Hyperspectral Super-Resolution: A Coupled Tensor Factorization Approach

Abstract

Hyperspectral super-resolution refers to the problem of fusing a hyperspectral image (HSI) and a multispectral image (MSI) to produce a super-resolution image (SRI) that has fine spatial and spectral resolution. State-of-the-art methods approach the problem via low-rank matrix approximations to the matricized HSI and MSI. These methods are effective to some extent, but a number of challenges remain. First, HSIs and MSIs are naturally third-order tensors (data "cubes") and thus matricization is prone to loss of structural information-which could degrade performance. Second, it is unclear whether or not these low-rank matrix-based fusion strategies can guarantee identifiability or exact recovery of the SRI. However, identifiability plays a pivotal role in estimation problems and usually has a significant impact on performance in practice. Third, the majority of the existing methods assume that there are known (or easily estimated) degradation operators applied to the SRI to form the corresponding HSI and MSI-which is hardly the case in practice. In this work, we propose to tackle the super-resolution problem from a tensor perspective. Specifically, we utilize the multidimensional structure of the HSI and MSI to propose a coupled tensor factorization framework that can effectively overcome the aforementioned issues. The proposed approach guarantees the identifiability of the SRI under mild and realistic conditions. Furthermore, it works with little knowledge of the degradation operators, which is clearly an advantage over the existing methods. Semi-real numerical experiments are included to show the effectiveness of the proposed approach.

Block-Randomized Stochastic Proximal Gradient for Low-Rank Tensor Factorization

“…In a lot of applications, some prior knowledge on the latent factors is known-e.g., in image processing, A (n) 's are normally assumed to be nonnegative [6]; in statistical machine learning, sometimes the columns of A (n) are assumed to be constrained within the probability simplex [36,42]; i.e., 1 A (n) = 1 , A (n) ≥ 0.…”

Tensor Completion From Regular Sub-Nyquist Samples

“…ThereforeŶ = A 2 , B 1 , C 1 reconstructs signal Y . The proof shares insights with that of Theorem 2 [19]. The basic difference lies in the fact that P are full row rank selection matrices instead of full rank dense matrices.…”

“…Lemma 1 [19] LetZ = QZ, where the elements of Z are drawn from an absolutely continuous joint distribution with respect to the Lebesgue measure in C IF and Q ∈ R I ×I is a row selection matrix with full row rank. Then the joint distribution of the elements inZ is absolutely continuous with respect to the Lebesgue measure in C I F It follows that P (1) 1 A, P (2) 3 C are drawn from non-singular absolutely continuous distributions.…”

Hyperspectral Super-Resolution via Global–Local Low-Rank Matrix Estimation

“…The tensor model is also a low-dimensional model, and it exploits not only the spectral-spatial structure but also the two-dimensional spatial structure. Tensor factorization is shown to exhibit favorable sufficient conditions on exact recovery guarantees [13,14]. Under the low-dimensional model, HSR can be seen as a problem of recovering a low-rank matrix from incomplete observations.…”

“…We identify a majorant for problem (9) by resorting to the following result. By applying Fact 1 to (13), and noting ψ p,τ (X, W ) ≥ ψ p,τ (X, W ) = φ p,τ (X) for any W ∈ S M ++ , we obtain the following majorant…”