Fusion from a spatially low resolution hyperspectral image (LR-HSI) and a spectrally low resolution multispectral image (MSI) to produce a high spatial-spectral HSI (HR-HSI), known as Hyperspectral super-resolution, has risen to a preferred topic for reinforcing the spatial-spectral resolution of HSI in recent years. In this work, we propose a new model, namely lowrank tensor ring decomposition based on tensor nuclear norm (LRTRTNN), for HSI-MSI fusion. Specifically, for each spectrally subspace cube, similar patches are grouped to exploit both the global low-rank property of LR-HSI and the non-local similarity of HR-MSI. Afterwards, a joint optimization of all groups via the presented LRTRTNN approximation is implemented in a unified cost function. With the introduced tensor nuclear norm (TNN) constraint, all 3D tensor ring factors are no longer unfolded to suit the matrix nuclear norm used in conventional methods, and the internal tensor structure can be naturally retained. The Alternating Direction Method of Multipliers (ADMM) is introduced for coefficients update. Numerical and visual experiments on real data show that our LRTRTNN method outperforms most state-of-the-art algorithms in terms of fusing performance.

Owing to the outstanding feature extraction capability, convolutional neural networks (CNNs) have been widely applied in hyperspectral image (HSI) classification problems and have achieved an impressive performance. However, it is well known that 2D convolution suffers from the absent consideration of spectral information, while 3D convolution requires a huge amount of computational cost. In addition, the cost of labeling and the limitation of computing resources make it urgent to improve the generalization performance of the model with scarcely labeled samples. To relieve these issues, we design an end-to-end 3D octave and 2D vanilla mixed CNN, namely Oct-MCNN-HS, based on the typical 3D-2D mixed CNN (MCNN). It is worth mentioning that two feature fusion operations are deliberately constructed to climb the top of the discriminative features and practical performance. That is, 2D vanilla convolution merges the feature maps generated by 3D octave convolutions along the channel direction, and homology shifting aggregates the information of the pixels locating at the same spatial position. Extensive experiments are conducted on four publicly available HSI datasets to evaluate the effectiveness and robustness of our model, and the results verify the superiority of Oct-MCNN-HS both in efficacy and efficiency.

The weighted nuclear norm minimization and truncated nuclear norm minimization are two well-known low-rank constraint for visual applications. By integrating their advantages into a unified formulation, we find a better weighting strategy, namely truncated reweighting norm minimization (TRNM), which provides better approximation to the target rank for some specific task. Albeit nonconvex and truncated, we prove that TRNM is equivalent to certain weighted quadratic programming problems, whose global optimum can be accessed by the newly presented reweighting singular value thresholding operator. More importantly, we design a computationally efficient optimization algorithm, namely momentum update and rank propagation (MURP), for the general TRNM regularized problems. The individual advantages of MURP include: (1) reducing iterations through non-monotonic search, and (2) mitigating computational cost by reducing the size of target matrix. Furthermore, the descent property and convergence of MURP are proven. Finally, two practical models, i.e., MCTRNM and SCTRNM, are presented for visual applications. Extensive experimental results show that our methods achieve better performance, both qualitatively and quantitatively, compared with several state-of-the-art algorithms.

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