Clustering-based quantisation for PDE-based image compression

Optimal known pixel data for inpainting in compression codecs based on partial differential equations is real-valued and thereby expensive to store. Thus, quantisation is required for efficient encoding. In this paper, we interpret the quantisation step as a clustering problem. Due to the global impact of each known pixel and correlations between spatial and tonal data, we investigate the central question, which kind of feature vectors should be used for clustering with popular strategies such as k-means. Our findings show that the number of colours can be reduced significantly without impacting the reconstruction quality. Surprisingly, these benefits are negated by an increased coding cost in compression applications.

[1]  Joachim Weickert,et al.  Beyond pure quality: Progressive modes, region of interest coding, and real time video decoding for PDE-based image compression , 2015, J. Vis. Commun. Image Represent..

[2]  Armin Iske,et al.  Scattered Data Coding in Digital Image Compression , 2002 .

[3]  R. Naik,et al.  Biological versus electronic adaptive coloration: how can one inform the other? , 2013, Journal of The Royal Society Interface.

[4]  Joachim Weickert,et al.  Evaluating the true potential of diffusion-based inpainting in a compression context , 2016, Signal Process. Image Commun..

[5]  R. Sundberg.  Maximum Likelihood Theory for Incomplete Data from an Exponential Family , 2016 .

[6]  G. Gendy,et al.  C-means clustering fuzzified by two membership relative entropy functions approach incorporating local data information for noisy image segmentation , 2016, Signal, Image and Video Processing.

[7]  Michael Breuß,et al.  Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian , 2017, SSVM.

[8]  Donald W. Bouldin,et al.  A Cluster Separation Measure , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Thomas Brox,et al.  iPiano: Inertial Proximal Algorithm for Nonconvex Optimization , 2014, SIAM J. Imaging Sci..

[10]  Yunjin Chen,et al.  A bi-level view of inpainting - based image compression , 2014, ArXiv.

[11]  N. Ahmed,et al.  A block coding technique for encoding sparse binary patterns , 1989, IEEE Trans. Acoust. Speech Signal Process..

[12]  Joachim Weickert,et al.  Why Does Non-binary Mask Optimisation Work for Diffusion-Based Image Compression? , 2015, EMMCVPR.

[13]  Concha Bielza,et al.  A comparison of clustering quality indices using outliers and noise , 2012, Intell. Data Anal..

[14]  R. Sundberg.  An iterative method for solution of the likelihood equations for incomplete data from exponential families , 1976 .

[15]  P. Rousseeuw.  Silhouettes: a graphical aid to the interpretation and validation of cluster analysis , 1987 .

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

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

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

[19]  T. Caliński,et al.  A dendrite method for cluster analysis , 1974 .

[20]  Joachim Weickert,et al.  An Optimal Control Approach to Find Sparse Data for Laplace Interpolation , 2013, EMMCVPR.

[21]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

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

[23]  Mohamed Abid,et al.  Lossless image compression using gradient based space filling curves (G-SFC) , 2015, Signal Image Video Process..

[24]  M. Breuß,et al.  Efficient Co-Domain Quantisation for PDE-Based Image Compression , 2016 .

[25]  J. H. Ward.  Hierarchical Grouping to Optimize an Objective Function , 1963 .

[26]  George A. Miller,et al.  Human memory and the storage of information , 1956, IRE Trans. Inf. Theory.

[27]  Robert Tibshirani,et al.  Estimating the number of clusters in a data set via the gap statistic , 2000 .

[28]  Carola-Bibiane Schönlieb,et al.  Partial Differential Equation Methods for Image Inpainting , 2015, Cambridge monographs on applied and computational mathematics.

[29]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[30]  Joachim Weickert,et al.  From Optimised Inpainting with Linear PDEs Towards Competitive Image Compression Codecs , 2015, PSIVT.

[31]  Michael Unser,et al.  Anisotropic Interpolation of Sparse Generalized Image Samples , 2013, IEEE Transactions on Image Processing.

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

[33]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[34]  A. Iske,et al.  Advances in Digital Image Compression by Adaptive Thinning , 2003 .

[35]  David Stutz.  IPIANO : INERTIAL PROXIMAL ALGORITHM FOR NON-CONVEX OPTIMIZATION , 2016 .

[36]  Armin Iske,et al.  Contextual Image Compression from Adaptive Sparse Data Representations , 2009 .

[37]  Joachim Weickert,et al.  Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion , 2014, International Journal of Computer Vision.

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