Justifying Tensor-Driven Diffusion from Structure-Adaptive Statistics of Natural Images

  • Pascal Peter
  • Joachim Weickert
  • Axel Munk
  • Tatyana Krivobokova
  • Housen Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8932)


Tensor-driven anisotropic diffusion and regularisation have been successfully applied to a wide range of image processing and computer vision tasks such as denoising, inpainting, and optical flow. Empirically it has been shown that anisotropic models with a diffusion tensor perform better than their isotropic counterparts with a scalar-valued diffusivity function. However, the reason for this superior performance is not well understood so far. Moreover, the specific modelling of the anisotropy has been carried out in a purely heuristic way. The goal of our paper is to address these problems. To this end, we use the statistics of natural images to derive a unifying framework for eight isotropic and anisotropic diffusion filters that have a corresponding variational formulation. In contrast to previous statistical models, we systematically investigate structure-adaptive statistics by analysing the eigenvalues of the structure tensor. With our findings, we justify existing successful models and assess the relationship between accurate statistical modelling and performance in the context of image denoising.


diffusion regularisation anisotropy diffusion tensor statistics of natural images image priors 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, A., Raskar, R., Chellappa, R.: What is the range of surface reconstructions from a gradient field? In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006, Part I. LNCS, vol. 3951, pp. 578–591. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Two deterministic half-quadratic regularization algorithms for computed imaging. In: Proc. 1994 IEEE International Conference on Image Processing, vol. 2, pp. 168–172. IEEE Computer Society Press, Austin (1994)Google Scholar
  3. 3.
    Cottet, G.H.: Diffusion approximation on neural networks and applications for image processing. In: Hodnett, F. (ed.) Proc. Sixth European Conference on Mathematics in Industry, pp. 3–9. Teubner, Stuttgart (1992)Google Scholar
  4. 4.
    Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Field, D.J.: Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America A 4(12), 2379–2394 (1987)CrossRefGoogle Scholar
  6. 6.
    Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.P.: Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision 31(2-3), 255–269 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gerig, G., Kübler, O., Kikinis, R., Jolesz, F.A.: Nonlinear anisotropic filtering of MRI data. IEEE Transactions on Medical Imaging 11, 221–232 (1992)CrossRefGoogle Scholar
  8. 8.
    Höcker, C., Fehmers, G.: Fast structural interpretation with structure-oriented filtering. The Leading Edge 21(3), 238–243 (2002)CrossRefGoogle Scholar
  9. 9.
    Huang, J., Mumford, D.: Image statistics for the British Aerospace segmented database. Tech. rep., Divison of Applied Math, Brown University, Providence (1999)Google Scholar
  10. 10.
    Huang, J., Mumford, D.: Statistics of natural images and models. In: Proc. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 1541–1547. IEEE Computer Society Press, Ft. Collins (1999)Google Scholar
  11. 11.
    Iijima, T.: Basic theory on normalization of pattern (in case of typical one-dimensional pattern). Bulletin of the Electrotechnical Laboratory 26, 368–388 (1962) (in Japanese)Google Scholar
  12. 12.
    Krajsek, K., Scharr, H.: Diffusion filtering without parameter tuning: Models and inference tools. In: Proc. 2010 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2536–2543. IEEE Computer Society Press, San Francisco (2010)Google Scholar
  13. 13.
    Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM Journal on Imaging Sciences 6(2), 938–983 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Manniesing, R., Viergever, M.A., Niessen, W.J.: Vessel enhancing diffusion: A scale space representation of vessel structures. Medical Image Analysis 10, 815–825 (2006)CrossRefGoogle Scholar
  15. 15.
    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: Proc. Eigth International Conference on Computer Vision, Vancouver, Canada, pp. 416–423 (July 2001)Google Scholar
  16. 16.
    Nagel, H.H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Transactions on Pattern Analysis and Machine Intelligence 8, 565–593 (1986)CrossRefGoogle Scholar
  17. 17.
    Olmos, A., Kingdom, F.: A biologically inspired algorithm for the recovery of shading and reflectance images. Perception 33(12), 1463–1473 (2004)CrossRefGoogle Scholar
  18. 18.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  19. 19.
    Pouli, T., Reinhard, E., Cunningham, D.W.: Image Statistics in Visual Computing. CRC Press, Boca Raton (2013)Google Scholar
  20. 20.
    Roth, S., Black, M.J.: Fields of experts: A framework for learning image priors. In: Proc. 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 860–867. IEEE Computer Society Press, San Diego (2005)Google Scholar
  21. 21.
    Roussos, A., Maragos, P.: Tensor-based image diffusions derived from generelizations of the total variation and Beltrami functionals. In: Proc. 17th IEEE International Conference on Image Processing, Hong Kong, pp. 4141–4144 (September 2010)Google Scholar
  22. 22.
    Scharr, H., Black, M.J., Haussecker, H.W.: Image statistics and anisotropic diffusion. In: Proc. Ninth International Conferance on Computer Vision, vol. 2, pp. 840–847. IEEE Computer Society Press, Nice (2003)CrossRefGoogle Scholar
  23. 23.
    Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. Journal of Mathematical Imaging and Vision 12(1), 43–63 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(4), 506–516 (2005)CrossRefGoogle Scholar
  25. 25.
    Weickert, J.: Anisotropic diffusion filters for image processing based quality control. In: Fasano, A., Primicerio, M. (eds.) Proc. Seventh European Conference on Mathematics in Industry, pp. 355–362. Teubner, Stuttgart (1994)Google Scholar
  26. 26.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  27. 27.
    Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Nashed, M.Z., Scherzer, O. (eds.) Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics, vol. 313, pp. 251–268. AMS, Providence (2002)CrossRefGoogle Scholar
  28. 28.
    Welk, M., Steidl, G., Weickert, J.: Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage. Applied and Computational Harmonic Analysis 24, 195–224 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Zhu, S.C., Mumford, D.: Prior learning and Gibbs reaction-diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(11), 1236–1250 (1997)CrossRefGoogle Scholar
  30. 30.
    Zhu, S.C., Wu, Y., Mumford, D.: Filters, random fields and maximum entropy (FRAME): Towards a unified theory for texture modeling. International Journal of Computer Vision 27(2), 107–126 (1998)CrossRefGoogle Scholar
  31. 31.
    Zimmer, H., Valgaerts, L., Bruhn, A., Breuß, M., Weickert, J., Rosenhahn, B., Seidel, H.P.: PDE-based anisotropic disparity-driven stereo vision. In: Deussen, O., Keim, D., Saupe, D. (eds.) Vision, Modelling, and Visualization 2008, pp. 263–272. AKA, Heidelberg (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pascal Peter
    • 1
  • Joachim Weickert
    • 1
  • Axel Munk
    • 2
  • Tatyana Krivobokova
    • 3
  • Housen Li
    • 2
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Felix-Bernstein-Chair for Mathematical StatisticsInstitute of Mathematical StochasticsGöttingenGermany
  3. 3.Statistical Methods Group, Courant Research Centre “Poverty, Equity and Growth”GöttingenGermany

Personalised recommendations