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Image Compression with Anisotropic Diffusion

Abstract

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec.

References

  1. 1.

    Alter, F., Durand, S., Froment, J.: Adapted total variation for artifact free decompression of JPEG images. J. Math. Imaging Vis. 23(2), 199–211 (2005)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Aly, H.A., Dubois, E.: Image up-sampling using total-variation regularization with a new observation model. IEEE Trans. Image Process. 14(10), 1647–1659 (2005)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147. Springer, New York (2002)

    MATH  Google Scholar 

  4. 4.

    Aurich, V., Daub, U.: Bilddatenkompression mit geplanten Verlusten und hoher Rate. In: Jähne, B., Geißler, P., Haußecker, H., Hering, F. (eds.) Mustererkennung 1996, pp. 138–146. Springer, Berlin (1996)

    Google Scholar 

  5. 5.

    Bae, E.: New PDE-based methods for surface and image reconstruction. Master’s thesis, Dept. of Mathematics, University of Bergen, Norway (2007)

  6. 6.

    Bajcsy, R., Kovačič, S.: Multiresolution elastic matching. Comput. Vis. Graph. Image Process. 46(1), 1–21 (1989)

    Article  Google Scholar 

  7. 7.

    Battiato, S., Gallo, G., Stanco, F.: Smart interpolation by anisotropic diffusion. In: Proc. Twelvth International Conference on Image Analysis and Processing, Montova, Italy, September 2003, pp. 572–577. IEEE Comput. Soc., Los Alamitos (2003)

    Chapter  Google Scholar 

  8. 8.

    Belahmidi, A., Guichard, F.: A partial differential equation approach to image zoom. In: Proc. 2004 IEEE International Conference on Image Processing, vol. 1, pp. 649–652. Singapore, October 2004

  9. 9.

    Belhachmi, Z., Bucur, D., Burgeth, B., Weickert, J.: How to choose interpolation data in images. Technical Report 205, Dept. of Mathematics, Saarland University, Saarbrücken, Germany, February 2008

  10. 10.

    Bertalmío, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. SIGGRAPH 2000, pp. 417–424. New Orleans, July 2000

  11. 11.

    Bornemann, F., März, T.: Fast image inpainting based on coherence transport. J. Math. Imaging Vis. 28(3), 259–278 (2007)

    Article  Google Scholar 

  12. 12.

    Bruckstein, A.M.: On image extrapolation. Technical Report CIS9316, Computer Science Department, Technion, Haifa, Israel, April 1993

  13. 13.

    Candés, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  Google Scholar 

  14. 14.

    Carlsson, S.: Sketch based coding of grey level images. Signal Process. 15, 57–83 (1988)

    Article  Google Scholar 

  15. 15.

    Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7(3), 376–386 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. 16.

    Chan, T.F., Shen, J.: Non-texture inpainting by curvature-driven diffusions (CDD). J. Vis. Commun. Image Represent. 12(4), 436–449 (2001)

    Article  Google Scholar 

  17. 17.

    Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)

    MATH  Google Scholar 

  18. 18.

    Chan, T.F., Zhou, H.M.: Feature preserving lossy image compression using nonlinear PDE’s. In: Luk, F.T. (ed.) Advanced Signal Processing Algorithms, Architectures, and Implementations VIII. Proceedings of SPIE, vol. 3461, pp. 316–327. SPIE Press, Bellingham (1998)

    Google Scholar 

  19. 19.

    Chan, T.F., Zhou, H.M.: Total variation improved wavelet thresholding in image compression. In: Proc. Seventh International Conference on Image Processing, vol. II, pp. 391–394. Vancouver, Canada, September 2000

  20. 20.

    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6(2), 298–311 (1997)

    Article  Google Scholar 

  21. 21.

    Dell, H.: Seed points in PDE-driven interpolation. Bachelor’s Thesis, Dept. of Computer Science, Saarland University, Saarbrücken, Germany (2006)

  22. 22.

    Demaret, L., Dyn, N., Iske, A.: Image compression by linear splines over adaptive triangulations. Signal Process. 86(7), 1604–1616 (2006)

    Article  MATH  Google Scholar 

  23. 23.

    Desai, U.Y., Mizuki, M.M., Masaki, I., Horn, B.K.P.: Edge and mean based image compression. Technical Report 1584 (A.I. Memo), Artificial Intelligence Lab., Massachusetts Institute of Technology, Cambridge, MA, USA, November 1996

  24. 24.

    Distasi, R., Nappi, M., Vitulano, S.: Image compression by B-tree triangular coding. IEEE Trans. Commun. 45(9), 1095–1100 (1997)

    Article  Google Scholar 

  25. 25.

    Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Math. Models Methods Appl. Sci. 10, 5–12 (1976)

    MathSciNet  Google Scholar 

  26. 26.

    Durand, S., Nikolova, M.: Restoration of wavelet coefficients by minimizing a specially designed objective function. In: Faugeras, O., Paragios, N. (eds.) Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision, Nice, France, October 2003. INRIA, Roequencourt (2003)

    Google Scholar 

  27. 27.

    Elder, J.H.: Are edges incomplete? Int. J. Comput. Vis. 34(2/3), 97–122 (1999)

    Article  Google Scholar 

  28. 28.

    Facciolo, G., Lecumberry, F., Almansa, A., Pardo, A., Caselles, V., Rougé, B.: Constrained anisotropic diffusion and some applications. In: Proc. 2006 British Machine Vision Conference, vol. 3, pp. 1049–1058. Edinburgh, Scotland, September 2006

  29. 29.

    Ford, G.E.: Application of inhomogeneous diffusion to image and video coding. In: Proc. 13th Asilomar Conference on Signals, Systems and Computers, vol. 2, pp. 926–930. Asilomar, CA, November 1996

  30. 30.

    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, vol. 3, pp. 165–168. San Francisco, CA, March 1992

  31. 31.

    Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38, 181–200 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. 32.

    Galić, I., Weickert, J., Welk, M., Bruhn, A., Belyaev, A., Seidel, H.-P.: Towards PDE-based image compression. In: Paragios, N., Faugeras, O., Chan, T., Schnörr, C. (eds.) Variational, Geometric and Level-Set Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3752, pp. 37–48. Springer, Berlin (2005)

    Chapter  Google Scholar 

  33. 33.

    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, vol. 2, pp. 455–458. Thessaloniki, Greece, October 2001

  34. 34.

    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 Methods in Computer Vision. Lecture Notes in Computer Science, vol. 2695, pp. 225–236. Springer, Berlin (2003)

    Chapter  Google Scholar 

  35. 35.

    Horn, B., Schunck, B.: Determining optical flow. Artif. Intell. 17, 185–203 (1981)

    Article  Google Scholar 

  36. 36.

    Huffman, D.A.: A method for the construction of minimum redundancy codes. Proc. IRE 40, 1098–1101 (1952)

    Article  Google Scholar 

  37. 37.

    Hummel, R., Moniot, R.: Reconstructions from zero-crossings in scale space. IEEE Trans. Acoust. Speech Signal Process. 37, 2111–2130 (1989)

    Article  Google Scholar 

  38. 38.

    Iijima, T.: Basic theory on normalization of pattern (in case of typical one-dimensional pattern). Bull. Electrotech. Lab. 26, 368–388 (1962). In Japanese

    Google Scholar 

  39. 39.

    Johansen, P., Skelboe, S., Grue, K., Andersen, J.D.: Representing signals by their toppoints in scale space. In: Proc. Eighth International Conference on Pattern Recognition, pp. 215–217. Paris, France, October 1986

  40. 40.

    Kanters, F.M.W., Lillholm, M., Duits, R., Jansen, B.J.P., Platel, B., Florack, L.M.J., ter Haar Romeny, B.M.: On image reconstruction from multiscale top points. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459, pp. 431–439. Springer, Berlin (2005)

    Google Scholar 

  41. 41.

    Kopilovic, I., Szirányi, T.: Artifact reduction with diffusion preprocessing for image compression. Opt. Eng. 44(2), 1–14 (2005)

    Article  Google Scholar 

  42. 42.

    Köstler, H., Stürmer, M., Freundl, C., Rüde, U.: PDE based video compression in real time. Technical Report 07-11, Lehrstuhl für Informatik 10, Univ. Erlangen–Nürnberg, Germany, 2007

  43. 43.

    Kunt, M., Ikonomopoulos, A., Kocher, M.: Second-generation image-coding techniques. Proc. IEEE 73(4), 549–574 (1985)

    Article  Google Scholar 

  44. 44.

    Lehmann, T., Gönner, C., Spitzer, K.: Survey: Interpolation methods in medical image processing. IEEE Trans. Med. Imag. 18(11), 1049–1075 (1999)

    Article  Google Scholar 

  45. 45.

    Lillholm, M., Nielsen, M., Griffin, L.D.: Feature-based image analysis. Int. J. Comput. Vis. 52(2/3), 73–95 (2003)

    Article  Google Scholar 

  46. 46.

    Liu, D., Sun, X., Wu, F., Li, S., Zhang, Y.-Q.: Image compression with edge-based inpainting. IEEE Trans. Circuits Syst. Video Technol. 17(10), 1273–1286 (2007)

    Article  Google Scholar 

  47. 47.

    Malgouyres, F., Guichard, F.: Edge direction preserving image zooming: a mathematical and numerical analysis. SIAM J. Numer. Anal. 39(1), 1–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. 48.

    Mallat, S., Zhong, S.: Characterisation of signals from multiscale edges. IEEE Trans. Pattern Anal. Mach. Intell. 14, 720–732 (1992)

    Article  Google Scholar 

  49. 49.

    March, R.: Computation of stereo disparity using regularization. Pattern Recogn. Lett. 8, 181–187 (1988)

    Article  Google Scholar 

  50. 50.

    Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: Proc. 1998 IEEE International Conference on Image Processing, vol. 3, pp. 259–263. Chicago, IL, October 1998

  51. 51.

    Meijering, E.: A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proc. IEEE 90(3), 319–342 (2002)

    Article  Google Scholar 

  52. 52.

    Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  53. 53.

    Mrázek, P.: Nonlinear diffusion for image filtering and monotonicity enhancement. Ph.D. thesis, Czech Technical University, Prague, Czech Republic, June 2001

  54. 54.

    Nagel, H.-H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. Pattern Anal. Mach. Intell. 8, 565–593 (1986)

    Article  Google Scholar 

  55. 55.

    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 

  56. 56.

    Pennebaker, W.B., Mitchell, J.L.: JPEG: Still Image Data Compression Standard. Springer, New York (1992)

    Google Scholar 

  57. 57.

    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  58. 58.

    Rane, S.D., Sapiro, G., Bertalmio, M.: Structure and texture filling-in of missing image blocks in wireless transmission and compression applications. IEEE Trans. Image Process. 12(3), 296–302 (2003)

    Article  MathSciNet  Google Scholar 

  59. 59.

    Rissanen, J., Langdon, G.G. Jr.: Arithmetic coding. IBM J. Res. Develop. 23(2), 149–162 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Roussos, A., Maragos, P.: Vector-valued image interpolation by an anisotropic diffusion-projection PDE. In: Sgallari, F., Murli, F., Paragios, N. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 4485, pp. 104–115. Springer, Berlin (2007)

    Chapter  Google Scholar 

  61. 61.

    Solé, A., Caselles, V., Sapiro, G., Arandiga, F.: Morse description and geometric encoding of digital elevation maps. IEEE Trans. Image Process. 13(9), 1245–1262 (2004)

    Article  MathSciNet  Google Scholar 

  62. 62.

    Strobach, P.: Quadtree-structured recursive plane decomposition coding of images. IEEE Trans. Signal Process. 39(6), 1380–1397 (1991)

    Article  Google Scholar 

  63. 63.

    Sullivan, G.J., Baker, R.J.: Efficient quadtree coding of images and video. IEEE Trans. Image Process. 3(3), 327–331 (1994)

    Article  Google Scholar 

  64. 64.

    Taubman, D.S., Marcellin, M.W. (eds.): JPEG 2000: Image Compression Fundamentals, Standards and Practice. Kluwer, Boston (2002)

    Google Scholar 

  65. 65.

    Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDEs: a common framework for different applications. IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 506–516 (2005)

    Article  Google Scholar 

  66. 66.

    Tsuji, H., Sakatani, T., Yashima, Y., Kobayashi, N.: A nonlinear spatio-temporal diffusion and its application to prefiltering in MPEG-4 video coding. In: Proc. 2002 IEEE International Conference on Image Processing, vol. 1, pp. 85–88. Rochester, NY, September 2002

  67. 67.

    Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Computing Suppl. 11, 221–236 (1996)

    Google Scholar 

  68. 68.

    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  69. 69.

    Weickert, J., Welk, M.: Tensor field interpolation with PDEs. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 315–325. Springer, Berlin (2006)

    Chapter  Google Scholar 

  70. 70.

    Xie, Z., Franklin, W.R., Cutler, B., Andrade, M.A., Inanc, M., Tracy, D.M.: Surface compression using over-determined Laplacian approximation. In: Luk, F.T. (ed.) Advanced Signal Processing Algorithms, Architectures, and Implementations XVII. Proceedings of SPIE, vol. 5266. SPIE Press, Bellingham (2007)

    Google Scholar 

  71. 71.

    Xiong, Z.W., Sun, X.Y., Wu, F., Li, S.P.: Image coding with parameter-assistant inpainting. In: Proc. 2007 IEEE International Conference on Image Processing, vol. 2, pp. 369–372. San Antonio, TX, September 2007

  72. 72.

    Yang, S., Hu, Y.-H.: Coding artifact removal using biased anisotropic diffusion. In: Proc. 1997 IEEE International Conference on Image Processing, vol. 2, pp. 346–349. Santa Barbara, CA, October 1997

  73. 73.

    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, pp. 691–694. Taipei, Taiwan, June 2004

  74. 74.

    Yokoya, N.: Surface reconstruction directly from binocular stereo images by multiscale-multistage regularization. In: Proc. Eleventh International Conference on Pattern Recognition, vol. 1, pp. 642–646. The Hague, The Netherlands, August 1992

  75. 75.

    Zeevi, Y., Rotem, D.: Image reconstruction from zero-crossings. IEEE Trans. Acoust. Speech Signal Process. 34, 1269–1277 (1986)

    Article  Google Scholar 

  76. 76.

    Zimmer, H.: PDE-based image compression using corner information. Master’s thesis, Dept. of Computer Science, Saarland University, Saarbrücken, Germany (2007)

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Galić, I., Weickert, J., Welk, M. et al. Image Compression with Anisotropic Diffusion. J Math Imaging Vis 31, 255–269 (2008). https://doi.org/10.1007/s10851-008-0087-0

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Keywords

  • Partial differential equations
  • Nonlinear diffusion
  • Image compression
  • Image inpainting