Second-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging

TL;DR

This paper introduces the $w$-product, a wavelet-based tensor multiplication method that enables fast, stable tensor decompositions and significantly accelerates hyperspectral image processing tasks while maintaining quality.

Contribution

The paper proposes the $w$-product and its sparse variant, offering a novel, efficient tensor multiplication scheme based on second-generation wavelets with theoretical and practical advantages.

Findings

Up to 92.21 times speedup in hyperspectral image reconstruction.

Comparable PSNR and SSIM metrics with faster processing.

Exponential acceleration with decomposition level compared to traditional methods.

Abstract

This paper introduces the $w$-product, a novel wavelet-based tensor multiplication scheme leveraging second-generation wavelet transforms to achieve linear transformation complexity while preserving essential algebraic properties. The $w$-product outperforms existing tensor multiplication approaches by enabling fast and numerically stable tensor decompositions by proposing ``$w$-svd'' and its sparse variant ``sp-$w$-svd'', for efficient low-rank approximations with significantly reduced computational costs. Experiments on low-rank hyperspectral image reconstruction demonstrate up to a $92.21$ times speedup compared to state-of-the-art ``$t$-svd'', with comparable PSNR and SSIM metrics. We discuss the Moore-Penrose inverse of tensors based on the $w$-product and examine its essential properties. Numerical examples are provided to support the theoretical results. Then, hyperspectral image deblurring experiments demonstrate up to $27.88$ times speedup with improved image quality. In particular, the $w$-product and the sp-$w$-product exhibit exponentially increasing acceleration with the decomposition level compared to the traditional approach of the $t$-product. This work provides a scalable framework for multidimensional data analysis, with future research directions including adaptive wavelet designs, higher-order tensor extensions, and real-time implementations.