Joint Spatial-Spectral Smoothing in a Minimum-Volume Simplex for Hyperspectral Image Super-Resolution

Abstract

The limitations of hyperspectral sensors usually lead to coarse spatial resolution of acquired images. A well-known fusion method called coupled non-negative matrix factorization (CNMF) often amounts to an ill-posed inverse problem with poor anti-noise performance. Moreover, from the perspective of matrix decomposition, the matrixing of remotely-sensed cubic data results in the loss of data’s structural information, which causes the performance degradation of reconstructed images. In addition to three-dimensional tensor-based fusion methods, Craig’s minimum-volume belief in hyperspectral unmixing can also be utilized to restore the data structure information for hyperspectral image super-resolution. To address the above difficulties simultaneously, this article incorporates the regularization of joint spatial-spectral smoothing in a minimum-volume simplex, and spatial sparsity—into the original CNMF, to redefine a bi-convex problem. After the convexification of the regularizers, the alternating optimization is utilized to decouple the regularized problem into two convex subproblems, which are then reformulated by separately vectorizing the variables via vector-matrix operators. The alternating direction method of multipliers is employed to split the variables and yield the closed-form solutions. In addition, in order to solve the bottleneck of high computational burden, especially when the size of the problem is large, complexity reduction is conducted to simplify the solutions with constructed matrices and tensor operators. Experimental results illustrate that the proposed algorithm outperforms state-of-the-art fusion methods, which verifies the validity of the new fusion approach in this article.

Low-Rank Tensor Decomposition With Smooth and Sparse Regularization for Hyperspectral and Multispectral Data Fusion

“…(3) Stopping rules of convergence As shown in Algorithm 1, the proposed LRTVS method decomposes the problem (7) into alternating iterations of three factor matrices and a core tensor. The stopping rule satisfies the relative difference threshold between the successive updates of the objective function F 0 (W, H, A, C) is less than 0.001 [44], [56], as shown in equation (43). The experimental results show that the increase in iterations has no significant effect on the convergence in the ADMM-based algorithms, so they need not run exhaustively.…”

“…(2) Spectral smooth regularization The endmember signature curve describes the spectral reflectance of the endmember, which is usually a smooth (or piecewise smooth) curve owing to the gradual change [44]. Therefore, the total variation is utilized to smooth the spectral signature.…”

“…Then, Algorithm 1 is taken to integrate the observed data Y and X , in which Algorithms 2, 3 and 4 are called to implement the factor matrices W, H, A and the core tensor C until convergence for reconstructing the HR-HSI Z. Thus, the similarity between Z and Z is measured by such quality evaluation metrics [29], [44] as degree of distortion (DD), reconstruction signal-to-noise ratio (RSNR), spectral angle mapper (SAM), root mean squared error (RMSE), erreur relative globale adimensionnelle de synthèse (ERGAS), and structural similarity index for measuring image quality (SSIM). The larger the value of RSNR and SSIM, the better the performance.…”

“…Because of symmetry, the weights of two spatial factor matrices satisfy λ w = λ h . Moreover, considering the gradual properties of the spectral signature [44], the total variation is adopted to vertically smooth the spectral factor matrix with the coefficient λ d . The weight of the core tensor is λ c .…”

Multiplicative Iterative Nonlinear Constrained Coupled Non-negative Matrix Factorization (MINC-CNMF) for Hyperspectral and Multispectral Image Fusion

“…Let Y h ∈ ℝ L h × N h be an observed LR-HSI with L h bands and N h pixels, and Y m ∈ ℝ L m × N m be an observed HR-MSI with L m bands and N m pixels, with L m < L h and N h <N m . Then data fusion of L h band from LR-HSI, Y h and N m pixels from HR-MSI, Y m to yield the desired high spectral and spatial resolution hyper-spectral image, Z ∈ ℝ L h × N m [9].…”