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

Aneesh Panchal, Ratikanta Behera

TL;DR

The paper introduces the w-product, a wavelet-inspired tensor multiplication based on second-generation (lifting) wavelets that achieves linear transformation complexity while preserving key tensor algebraic properties. It develops the w-SVD and a sparse variant sp-w-SVD, enabling fast, stable low-rank tensor decompositions with substantial speedups over FFT-based methods in hyperspectral imaging tasks. The authors establish associativity, Moore-Penrose inverse formulations, and trace properties within the w-product framework, and demonstrate dramatic computational gains (up to 92.21x for reconstruction and 27.88x for deblurring) without sacrificing reconstruction quality. This work provides a scalable, multiresolution approach for multidimensional data analysis and points to adaptive wavelet designs and real-time hardware implementations as future directions.

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.

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

TL;DR

The paper introduces the w-product, a wavelet-inspired tensor multiplication based on second-generation (lifting) wavelets that achieves linear transformation complexity while preserving key tensor algebraic properties. It develops the w-SVD and a sparse variant sp-w-SVD, enabling fast, stable low-rank tensor decompositions with substantial speedups over FFT-based methods in hyperspectral imaging tasks. The authors establish associativity, Moore-Penrose inverse formulations, and trace properties within the w-product framework, and demonstrate dramatic computational gains (up to 92.21x for reconstruction and 27.88x for deblurring) without sacrificing reconstruction quality. This work provides a scalable, multiresolution approach for multidimensional data analysis and points to adaptive wavelet designs and real-time hardware implementations as future directions.

Abstract

This paper introduces the -product, a novel wavelet-based tensor multiplication scheme leveraging second-generation wavelet transforms to achieve linear transformation complexity while preserving essential algebraic properties. The -product outperforms existing tensor multiplication approaches by enabling fast and numerically stable tensor decompositions by proposing ``-svd'' and its sparse variant ``sp--svd'', for efficient low-rank approximations with significantly reduced computational costs. Experiments on low-rank hyperspectral image reconstruction demonstrate up to a times speedup compared to state-of-the-art ``-svd'', with comparable PSNR and SSIM metrics. We discuss the Moore-Penrose inverse of tensors based on the -product and examine its essential properties. Numerical examples are provided to support the theoretical results. Then, hyperspectral image deblurring experiments demonstrate up to times speedup with improved image quality. In particular, the -product and the sp--product exhibit exponentially increasing acceleration with the decomposition level compared to the traditional approach of the -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.
Paper Structure (14 sections, 83 equations, 6 figures, 5 tables, 2 algorithms)

This paper contains 14 sections, 83 equations, 6 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction and Motivations
  2. Definitions and Theorems
  3. Second-generation wavelet transforms
  4. Second-generation wavelet-inspired tensor product ($w$-product)
  5. Complexity Analysis and Discussions
  6. Applications to Hyperspectral Image Analysis
  7. Experimental setup
  8. Application to low-rank hyperspectral image reconstruction
  9. Computational analysis
  10. Experimental results
  11. Application to hyperspectral image deblurring
  12. Computational analysis
  13. Experimental results
  14. Conclusions and Future Directions

Figures (6)

  • Figure 1: Visual representation of a simple second-generation wavelet decomposition and reconstruction.
  • Figure 2: Flowchart explaining the working of the proposed wavelet product for tensors with assumption $p=2^3$.
  • Figure 3: Analytical complexity (Number of operations).
  • Figure 4: Experimental time taken in multiplication.
  • Figure 5: Number of operations required for SVD using various products.
  • ...and 1 more figures

Theorems & Definitions (9)

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