Towards PDE-Based Image Compression

  • Irena Galić
  • Joachim Weickert
  • Martin Welk
  • Andrés Bruhn
  • Alexander Belyaev
  • Hans-Peter Seidel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3752)


While methods based on partial differential equations (PDEs) and variational techniques are powerful tools for denoising and inpainting digital images, their use for image compression was mainly focussing on pre- or postprocessing so far. In our paper we investigate their potential within the decoding step. We start with the observation that edge-enhancing diffusion (EED), an anisotropic nonlinear diffusion filter with a diffusion tensor, is well-suited for scattered data interpolation: Even when the interpolation data are very sparse, good results are obtained that respect discontinuities and satisfy a maximum–minimum principle. This property is exploited in our studies on PDE-based image compression. We use an adaptive triangulation method based on B-tree coding for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the EED process. They can be coded in a compact and elegant way that reflects the B-tree structure. Our experiments illustrate that for high compression rates and non-textured images, this PDE-based approach gives visually better results than the widely-used JPEG coding.


Image Compression Interpolation Point Image Code Average Absolute Error Homogeneous Diffusion 
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.
    Alter, F., Durand, S., Froment, J.: Adapted total variation for artifact free decompression of JPEG images. Journal of Mathematical Imaging and Vision 23(2), 199–211 (2005)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. SIGGRAPH 2000, New Orleans, LI, July 2000, pp. 417–424 (2000)Google Scholar
  3. 3.
    Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Transactions on Image Processing 7(3), 376–386 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chan, T.F., Shen, J.: Non-texture inpainting by curvature-driven diffusions (CDD). Journal of Visual Communication and Image Representation 12(4), 436–449 (2001)CrossRefGoogle Scholar
  5. 5.
    Chan, T.F., Zhou, H.M.: Total variation improved wavelet thresholding in image compression. In: Proc. Seventh International Conference on Image Processing, Vancouver, Canada, September 2000, vol. II, pp. 391–394 (2000)Google Scholar
  6. 6.
    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Transactions on Image Processing 6(2), 298–311 (1997)CrossRefGoogle Scholar
  7. 7.
    Demaret, L., Dyn, N., Iske, A.: Image compression by linear splines over adaptive triangulations. Technical report, Dept. of Mathematics, University of Leicester, UK (January 2005)Google Scholar
  8. 8.
    Distasi, R., Nappi, M., Vitulano, S.: Image compression by B-tree triangular coding. IEEE Transactions on Communications 45(9), 1095–1100 (1997)CrossRefGoogle Scholar
  9. 9.
    Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Mathematical Models and Numerical Analysis 10, 5–12 (1976)MathSciNetGoogle Scholar
  10. 10.
    Ford, G.E., Estes, R.R., Chen, H.: Scale-space analysis for image sampling and interpolation. In: Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, San Francisco, CA, March 1992, vol. 3, pp. 165–168 (1992)Google Scholar
  11. 11.
    Franke, R.: Scattered data interpolation: Tests of some methods. Mathematics of Computation 38, 181–200 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gothandaraman, A., Whitaker, R., Gregor, J.: Total variation for the removal of blocking effects in DCT based encoding. In: Proc. 2001 IEEE International Conference on Image Processing, Thessaloniki, Greece, October 2001, vol. 2, pp. 455–458 (2001)Google Scholar
  13. 13.
    Grossauer, H., Scherzer, O.: Using the complex Ginzburg–Landau equation for digital impainting in 2D and 3D. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 225–236. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Huffman, D.A.: A method for the construction of minimum redundancy codes. Proceedings of the IRE 40, 1098–1101 (1952)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Kopilovic, I., Szirányi, T.: Artifact reduction with diffusion preprocessing for image compression. Optical Engineering 44(2), 1–14 (2005)CrossRefGoogle Scholar
  17. 17.
    Lehmann, T., Gönner, C., Spitzer, K.: Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging 18(11), 1049–1075 (1999)CrossRefGoogle Scholar
  18. 18.
    Malgouyres, F., Guichard, F.: Edge direction preserving image zooming: A mathematical and numerical analysis. SIAM Journal on Numerical Analysis 39(1), 1–37 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: Proc. 1998 IEEE International Conference on Image Processing, Chicago, IL, October 1998, vol. 3, pp. 259–263 (1998)Google Scholar
  20. 20.
    Meijering, E.: A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE 90(3), 319–342 (2002)CrossRefGoogle Scholar
  21. 21.
    Mrázek, P.: Nonlinear Diffusion for Image Filtering and Monotonicity Enhancement. PhD thesis, Czech Technical University, Prague, Czech Republic (June 2001)Google Scholar
  22. 22.
    Nielson, G.M., Tvedt, J.: Comparing methods of interpolation for scattered volumetric data. In: Rogers, D.F., Earnshaw, R.A. (eds.) State of the Art in Computer Graphics: Aspects of Visualization, pp. 67–86. Springer, New York (1994)Google Scholar
  23. 23.
    Pennebaker, W.B., Mitchell, J.L.: JPEG: Still Image Data Compression Standard. Springer, New York (1992)Google Scholar
  24. 24.
    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
  25. 25.
    Solé, A., Caselles, V., Sapiro, G., Arandiga, F.: Morse description and geometric encoding of digital elevation maps. IEEE Transactions on Image Processing 13(9), 1245–1262 (2004)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Taubman, D.S., Marcellin, M.W. (eds.): JPEG 2000: Image Compression Fundamentals, Standards and Practice. Kluwer, Boston (2002)Google Scholar
  27. 27.
    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
  28. 28.
    Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Computing Supplement 11, 221–236 (1996)Google Scholar
  29. 29.
    Yang, S., Hu, Y.-H.: Coding artifact removal using biased anisotropic diffusion. In: Proc. 1997 IEEE International Conference on Image Processing, Santa Barbara, CA, October 1997, vol. 2, pp. 346–349 (1997)Google Scholar
  30. 30.
    Yao, S., Lin, W., Lu, Z., Ong, E.P., Yang, X.: Adaptive nonlinear diffusion processes for ringing artifacts removal on JPEG 2000 images. In: Proc. 2004 IEEE International Conference on Multimedia and Expo, Taipei, Taiwan, June 2004, pp. 691–694 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Irena Galić
    • 1
    • 2
  • Joachim Weickert
    • 1
  • Martin Welk
    • 1
  • Andrés Bruhn
    • 1
  • Alexander Belyaev
    • 2
  • Hans-Peter Seidel
    • 2
  1. 1.Mathematical Image Analysis Group, Faculty of Math. and Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Max-Planck Institute for InformaticsSaarbrückenGermany

Personalised recommendations