Towards PDE-Based Image Compression

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.

[1]  Jean-Michel Morel,et al.  An axiomatic approach to image interpolation , 1997, Proceedings of International Conference on Image Processing.

[2]  Michael W. Marcellin,et al.  JPEG2000 - image compression fundamentals, standards and practice , 2002, The Kluwer International Series in Engineering and Computer Science.

[3]  Nira Dyn,et al.  Image compression by linear splines over adaptive triangulations , 2006, Signal Process..

[4]  A. Gothandaraman,et al.  Total variation for the removal of blocking effects in DCT based encoding , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[5]  David A. Huffman,et al.  A method for the construction of minimum-redundancy codes , 1952, Proceedings of the IRE.

[6]  Tony F. Chan,et al.  Non-texture inpainting by curvature-driven diffusions (CDD) , 2001 .

[7]  Hong Chen,et al.  Space scale analysis for image sampling and interpolation , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[8]  Joachim Weickert,et al.  Theoretical Foundations of Anisotropic Diffusion in Image Processing , 1994, Theoretical Foundations of Computer Vision.

[9]  E. Meijering.  A chronology of interpolation: from ancient astronomy to modern signal and image processing , 2002, Proc. IEEE.

[10]  Rae A. Earnshaw,et al.  State of the art in computer graphics: aspects of visualization , 1994 .

[11]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  Jacques Froment,et al.  Adapted Total Variation for Artifact Free Decompression of JPEG Images , 2005, Journal of Mathematical Imaging and Vision.

[13]  Otmar Scherzer,et al.  Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D , 2003, Scale-Space.

[14]  Gregory M. Nielson,et al.  Comparing methods of inter-polation for scattered volumetric data , 1994 .

[15]  Yu Hen Hu,et al.  Coding artifacts removal using biased anisotropic diffusion , 1997, Proceedings of International Conference on Image Processing.

[16]  Jean Duchon,et al.  Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces , 1976 .

[17]  Riccardo Distasi,et al.  Image compression by B-tree triangular coding , 1997, IEEE Trans. Commun..

[18]  Thomas Martin Deserno,et al.  Survey: interpolation methods in medical image processing , 1999, IEEE Transactions on Medical Imaging.

[19]  Tony F. Chan,et al.  Total variation improved wavelet thresholding in image compression , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[20]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

[21]  Guillermo Sapiro,et al.  Image inpainting , 2000, SIGGRAPH.

[22]  Tamás Szirányi,et al.  Artifact reduction with diffusion preprocessing for image compression , 2005 .

[23]  E. Meijering,et al.  A chronology of interpolation: from ancient astronomy to modern signal and image processing , 2002, Proc. IEEE.

[24]  Guillermo Sapiro,et al.  Morse description and geometric encoding of digital elevation maps , 2004, IEEE Transactions on Image Processing.

[25]  Lewis D. Griffin,et al.  Scale Space Methods in Computer Vision , 2003, Lecture Notes in Computer Science.

[26]  Rachid Deriche,et al.  Vector-valued image regularization with PDEs: a common framework for different applications , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  Weisi Lin,et al.  Adaptive nonlinear diffusion processes for ringing artifacts removal on JPEG 2000 images , 2004, 2004 IEEE International Conference on Multimedia and Expo (ICME) (IEEE Cat. No.04TH8763).

[28]  Jean-Michel Morel,et al.  Level lines based disocclusion , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[29]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[30]  Joan L. Mitchell,et al.  JPEG: Still Image Data Compression Standard , 1992 .

[31]  Tony F. Chan,et al.  Nontexture Inpainting by Curvature-Driven Diffusions , 2001, J. Vis. Commun. Image Represent..

[32]  François Malgouyres,et al.  Edge Direction Preserving Image Zooming: A Mathematical and Numerical Analysis , 2001, SIAM J. Numer. Anal..