Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

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.

[1]  Folkmar Bornemann,et al.  Fast Image Inpainting Based on Coherence Transport , 2007, Journal of Mathematical Imaging and Vision.

[2]  Ichiro Masaki,et al.  Edge and Mean based Image Compression , 1996 .

[3]  Stefan Carlsson,et al.  Sketch based coding of grey level images , 1988 .

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

[5]  Guillermo Sapiro,et al.  Structure and texture filling-in of missing image blocks in wireless transmission and compression applications , 2003, IEEE Trans. Image Process..

[6]  Volker Aurich,et al.  Bilddatenkompression mit geplanten Verlusten und hoher Rate , 1996, DAGM-Symposium.

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

[8]  Laurent D. Cohen,et al.  Image compression with anisotropic triangulations , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[9]  Eric Dubois,et al.  Image up-sampling using total-variation regularization with a new observation model , 2005, IEEE Transactions on Image Processing.

[10]  Omer Demirkaya,et al.  Nonlinear Diffusion Filtering , 2008 .

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

[12]  Joachim Weickert,et al.  Edge-Based Image Compression with Homogeneous Diffusion , 2009, CAIP.

[13]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[14]  A. R. Gourlay,et al.  Hopscotch: a Fast Second-order Partial Differential Equation Solver , 1970 .

[15]  James H. Elder,et al.  Are Edges Incomplete? , 1999, International Journal of Computer Vision.

[16]  Otmar Scherzer,et al.  Analysis of Iterative Methods for Solving a Ginzburg-Landau Equation , 2005, International Journal of Computer Vision.

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

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

[19]  Jorma Rissanen,et al.  Generalized Kraft Inequality and Arithmetic Coding , 1976, IBM J. Res. Dev..

[20]  Luc Florack,et al.  On Image Reconstruction from Multiscale Top Points , 2005, Scale-Space.

[21]  T. Brox,et al.  Diffusion and regularization of vector- and matrix-valued images , 2002 .

[22]  Gary J. Sullivan,et al.  Efficient quadtree coding of images and video , 1994, IEEE Trans. Image Process..

[23]  Atsushi Imiya,et al.  Linear Scale-Space has First been Proposed in Japan , 1999, Journal of Mathematical Imaging and Vision.

[24]  Christian Chatellier,et al.  MPEG-4 compression artifacts removal on color video sequences using 3D nonlinear diffusion , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[25]  Joachim Weickert,et al.  Editorial: Special issue for the 5th International Conference on Scale-Space and PDE Methods in Computer Vision , 2006, International Journal of Computer Vision.

[26]  Hans-Peter Seidel,et al.  Towards PDE-Based Image Compression , 2005, VLSM.

[27]  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.

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

[29]  H. Tsuji,et al.  A nonlinear spatio-temporal diffusion and its application to prefiltering in MPEG-4 video coding , 2002, Proceedings. International Conference on Image Processing.

[30]  Hans-Peter Seidel,et al.  Image Compression with Anisotropic Diffusion , 2008, Journal of Mathematical Imaging and Vision.

[31]  Michael W. Marcellin,et al.  JPEG2000 - image compression fundamentals, standards and practice , 2013, The Kluwer international series in engineering and computer science.

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

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

[34]  Sebastiano Battiato,et al.  Smart interpolation by anisotropic diffusion , 2003, 12th International Conference on Image Analysis and Processing, 2003.Proceedings..

[35]  T D Cradduck,et al.  National electrical manufacturers association , 1983, Journal of the A.I.E.E..

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

[37]  Mads Nielsen,et al.  Feature-Based Image Analysis , 2003, International Journal of Computer Vision.

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

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

[40]  Yehoshua Y. Zeevi,et al.  Image reconstruction from zero crossings , 1986, IEEE Trans. Acoust. Speech Signal Process..

[41]  G.E. Ford,et al.  Application of inhomogeneous diffusion to image and video coding , 1996, Conference Record of The Thirtieth Asilomar Conference on Signals, Systems and Computers.

[42]  Frank Neumann,et al.  Optimising Spatial and Tonal Data for Homogeneous Diffusion Inpainting , 2011, SSVM.

[43]  Tolga Acar,et al.  Image coding using a weak membrane model of images , 1994, Other Conferences.

[44]  Peter Strobach,et al.  Quadtree-structured recursive plane decomposition coding of images , 1991, IEEE Trans. Signal Process..

[45]  Joachim Weickert,et al.  How to Choose Interpolation Data in Images , 2009, SIAM J. Appl. Math..

[46]  Marie Weisz,et al.  Fundamentals Of Digital Imaging In Medicine , 2016 .

[47]  Masayuki Nakajima,et al.  Spatial prefiltering scheme based on anisotropic diffusion in low-bitrate video coding , 2007, Systems and Computers in Japan.

[48]  Zhiwei Xiong,et al.  Image Coding with Parameter-Assistant Inpainting , 2007, 2007 IEEE International Conference on Image Processing.

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

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

[51]  Yu Sun,et al.  Two Image Compression Schemes Based on Image Inpainting , 2009, 2009 International Joint Conference on Computational Sciences and Optimization.

[52]  Federico Lecumberry,et al.  Constrained Anisotropic Diffusion and some Applications , 2006, BMVC.

[53]  G. Aronsson.  Extension of functions satisfying lipschitz conditions , 1967 .

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

[55]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[56]  H. Köstler,et al.  PDE based Video Compression in Real Time , 2007 .

[57]  Oleg S. Pianykh,et al.  Digital Imaging and Communications in Medicine (DICOM) , 2017, Radiopaedia.org.

[58]  Stéphane Mallat,et al.  Characterization of Signals from Multiscale Edges , 2011, IEEE Trans. Pattern Anal. Mach. Intell..

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

[60]  Frédéric Guichard,et al.  A partial differential equation approach to image zoom , 2004, 2004 International Conference on Image Processing, 2004. ICIP '04..

[61]  J. Weickert,et al.  A Confidence Measure for Variational Optic flow Methods , 2006 .

[62]  Alfred M. Bruckstein,et al.  ON IMAGE EXTRAPOLATION. , 1993 .

[63]  Annett Baier,et al.  Geometric Properties For Incomplete Data , 2016 .

[64]  Joachim Weickert,et al.  Beating the Quality of JPEG 2000 with Anisotropic Diffusion , 2009, DAGM-Symposium.

[65]  Matthew V. Mahoney,et al.  Adaptive weighing of context models for lossless data compression , 2005 .

[66]  Barbara Cutler,et al.  Surface compression using over-determined Laplacian approximation , 2007, SPIE Optical Engineering + Applications.

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

[68]  Joachim Weickert,et al.  Partial Differential Equations for Interpolation and Compression of Surfaces , 2008, MMCS.

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

[70]  M. Kunt,et al.  Second-generation image-coding techniques , 1985, Proceedings of the IEEE.

[71]  David Tschumperlé,et al.  Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE's , 2006, International Journal of Computer Vision.

[72]  Petros Maragos,et al.  Vector-Valued Image Interpolation by an Anisotropic Diffusion-Projection PDE , 2007, SSVM.

[73]  Dong Liu,et al.  Image Compression With Edge-Based Inpainting , 2007, IEEE Transactions on Circuits and Systems for Video Technology.

[74]  Terry A. Welch,et al.  A Technique for High-Performance Data Compression , 1984, Computer.

[75]  Ian Lewis,et al.  Proceedings of the SPIE , 2012 .

[76]  Joachim Weickert,et al.  Tensor Field Interpolation with PDEs , 2006, Visualization and Processing of Tensor Fields.

[77]  Alfred M. Bruckstein,et al.  Scale Space and Variational Methods in Computer Vision , 2011, Lecture Notes in Computer Science.

[78]  Pascal Peter,et al.  Three-Dimensional Data Compression with Anisotropic Diffusion , 2013, GCPR.

[79]  Alexandru Telea,et al.  An Image Inpainting Technique Based on the Fast Marching Method , 2004, J. Graphics, GPU, & Game Tools.

[80]  Robert Hummel,et al.  Reconstructions from zero crossings in scale space , 1989, IEEE Trans. Acoust. Speech Signal Process..

[81]  G. Koepfler,et al.  A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets , 1995 .

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