image priors. In: CVPR, vol. 2, pp. 860–867 (2005)Google Scholar
  18. 18.
    Scharr, H., Black, M.J., Haussecker, H.W.: Image statistics and anisotropic diffusion. In: ICCV 2003, Nice, France (2003)Google Scholar
  19. 19.
    Scharr, H., Spies, H.: Accurate optical flow in noisy image sequences using flow adapted anisotropic diffusion. Signal Processing: Image Communication (2005)Google Scholar
  20. 20.
    Schnörr, C.: A study of a convex variational diffusion approach for image segmentation and feature extraction. J. Math. Im. and Vis. 8(3), 271–292 (1998)zbMATHCrossRefGoogle Scholar
  21. 21.
    Shizawa, M., Mase, K.: Simultaneous multiple optical flow estimation. In: ICPR 1990, pp. 274–278 (1990)Google Scholar
  22. 22.
    Stuke, I., Aach, T., Barth, E., Mota, C.: Analysing superimposed oriented patterns. In: 6th IEEE SSIAI, pp. 133–137 (2004)Google Scholar
  23. 23.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: ICCV, Bombay, India (1998)Google Scholar
  24. 24.
    Tschumperle, D., Deriche, R.: Vector-valued image regularization with pde’s: A common framework for different applications. In: CVPR (2003)Google Scholar
  25. 25.
    Weickert, J.: Anisotropic diffusion in image processing. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  26. 26.
    Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in pde-based computation of image motion. IJCV, 245–264 (December 2001)Google Scholar
  27. 27.
    Zhu, S.C., Mumford, D.: Prior learning and gibbs reaction-diffusion. PAMI 19(11), 1236–1250 (1997)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hanno Scharr
    • 1
  1. 1.ICG III, Research Center JülichJülichGermany

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