Advertisement

International Journal of Computer Vision

, Volume 108, Issue 3, pp 222–240 | Cite as

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

  • Christian SchmaltzEmail author
  • Pascal Peter
  • Markus Mainberger
  • Franziska Ebel
  • Joachim Weickert
  • Andrés Bruhn
Article

Abstract

Galić et al. (Journal of Mathematical Imaging and Vision 31:255–269, 2008) have shown that compression based on edge-enhancing anisotropic diffusion (EED) can outperform the quality of JPEG for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation. However, the reasons for the good performance of EED remained unclear, and they could not outperform the more advanced JPEG 2000. The goals of the present paper are threefold: Firstly, we investigate the compression qualities of various partial differential equations. This sheds light on the favourable properties of EED in the context of image compression. Secondly, we demonstrate that it is even possible to beat the quality of JPEG 2000 with EED if one uses specific subdivisions on rectangles and several important optimisations. These amendments include improved entropy coding, brightness and diffusivity optimisation, and interpolation swapping. Thirdly, we demonstrate how to extend our approach to 3-D and shape data. Experiments on classical test images and 3-D medical data illustrate the high potential of our approach.

Keywords

Image compression Edge-enhancing anisotropic diffusion (EED) Partial differential equations (PDEs) Subdivision JPEG 2000 Kanizsa triangle 

Notes

Acknowledgments

We thank Irena Galić (University of Osijek, Croatia) for fruitful discussions and for providing two images and code of her compression algorithm, David Tschumperlé (French National Center for Scientific Research, France) for providing the image in Fig. 1, and Wiro Niessen (University Medical Center Rotterdam, The Netherlands) for providing the test image in Fig. 12.

References

  1. Acar, T., & Gökmen, M. (1994). Image coding using weak membrane model of images. In A. K. Katsaggelos (Ed.), Proceedings of SPIE on Visual Communications and Image Processing ’94 (Vol. 2308). Bellingham: SPIE Press.Google Scholar
  2. Alter, F., Durand, S., & Froment, J. (2005). Adapted total variation for artifact free decompression of JPEG images. Journal of Mathematical Imaging and Vision, 23(2), 199–211.CrossRefMathSciNetGoogle Scholar
  3. Aly, H. A., & Dubois, E. (2005). Image up-sampling using total-variation regularization with a new observation model. IEEE Transactions on Image Processing, 14(10), 1647–1659.CrossRefMathSciNetGoogle Scholar
  4. Aronsson, G. (1967). Extension of functions satisfying Lipschitz conditions. Arkiv för Matematik, 6(6), 551–561.CrossRefzbMATHMathSciNetGoogle Scholar
  5. Aurich, V., & Daub, U. (1996). Bilddatenkompression mit geplanten Verlusten und hoher Rate. In B. Jähne, P. Geißler, H. Haußecker, & F. Hering (Eds.), Mustererkennung 1996 (pp. 138–146). Berlin: Springer.CrossRefGoogle Scholar
  6. Bae, E., & Weickert, J. (2010). Partial differential equations for interpolation and compression of surfaces. In M. Daehlen, M. Floater, T. Lyche, J. L. Merrien, K. Mørken, & L. L. Schumaker (Eds.), Mathematical methods for curves and surfaces. Lecture notes in computer science (Vol. 5862, pp. 1–14). Berlin: Springer.Google Scholar
  7. Battiato, S., Gallo, G., & Stanco, F. (2003). Smart interpolation by anisotropic diffusion. In Proceedings of Twelvth International Conference on Image Analysis and Processing (pp. 572–577). Montova: IEEE Computer Society Press.Google Scholar
  8. Belahmidi, A., & Guichard, F. (2004). A partial differential equation approach to image zoom. In Procroceedings of 2004 IEEE International Conference on Image Processing (Vol. 1, pp. 649–652), Singapore.Google Scholar
  9. Belhachmi, Z., Bucur, D., Burgeth, B., & Weickert, J. (2009). How to choose interpolation data in images. SIAM Journal on Applied Mathematics, 70(1), 333–352.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Bertalmío, M., Sapiro, G., Caselles, V., & Ballester, C. (2000). Image inpainting. In Proceedings of SIGGRAPH 2000 (pp. 417–424), New Orleans.Google Scholar
  11. Bornemann, F., & März, T. (2007). Fast image inpainting based on coherence transport. Journal of Mathematical Imaging and Vision, 28(3), 259–278.CrossRefMathSciNetGoogle Scholar
  12. Borzì, A., Grossauer, H., & Scherzer, O. (2005). Analysis of iterative methods for solving a Ginzburg–Landau equation. International Journal of Computer Vision, 64(2–3), 203–219.CrossRefGoogle Scholar
  13. Bougleux, S., Peyré, G., & Cohen, L. (2009). Image compression with anisotropic triangulations. In Proceedings of Tenth International Conference on Computer Vision, Kyoto.Google Scholar
  14. Bourdon, P., Augereau, B., Chatellier, C., & Olivier, C. (2004). MPEG-4 compression artifacts removal on color video sequences using 3D nonlinear diffusion. In Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Vol. 3, pp 729–732). Montreal: IEEE Computer Society Press.Google Scholar
  15. Bourne, R. (2010). Fundamentals of digital imaging in medicine. London: Springer.CrossRefGoogle Scholar
  16. Bruckstein, A. M. (1993). On image extrapolation. Technical Report on CIS9316. Haifa: Computer Science Department, Technion.Google Scholar
  17. Bruhn, A., & Weickert, J. (2006). A confidence measure for variational optic flow methods. In R. Klette, R. Kozera, L. Noakes, & J. Weickert (Eds.), Geometric properties from incomplete data. Computational imaging and vision (Vol. 31, pp. 283–297). Dordrecht: Springer. Google Scholar
  18. Candés, E., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.CrossRefzbMATHGoogle Scholar
  19. Carlsson, S. (1988). Sketch based coding of grey level images. Signal Processing, 15, 57–83.CrossRefGoogle Scholar
  20. Caselles, V., Morel, J. M., & Sbert, C. (1998). An axiomatic approach to image interpolation. IEEE Transactions on Image Processing, 7(3), 376–386.CrossRefzbMATHMathSciNetGoogle Scholar
  21. Catté, F., Lions, P. L., Morel, J. M., & Coll, T. (1992). Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis, 32, 1895–1909.CrossRefGoogle Scholar
  22. Chan, T. F., & Shen, J. (2001). Non-texture inpainting by curvature-driven diffusions (CDD). Journal of Visual Communication and Image Representation, 12(4), 436–449.CrossRefGoogle Scholar
  23. Chan, T. F., & Zhou, H. M. (2000). Total variation improved wavelet thresholding in image compression. In Proceedings of Seventh International Conference on Image Processing (Vol. 2, pp. 391–394). Vancouver, Canada.Google Scholar
  24. Charbonnier, P., Blanc-Féraud, L., Aubert, G., & Barlaud, M. (1997). Deterministic edge-preserving regularization in computed imaging. IEEE Transactions on Image Processing, 6(2), 298–311.CrossRefGoogle Scholar
  25. Demaret, L., Dyn, N., & Iske, A. (2006). Image compression by linear splines over adaptive triangulations. Signal Processing, 86(7), 1604–1616.CrossRefzbMATHGoogle Scholar
  26. Desai, U. Y., Mizuki, M. M., Masaki, I., & Horn, B. K. P. (1996). Edge and mean based image compression. Technical Report 1584 (A.I. Memo). Cambridge, MA: Artificial Intelligence Lab., Massachusetts Institute of Technology.Google Scholar
  27. Dipperstein, M. (2009). Michael Dipperstein’s page o’stuff. http://michael.dipperstein.com/index.html.
  28. Distasi, R., Nappi, M., & Vitulano, S. (1997). Image compression by B-tree triangular coding. IEEE Transactions on Communications, 45(9), 1095–1100.CrossRefGoogle Scholar
  29. Elder, J. H. (1999). Are edges incomplete? International Journal of Computer Vision, 34(2/3), 97–122.CrossRefGoogle Scholar
  30. Facciolo, G., Lecumberry, F., Almansa, A., Pardo, A., Caselles, V., & Rougé, B. (2006). Constrained anisotropic diffusion and some applications. In: Proceeings of 2006 British Machine Vision Conference (Vol. 3, pp. 1049–1058), Edinburgh.Google Scholar
  31. Ford, G. E. (1996). Application of inhomogeneous diffusion to image and video coding. In Proceedings of 13th Asilomar Conference on Signals, Systems and Computers (Vol. 2, pp. 926–930), Asilomar, CA.Google Scholar
  32. Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., & Seidel, H. P. (2005). Towards PDE-based image compression. In N. Paragios, O. Faugeras, T. Chan, & C. Schnörr (Eds.), Variational, Geometric and Level-Set Methods in Computer Vision. Lecture Notes in Computer Science (Vol. 3752, pp. 37–48). Berlin: Springer.Google Scholar
  33. Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., & Seidel, H. P. (2008). Image compression with anisotropic diffusion. Journal of Mathematical Imaging and Vision, 31(2–3), 255–269.MathSciNetGoogle Scholar
  34. Gourlay, A. R. (1970). Hopscotch: A fast second-order partial differential equation solver. IMA Journal of Applied Mathematics, 6(4), 375–390.CrossRefzbMATHMathSciNetGoogle Scholar
  35. Huffman, D. A. (1952). A method for the construction of minimum redundancy codes. Proceedings of the IRE, 40, 1098–1101.CrossRefGoogle Scholar
  36. Hummel, R., & Moniot, R. (1989). Reconstructions from zero-crossings in scale space. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37, 2111–2130.CrossRefGoogle Scholar
  37. Iijima, T. (1959). Basic theory of pattern observation. In Papers of Technical Group on Automata and Automatic Control, Japan: IECE (in Japanese).Google Scholar
  38. Johansen, P., Skelboe, S., Grue, K., & Andersen, J. D. (1986). Representing signals by their toppoints in scale space. In Proceedings of Eighth International Conference on Pattern Recognition (pp 215–217). Paris.Google Scholar
  39. Kanters, F. M. W., Lillholm, M., Duits, R., Jansen, B. J. P., Platel, B., Florack, L., et al. (2005). On image reconstruction from multiscale top points. In R. Kimmel, N. Sochen, & J. Weickert (Eds.), Scale Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science (Vol. 3459, pp. 431–439). Berlin: Springer.Google Scholar
  40. Kopilovic, I., & Szirányi, T. (2005). Artifact reduction with diffusion preprocessing for image compression. Optical Engineering, 44(2), 1–14.Google Scholar
  41. Köstler, H., Stürmer, M., Freundl, C., & Rüde, U. (2007). PDE based video compression in real time. Technical Report 07–11, Lehrstuhl für Informatik 10, Univ. Erlangen-Nürnberg, Germany.Google Scholar
  42. Kunt, M., Ikonomopoulos, A., & Kocher, M. (1985). Second-generation image-coding techniques. Proceedings of the IEEE, 73(4), 549–574.CrossRefGoogle Scholar
  43. Lillholm, M., Nielsen, M., & Griffin, L. D. (2003). Feature-based image analysis. International Journal of Computer Vision, 52(2/3), 73–95.CrossRefGoogle Scholar
  44. Liu, D., Sun, X., Wu, F., Li, S., & Zhang, Y. Q. (2007). Image compression with edge-based inpainting. IEEE Transactions on Circuits, Systems and Video Technology, 17(10), 1273–1286.CrossRefGoogle Scholar
  45. Mahoney, M. (2005). Adaptive weighing of context models for lossless data compression. Technical Report CS-2005-16. Melbourne, FL: Florida Institute of Technology.Google Scholar
  46. Mahoney, M. (2009). Data compression programs. http://mattmahoney.net/dc/.
  47. Mainberger, M., & Weickert, J. (2009). Edge-based image compression with homogeneous diffusion. In X. Jiang & N. Petkov (Eds.), Computer Analysis of Images and Patterns. Lecture Notes in Computer Science (Vol. 5702, pp. 476–483). Berlin: Springer.Google Scholar
  48. Mainberger, M., Hoffmann, S., Weickert, J., Tang, C. H., Johannsen, D., Neumann, F., et al. (2012). Optimising spatial and tonal data for homogeneous diffusion inpainting. In A. M. Bruckstein, t B. er Haar Romeny, A. M. Bronstein, & M. M. Bronstein (Eds.), Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science (pp. 26–37). Berlin: Springer.Google Scholar
  49. Malgouyres, F., & Guichard, F. (2001). Edge direction preserving image zooming: A mathematical and numerical analysis. SIAM Journal on Numerical Analysis, 39(1), 1–37.CrossRefzbMATHMathSciNetGoogle Scholar
  50. Mallat, S., & Zhong, S. (1992). Characterisation of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 720–732.CrossRefGoogle Scholar
  51. Masnou, S., & Morel, J. M. (1998). Level lines based disocclusion. In Proceedings of the 1998 IEEE International Conference on Image Processing (Vol. 3, pp. 259–263), Chicago, IL.Google Scholar
  52. National Electrical Manufacturers Association. (2004). Digital Imaging and Communications in Medicine (DICOM): Part 5 Data Structures and Encoding. PS 3.5-2004.Google Scholar
  53. Pennebaker, W. B., & Mitchell, J. L. (1992). JPEG: Still image data compression standard. New York: Springer.Google Scholar
  54. Perona, P., & Malik, J. (1990). Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 629–639.CrossRefGoogle Scholar
  55. Peter, P. (2012). Three-dimensional data compression with anisotropic diffusion. In Proceedings of DAGM-OAGM 2012 Symposium for Pattern Recognition, Young Researchers Forum. Berlin: Springer.Google Scholar
  56. Rane, S. D., Sapiro, G., & Bertalmio, M. (2003). Structure and texture filling-in of missing image blocks in wireless transmission and compression applications. IEEE Transactions on Image Processing, 12(3), 296–302.CrossRefMathSciNetGoogle Scholar
  57. Rissanen, J. J. (1976). Generalized Kraft inequality and arithmetic coding. IBM Journal of Research and Development, 20(3), 198–203. Google Scholar
  58. Roussos, A., & Maragos, P. (2007). Vector-valued image interpolation by an anisotropic diffusion-projection PDE. In F. Sgallari, F. Murli, & N. Paragios (Eds.), Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science (Vol. 4485, pp. 104–115). Berlin: Springer.Google Scholar
  59. Schmaltz, C., Weickert, J., & Bruhn, A. (2009). Beating the quality of JPEG 2000 with anisotropic diffusion. In J. Denzler, G. Notni, & H. Süße (Eds.), Pattern Recognition. Lecture Notes in Computer Science (Vol. 5748, pp. 452–461). Berlin: Springer.Google Scholar
  60. Solé, A., Caselles, V., Sapiro, G., & Arandiga, F. (2004). Morse description and geometric encoding of digital elevation maps. IEEE Transactions on Image Processing, 13(9), 1245–1262.CrossRefMathSciNetGoogle Scholar
  61. Strobach, P. (1991). Quadtree-structured recursive plane decomposition coding of images. IEEE Transactions on Signal Processing, 39(6), 1380–1397.CrossRefGoogle Scholar
  62. Sullivan, G. J., & Baker, R. J. (1994). Efficient quadtree coding of images and video. IEEE Transactions on Image Processing, 3(3), 327–331.CrossRefGoogle Scholar
  63. Taubman, D. S., & Marcellin, M. W. (Eds.). (2002). JPEG 2000: Image compression fundamentals. Standards and practice. Boston: Kluwer.Google Scholar
  64. Telea, A. (2004). An image inpainting technique based on the fast marching method. Journal of Graphics Tools, 9(1), 23–34.CrossRefGoogle Scholar
  65. Tschumperlé, D. (2006). Fast anisotropic smoothing of multi-valued images using curvature-preserving PDE’s. International Journal of Computer Vision, 68(1), 65–82.CrossRefGoogle Scholar
  66. Tschumperlé, D., & Deriche, R. (2005). Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(4), 506–516.CrossRefGoogle Scholar
  67. Tsuji, H., Sakatani, T., Yashima, Y., & Kobayashi, N. (2002). A nonlinear spatio-temporal diffusion and its application to prefiltering in MPEG-4 video coding. In Proceedings of 2002 IEEE International Conference on Image Processing (Vol. 1, pp. 85–88), Rochester, NY.Google Scholar
  68. Tsuji, H., Tokumasu, S., Takahashi, H., & Nakajima, M. (2007). Spatial prefiltering scheme based on anisotropic diffusion in low-bitrate video coding. Systems and Computers in Japan, 38(10), 34–45.CrossRefGoogle Scholar
  69. Weickert, J. (1996). Theoretical foundations of anisotropic diffusion in image processing. Computing Supplement, 11, 221–236.CrossRefGoogle Scholar
  70. Weickert, J. (1999). Nonlinear diffusion filtering. In B. Jähne, H. Haußecker, & P. Geißler (Eds.), Handbook on computer vision and applications. Signal processing and pattern recognition (Vol. 2, pp. 423–450). San Diego: Academic Press.Google Scholar
  71. Weickert, J., & Brox, T. (2002). Diffusion and regularization of vector- and matrix-valued images. In M. Z. Nashed & O. Scherzer (Eds.), Inverse problems, image analysis, and medical imaging, contemporary mathematics (Vol. 313, pp. 251–268). Providence: AMS.CrossRefGoogle Scholar
  72. Weickert, J., & Welk, M. (2006). Tensor field interpolation with PDEs. In J. Weickert & H. Hagen (Eds.), Visualization and processing of tensor fields (pp. 315–325). Berlin: Springer.CrossRefGoogle Scholar
  73. Weickert, J., Ishikawa, S., & Imiya, A. (1999). Linear scale-space has first been proposed in Japan. Journal of Mathematical Imaging and Vision, 10(3), 237–252.CrossRefzbMATHMathSciNetGoogle Scholar
  74. Welch, T. A. (1984). A technique for high-performance data compression. Computer, 17(6), 8–19.CrossRefGoogle Scholar
  75. Wu, Y., Zhang, H., Sun, Y., & Guo, H. (2009) Two image compression schemes based on image inpainting. In: Proceedings of the 2009 International Joint Conference on Computational Sciences and Optimization (pp. 816–820). IEEE Computer Society Press: San Francisco, CA.Google Scholar
  76. Xie, Z., Franklin, W. R., Cutler, B., Andrade, M. A., Inanc, M., & Tracy, D. M. (2007). Surface compression using over-determined Laplacian approximation. In F. T. Luk (Ed.), Advanced Signal Processing Algorithms, Architectures, and Implementations XVII. Proceedings of SPIE (Vol. 6697). Bellingham: SPIE Press.Google Scholar
  77. Xiong, Z. W., Sun, X. Y., Wu, F., & Li, S. P. (2007) Image coding with parameter-assistant inpainting. In: Proceedings of the 2007 IEEE International Conference on Image Processing (Vol. 2, pp. 369–372), San Antonio, TX. Google Scholar
  78. Zeevi, Y., & Rotem, D. (1986). Image reconstruction from zero-crossings. IEEE Transactions on Acoustics, Speech, and Signal Processing, 34, 1269–1277.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Christian Schmaltz
    • 1
    Email author
  • Pascal Peter
    • 1
  • Markus Mainberger
    • 1
  • Franziska Ebel
    • 1
  • Joachim Weickert
    • 1
  • Andrés Bruhn
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceSaarland University SaarbrückenGermany

Personalised recommendations