# American Institute of Mathematical Sciences

April  2020, 14(2): 339-361. doi: 10.3934/ipi.2020015

## Non-local blind hyperspectral image super-resolution via 4d sparse tensor factorization and low-rank

 1 Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2 Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

* Corresponding author: Weihong Guo

Received  June 2019 Revised  October 2019 Published  February 2020

Hyperspectral image (HSI) super-resolution is a technique to improve the spatial resolution of a HSI for better visual perception and down stream applications. This is a very ill-posed inverse problem and is often solved by fusing the low-resolution (LR) HSI with a high-resolution (HR) multispectral image (MSI). It is more challenging for blind HSI super-resolution, i.e., when the spatial degradation operators are completely unknown. In this paper, we propose a novel sparse tensor factorization model for the task of blind HSI super-resolution using the spatial non-local self-similarity and spectral global correlation of HSIs. Image clustering method is employed to collect some similar 3D cubes of HSIs which can be formed as some 4D image clusters with high correlation. We conduct cluster wise computation to not only save computation time but also to introduce a non-local regularity originated from the redundancy of cubes. By using the sparsity of tensor decomposition and the low-rank in non-local self-similarity direction underlying 4D similar clusters, we design a sparse tensor regularization term, which preserves the spatial-spectral structural correlation of HSIs. In addition, we present a proximal alternating direction method of multipliers (ADMM) based algorithm to efficiently solve the proposed model. Numerical experiments demonstrate that the proposed model outperforms many state-of-the-art HSI super-resolution methods.

Citation: Wei Wan, Weihong Guo, Jun Liu, Haiyang Huang. Non-local blind hyperspectral image super-resolution via 4d sparse tensor factorization and low-rank. Inverse Problems & Imaging, 2020, 14 (2) : 339-361. doi: 10.3934/ipi.2020015
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##### References:
Illustration of the hyperspectral image super-resolution task
The SAMs of all competing methods on CAVE dataset with uniform blur
The reconstructed images and the corresponding error images of fake and real peppers at the $8^{th}$ band and flowers at the $14^{th}$ band with Gaussian blur and scaling factor 8
The reconstructed images and the corresponding error images of the Indian Pines image at the $60^{th}$ and $130^{th}$ bands with uniform blur and scaling factor 8
The functional energy curve for the proposed algorithm
The reconstructed images and the corresponding error images of the Pavia image at the $45^{th}$ and $75^{th}$ bands with uniform blur and scaling factor 8
Comparison on the average values of each of the five quantitative measures on 32 scenes from CAVE dataset with scaling factor 8
 Methods PSNR SSIM RMSE SAM ERGAS Uniform blur CNMF[37] 43.052 0.985 2.299 5.934 1.224 Hysure[24] 41.461 0.975 2.702 10.737 1.497 STEREO[14] 39.680 0.946 3.611 12.212 1.898 Proposed 45.332 0.987 1.822 4.969 0.984 Gaussian blur CNMF[37] 42.258 0.981 2.511 6.578 1.329 Hysure[24] 41.961 0.977 2.541 10.310 1.425 STEREO[14] 39.542 0.945 3.658 12.251 1.923 Proposed 45.206 0.987 1.847 4.983 0.995
 Methods PSNR SSIM RMSE SAM ERGAS Uniform blur CNMF[37] 43.052 0.985 2.299 5.934 1.224 Hysure[24] 41.461 0.975 2.702 10.737 1.497 STEREO[14] 39.680 0.946 3.611 12.212 1.898 Proposed 45.332 0.987 1.822 4.969 0.984 Gaussian blur CNMF[37] 42.258 0.981 2.511 6.578 1.329 Hysure[24] 41.961 0.977 2.541 10.310 1.425 STEREO[14] 39.542 0.945 3.658 12.251 1.923 Proposed 45.206 0.987 1.847 4.983 0.995
The performance comparison of the methods on the Indian Pines image with scaling factor 8
 Methods PSNR SSIM RMSE SAM ERGAS Uniform blur CNMF[37] 42.866 0.978 2.557 2.394 9.406 Hysure[24] 45.232 0.984 1.891 1.701 9.389 STEREO[14] 48.212 0.985 1.267 1.225 9.405 Proposed 51.441 0.993 0.928 0.888 9.368 Gaussian blur CNMF[37] 42.740 0.977 2.599 2.430 9.409 Hysure[24] 45.927 0.986 1.703 1.547 9.384 STEREO[14] 47.050 0.981 1.510 1.413 9.438 Proposed 51.326 0.993 0.950 0.904 9.369
 Methods PSNR SSIM RMSE SAM ERGAS Uniform blur CNMF[37] 42.866 0.978 2.557 2.394 9.406 Hysure[24] 45.232 0.984 1.891 1.701 9.389 STEREO[14] 48.212 0.985 1.267 1.225 9.405 Proposed 51.441 0.993 0.928 0.888 9.368 Gaussian blur CNMF[37] 42.740 0.977 2.599 2.430 9.409 Hysure[24] 45.927 0.986 1.703 1.547 9.384 STEREO[14] 47.050 0.981 1.510 1.413 9.438 Proposed 51.326 0.993 0.950 0.904 9.369
The performance comparison of noisy cases on the Pavia image with uniform blur and scaling factor 8
 Methods PSNR SSIM RMSE SAM ERGAS SNRh=35dB, SNRm=40dB CNMF[37] 37.958 0.982 3.335 2.790 0.958 Hysure[24] 42.573 0.987 2.278 2.420 0.584 STEREO[14] 37.307 0.951 3.715 4.022 1.039 Proposed 43.731 0.989 1.848 1.944 0.497 SNRh=30dB, SNRm=35dB CNMF[37] 37.694 0.979 3.439 2.885 0.979 Hysure[24] 41.541 0.982 2.338 2.553 0.644 STEREO[14] 35.584 0.932 4.401 4.829 1.252 Proposed 42.097 0.981 2.158 2.349 0.604
 Methods PSNR SSIM RMSE SAM ERGAS SNRh=35dB, SNRm=40dB CNMF[37] 37.958 0.982 3.335 2.790 0.958 Hysure[24] 42.573 0.987 2.278 2.420 0.584 STEREO[14] 37.307 0.951 3.715 4.022 1.039 Proposed 43.731 0.989 1.848 1.944 0.497 SNRh=30dB, SNRm=35dB CNMF[37] 37.694 0.979 3.439 2.885 0.979 Hysure[24] 41.541 0.982 2.338 2.553 0.644 STEREO[14] 35.584 0.932 4.401 4.829 1.252 Proposed 42.097 0.981 2.158 2.349 0.604
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