dorsal/arxiv
View SchemaSecond-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging
| Authors | Aneesh Panchal, Ratikanta Behera |
|---|---|
| Categories | |
| ArXiv ID | 2601.08228vv1 |
| URL | https://arxiv.org/abs/2601.08228 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
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"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\u0027\u0027 and its sparse variant ``sp-$w$-svd\u0027\u0027, 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\u0027\u0027, 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.",
"arxiv_id": "2601.08228",
"authors": [
"Aneesh Panchal",
"Ratikanta Behera"
],
"categories": [
"math.NA",
"cs.NA"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Second-Generation Wavelet-inspired Tensor Product with Applications in Hyperspectral Imaging",
"url": "https://arxiv.org/abs/2601.08228",
"version": "v1"
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