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Non-negative Einstein tensor factorization for unmixing hyperspectral images

Anas El Hachimi, Khalide Jbilou, Ahmed Ratnani

TL;DR

The paper tackles denoising and linear unmixing of hyperspectral images using a tensor-based non-negative factorization framework built on the Einstein product. It formulates an optimization problem with non-negativity and abundance-sum constraints, and promotes sparsity via an $\ell_1$ surrogate, solved by a tensorized multiplicative-updates scheme augmented with surrogate variables. The approach introduces the Einstein non-negative rank, analyzes uniqueness properties, and extends to HI-specific factorization with practical constraints. To accelerate convergence, the authors adapt tensor extrapolation methods (RRE and TET) and demonstrate superior denoising and unmixing performance on synthetic and real HI datasets, with acceleration methods reducing iteration counts while maintaining accuracy.

Abstract

In this manuscript, we introduce a tensor-based approach to Non-Negative Tensor Factorization (NTF). The method entails tensor dimension reduction through the utilization of the Einstein product. To maintain the regularity and sparsity of the data, certain constraints are imposed. Additionally, we present an optimization algorithm in the form of a tensor multiplicative updates method, which relies on the Einstein product. To guarantee a minimum number of iterations for the convergence of the proposed algorithm, we employ the Reduced Rank Extrapolation (RRE) and the Topological Extrapolation Transformation Algorithm (TEA). The efficacy of the proposed model is demonstrated through tests conducted on Hyperspectral Images (HI) for denoising, as well as for Hyperspectral Image Linear Unmixing. Numerical experiments are provided to substantiate the effectiveness of the proposed model for both synthetic and real data.

Non-negative Einstein tensor factorization for unmixing hyperspectral images

TL;DR

The paper tackles denoising and linear unmixing of hyperspectral images using a tensor-based non-negative factorization framework built on the Einstein product. It formulates an optimization problem with non-negativity and abundance-sum constraints, and promotes sparsity via an surrogate, solved by a tensorized multiplicative-updates scheme augmented with surrogate variables. The approach introduces the Einstein non-negative rank, analyzes uniqueness properties, and extends to HI-specific factorization with practical constraints. To accelerate convergence, the authors adapt tensor extrapolation methods (RRE and TET) and demonstrate superior denoising and unmixing performance on synthetic and real HI datasets, with acceleration methods reducing iteration counts while maintaining accuracy.

Abstract

In this manuscript, we introduce a tensor-based approach to Non-Negative Tensor Factorization (NTF). The method entails tensor dimension reduction through the utilization of the Einstein product. To maintain the regularity and sparsity of the data, certain constraints are imposed. Additionally, we present an optimization algorithm in the form of a tensor multiplicative updates method, which relies on the Einstein product. To guarantee a minimum number of iterations for the convergence of the proposed algorithm, we employ the Reduced Rank Extrapolation (RRE) and the Topological Extrapolation Transformation Algorithm (TEA). The efficacy of the proposed model is demonstrated through tests conducted on Hyperspectral Images (HI) for denoising, as well as for Hyperspectral Image Linear Unmixing. Numerical experiments are provided to substantiate the effectiveness of the proposed model for both synthetic and real data.
Paper Structure (12 sections, 9 theorems, 56 equations, 8 figures, 6 tables, 6 algorithms)

This paper contains 12 sections, 9 theorems, 56 equations, 8 figures, 6 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. NTF based on the Einstein product (ENTF)
  4. NTF by the Einstein product based on HI
  5. Tensor extrapolation methods
  6. Numerical experiments
  7. Denoising Hyperspectral images
  8. Hyperspectral unmixing
  9. Synthetic data
  10. Real data
  11. Results of the acceleration methods
  12. Conclusion

Key Result

Proposition 2.3

Let $\mathcal{A}\in \mathbb{R}^{I_1\times \ldots \times I_N\times J_1 \times \ldots \times J_M}$ and $\mathcal{B}\in \mathbb{R}^{J_1\times \ldots \times J_M \times L_1 \times \ldots \times L_K}$, then

Figures (8)

  • Figure 6.1: The mapping of the five abundance maps of the Legendre data with SNR=$20$ using the methods NMF-$\ell_{1}$, NMF-$\ell_{2}$, NMF-$\ell_{1/2}$, MV-NTF-S, and ENTF.
  • Figure 6.2: The five abundance maps of the DC1 dataset obtained through the utilization of the methods NMF-$\ell_{1}$, NMF-$\ell_{2}$, NMF-$\ell_{1/2}$, MV-NTF-S, and ENTF.
  • Figure 6.3: The mapping of the abundances of the endmembers Trees, Soil, Water, and Road for the Jasper Ridge dataset using the FCLS, SCLSU, MESMA, RUSAL, MV-NTF-S, and ENTF methods.
  • Figure 6.4: The mapping of the abundances of the endmembers Water, Trees, and Soil for the Samson dataset employing the FCLS, SCLSU, MESMA, RUSAL, MV-NTF-S, and ENTF methods.
  • Figure 6.5: Jasper Ridge endmembers obtained by the methods MESMA, MV-NTF-S, and ENTF.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Definition 2.1: kolda
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 3.1: ENTF
  • Definition 3.2: Einstein non-negative rank
  • Definition 3.3
  • Lemma 3.4
  • ...and 11 more