Rethinking Coupled Tensor Analysis for Hyperspectral Super-Resolution: Recoverable Modeling Under Endmember Variability

TL;DR

The paper tackles hyperspectral super-resolution under endmember variability by introducing an LMN block-term tensor model that extends CPD, Tucker, and LL1 while preserving interpretability and enabling EV modeling. It proves recoverability guarantees for the LMN-based CTD framework, introduces the CLIMB algorithm with physics-inspired regularization, and demonstrates superior performance on semi-real and real datasets with pronounced EV. The approach balances expressiveness and interpretability, offering a robust tool for HSR under realistic spectral variability. Overall, LMN-CTD provides both theoretical guarantees and practical gains over existing tensor-CTD methods in EV-rich fusion tasks.

Abstract

This work revisits the hyperspectral super-resolution (HSR) problem, i.e., fusing a pair of spatially co-registered hyperspectral (HSI) and multispectral (MSI) images to recover a super-resolution image (SRI) that enhances the spatial resolution of the HSI. Coupled tensor decomposition (CTD)-based methods have gained traction in this domain, offering recoverability guarantees under various assumptions. Existing models such as canonical polyadic decomposition (CPD) and Tucker decomposition provide strong expressive power but lack physical interpretability. The block-term decomposition model with rank-$(L_r, L_r, 1)$ terms (the LL1 model) yields interpretable factors under the linear mixture model (LMM) of spectral images, but LMM assumptions are often violated in practice -- primarily due to nonlinear effects such as endmember variability (EV). To address this, we propose modeling spectral images using a more flexible block-term tensor decomposition with rank-$(L_r, M_r, N_r)$ terms (the LMN model). This modeling choice retains interpretability, subsumes CPD, Tucker, and LL1 as special cases, and robustly accounts for non-ideal effects such as EV, offering a balanced tradeoff between expressiveness and interpretability for HSR. Importantly, under the LMN model for HSI and MSI, recoverability of the SRI can still be established under proper conditions -- providing strong theoretical support. Extensive experiments on synthetic and real datasets further validate the effectiveness and robustness of the proposed method compared with existing CTD-based approaches.

Figures (17)

• Figure 1: Illustration of spatial and spectral degradations from SRI ($\underline{\bm{Y}}_{\rm S}$) to HSI ($\underline{\bm{Y}}_{\rm H}$) and MSI ($\underline{\bm{Y}}_{\rm M}$), respectively; figure adapted from Kanatsoulis2018HSRDing2021HSR.
• Figure 1: Illustration of the mode-3 fibers of $\underline{\bm{T}}_r$, which correspond to the spectral signatures of the same materials across the $I_M\times J_M$ space. The key postulate is that, as all of the spectra are variations of the same endmember, all the fibers are similar to each other. Therefore, each $\underline{\bm{T}}_r$ has a low multi-linear rank.
• Figure 1: Comparison of sparsity (first row) and spatial smoothness (second row to fourth row: mode-1, mode-2, and mode-3, respectively) exhibited by original $\underline{\bm Y}_{\rm S}$ and each $\underline{\bm T}_r$ for Jasper Ridge dataset with four materials.
• Figure 1: Illustration of endmember variability of the subimages of Pavia University, Washington DC, and Jasper Ridge datasets. (The spectral signatures of materials of each dataset are manually selected pure pixels.)
• Figure 1: The singular values of unfolding matrices (top to bottom: mode-1, mode-2, and mode-3, respectively) of each $\underline{\bm{T}}_r$ for Jasper Ridge dataset.

Theorems & Definitions (12)

• Definition 3.1: Essential Uniqueness
• Theorem 3.2: Identifiability of LMN
• Proof 1
• Theorem 3.3
• Theorem 3.4
• Remark 3.5
• Remark 4.1
• Remark 4.2
• Proposition 4.3
• Lemma 2.1: Corollary 2.6 Domanov2024BTD