Lossy volume compression using Tucker truncation and thresholding

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. 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.

TTHRESH: Tensor Compression for Multidimensional Visual Data

“…Progressive tensor rank reduction (the so-called truncation; see later sections) has been shown to reveal features and structural details at different scales in volume data [21]. Further recent efforts in the context of tensor compression include [2,9,[22][23][24] for interactive volume rendering and visualization, [25] for 3D displays, [26] for integral histograms of images and volumes, and [27][28][29][30] for reflectance fields, among others. The large-scale renderer TAMRESH [23] resembles block-transform coding in that the input volume is partitioned in small multiresolution cubic bricks; each brick is then compressed as a separate HOSVD core.…”

“…The large-scale renderer TAMRESH [23] resembles block-transform coding in that the input volume is partitioned in small multiresolution cubic bricks; each brick is then compressed as a separate HOSVD core. Recently, Tucker core hard thresholding combined with factor matrix quantization was shown [2] to yield better compression rate than slice-wise truncating the core. These points have motivated the compressor proposed here.…”

“…[20], [22], [23] and [24]. Right: the full core approach first considered in [2] and here extended into a full-fledged compressor with adaptive thresholding and bit-plane coding.…”

Data Reduction Techniques for Simulation, Visualization and Data Analysis

“…The work by Ballester‐Ripoll and Pajarola [BRP16] aimed to improve the standard Tucker decomposition. They first presented core truncation, where they discarded the least significant ranks, or segments of the basis function matrices, and corresponding segments of the reduced core tensor.…”

“…The work by Ballester-Ripoll and Pajarola [BRP16] aimed to improve the standard Tucker decomposition. They first presented core truncation, where they discarded the least significant ranks, or Given the input data/tensor X , the least square fittings results in a core tensor G and three basis factor matrices A, B and C. G is considered an approximation of X .…”