Lossy volume compression using Tucker truncation and thresholding
Tensor decompositions, in particular the Tucker model, are a powerful family of techniques for dimensionality reduction and are being increasingly used for compactly encoding large multidimensional arrays, images and other visual data sets. In interactive applications, volume data often needs to be...
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| Published in: | The Visual computer Vol. 32; no. 11; pp. 1433 - 1446 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2016
Springer Nature B.V |
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| ISSN: | 0178-2789, 1432-2315 |
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| Abstract | Tensor decompositions, in particular the Tucker model, are a powerful family of techniques for dimensionality reduction and are being increasingly used for compactly encoding large multidimensional arrays, images and other visual data sets. In interactive applications, volume data often needs to be decompressed and manipulated dynamically; when designing data reduction and reconstruction methods, several parameters must be taken into account, such as the achievable compression ratio, approximation error and reconstruction speed. Weighing these variables in an effective way is challenging, and here we present two main contributions to solve this issue for Tucker tensor decompositions. First, we provide algorithms to efficiently compute, store and retrieve good choices of tensor rank selection and decompression parameters in order to optimize memory usage, approximation quality and computational costs. Second, we propose a Tucker compression alternative based on coefficient thresholding and zigzag traversal, followed by logarithmic quantization on both the transformed tensor core and its factor matrices. In terms of approximation accuracy, this approach is theoretically and empirically better than the commonly used tensor rank truncation method. |
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| AbstractList | Tensor decompositions, in particular the Tucker model, are a powerful family of techniques for dimensionality reduction and are being increasingly used for compactly encoding large multidimensional arrays, images and other visual data sets. In interactive applications, volume data often needs to be decompressed and manipulated dynamically; when designing data reduction and reconstruction methods, several parameters must be taken into account, such as the achievable compression ratio, approximation error and reconstruction speed. Weighing these variables in an effective way is challenging, and here we present two main contributions to solve this issue for Tucker tensor decompositions. First, we provide algorithms to efficiently compute, store and retrieve good choices of tensor rank selection and decompression parameters in order to optimize memory usage, approximation quality and computational costs. Second, we propose a Tucker compression alternative based on coefficient thresholding and zigzag traversal, followed by logarithmic quantization on both the transformed tensor core and its factor matrices. In terms of approximation accuracy, this approach is theoretically and empirically better than the commonly used tensor rank truncation method. |
| Author | Ballester-Ripoll, Rafael Pajarola, Renato |
| Author_xml | – sequence: 1 givenname: Rafael surname: Ballester-Ripoll fullname: Ballester-Ripoll, Rafael email: rballester@ifi.uzh.ch organization: Visualization and MultiMedia Lab, Department of Informatics, University of Zürich – sequence: 2 givenname: Renato surname: Pajarola fullname: Pajarola, Renato organization: Visualization and MultiMedia Lab, Department of Informatics, University of Zürich |
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| Cites_doi | 10.1145/2167076.2167077 10.1007/978-3-642-28027-6 10.1145/1015706.1015725 10.1016/j.cag.2014.10.002 10.1109/TCOM.1976.1093309 10.1109/TPAMI.2012.140 10.1137/S0895479898346995 10.1109/TVCG.2011.214 10.1109/TVCG.2007.70406 10.1137/07070111X 10.1137/S0895479896305696 10.1111/j.1467-8659.2011.02072.x 10.1145/1073204.1073224 10.1145/1141911.1141981 10.1109/IGARSS.2012.6350833 10.1111/cgf.12102 10.1109/ICIP.2000.899404 10.1088/0266-5611/27/2/025010 10.1109/ICIP.2007.4379951 10.1007/s10543-013-0455-z 10.1137/110836067 10.1016/j.neucom.2012.03.039 10.1145/1276377.1276411 10.1109/SIBGRAPI.2001.963051 10.1109/TVCG.2012.274 10.1109/ICIP.2008.4712383 10.1007/BF02310791 |
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| References_xml | – reference: KressnerDSteinlechnerMVandereyckenBLow-rank tensor completion by Riemannian optimizationBIT Numer. Math.2014542447468322351010.1007/s10543-013-0455-z1300.65040 – reference: Tsai, Y.T., Shih, Z.C.: K-clustered tensor approximation: a sparse multilinear model for real-time rendering. ACM Trans. Graph. 31(3) (2012) – reference: Wu, Q., Chen, C., Yu, Y.: Wavelet-based hybrid multilinear models for multidimensional image approximation. In: Proceedings IEEE International Conference on Image Processing, pp. 2828–2831 (2008) – reference: Zaid, A., Olivier, C., Alata, O., Marmoiton, F.: Transform image coding with global thresholding: application to baseline jpeg. In: Proceedings of the XIV Brazilian Symposium on Computer Graphics and Image Processing, pp. 164–171 (2001) – reference: Bader, B.W., Kolda, T.G. et al.: MATLAB tensor toolbox version 2.5. (2012). http://www.sandia.gov/tgkolda/TensorToolbox/ – reference: RajwadeARangarajanABanerjeeAImage denoising using the higher order singular value decompositionIEEE Trans. Pattern Anal. Mach. Intell.201335484986210.1109/TPAMI.2012.140 – reference: Wu, Q., Xia, T., Yu, Y.: Hierarchical tensor approximation of multidimensional images. In: Proceedings of the IEEE International Conference in Image Processing, vol. 4, pp. IV-49–IV-52 (2007) – reference: TreibMBürgerKReichlFMeneveauCSzalayAWestermannRTurbulence visualization at the terascale on desktop PCsIEEE Trans. Vis. Comput. Graph. (Proc. Scientific Visualization 2012)201218122169217710.1109/TVCG.2012.274 – reference: Chen, H., Lei, W., Zhou, S., Zhang, Y.: An optimal-truncation-based tucker decomposition method for hyperspectral image compression. In: IGARSS, pp. 4090–4093 (2012) – reference: Real world medical data sets. (2014). http://volvis.org/ – reference: Sun, X., Zhou, K., Chen, Y., Lin, S., Shi, J., Guo, B.: Interactive relighting with dynamic brdfs. ACM Trans. Graph. 26(3) (2007) – reference: Rövid, A., Rudas, I.J., Sergyán, S., Szeidl, L.: Hosvd based image processing techniques. In: Proceedings of the 10th WSEAS International Conference on Artificial Intelligence, Knowledge Engineering and Data Bases, AIKED’11, pp. 297–302. World Scientific and Engineering Academy and Society (WSEAS), Stevens Point (2011) – reference: VannieuwenhovenNVandebrilRMeerbergenKA new truncation strategy for the higher-order singular value decompositionSIAM J. Sci. Comput.201234210271052291431410.1137/1108360671247.65055 – reference: CarrollJDChangJJAnalysis of individual differences in multidimensional scaling via an n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-way generalization of “Eckart-Young” decompositionsPsychometrika197035328331910.1007/BF023107910202.19101 – reference: KoldaTGBaderBWTensor decompositions and applicationsSIAM Rev.2009513455500253505610.1137/07070111X1173.65029 – reference: de LathauwerLde MoorBVandewalleJOn the best rank-1 and rank-(R1,R2,...,RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R_1, R_2, ..., R_N$$\end{document}) approximation of higher-order tensorsSIAM J. 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