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In this paper, the quaternion matrix UTV (QUTV) decomposition and quaternion tensor UTV (QTUTV) decomposition are proposed. To begin, the terms QUTV and QTUTV are defined, followed by the algorithms. Subsequently, by employing random sampling from the quaternion normal distribution, randomized QUTV and randomized QTUTV are generated to provide enhanced algorithmic efficiency. These techniques produce decompositions that are straightforward 9 to understand and require minimal cost. Furthermore, theoretical analysis is discussed. Specifically, the upper bounds for approximating QUTV on the rank-K and QTUTV on the TQt-rank K errors are provided, followed by deterministic error bounds and average-case error bounds for the randomized situations, which demonstrate the correlation between the accuracy of the low-rank approximation and the singular values. Finally, numerous numerical experiments are presented to verify that the proposed algorithms work more efficiently and with similar relative errors compared to other comparable decomposition methods. For the novel decompositions, the theory analysis offers a solid theoretical basis and the experiments show significant potential for the associated processing tasks of color images and color videos.

The convolution operation is a powerful tool for feature extraction and plays a prominent role in the field of computer vision. However, when targeting the pixel-wise tasks like image fusion, it would not fully perceive the particularity of each pixel in the image if the uniform convolution kernel is used on different patches. In this paper, we propose a local adaptive convolution (LAConv), which is dynamically adjusted to different spatial locations. LAConv enables the network to pay attention to every specific local area in the learning process. Besides, the dynamic bias (DYB) is introduced to provide more possibilities for the depiction of features and make the network more flexible. We further design a residual structure network equipped with the proposed LAConv and DYB modules, and apply it to two image fusion tasks. Experiments for pansharpening and hyperspectral image super-resolution (HISR) demonstrate the superiority of our method over other state-of-the-art methods. It is worth mentioning that LAConv can also be competent for other super-resolution tasks with less computation effort.

In this paper, we study multi-dimensional image recovery. Recently, transform-based tensor nuclear norm minimization methods are considered to capture low-rank tensor structures to recover third-order tensors in multi-dimensional image processing applications. The main characteristic of such methods is to perform the linear transform along the third mode of third-order tensors, and then compute tensor nuclear norm minimization on the transformed tensor so that the underlying low-rank tensors can be recovered. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform via the observed tensor data under self-supervision. The proposed network makes use of low-rank representation of transformed tensors and data-fitting between the observed tensor and the reconstructed tensor to construct the nonlinear transformation. Extensive experimental results on tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging are presented to demonstrate that the performance of the proposed method is better than that of state-of-the-art methods.

In this paper, we propose a novel tensor learning and coding model for third-order data completion. Our model is to learn a data-adaptive dictionary from the given observations, and determine the coding coefficients of third-order tensor tubes. In the completion process, we minimize the low-rankness of each tensor slice containing the coding coefficients. By comparison with the traditional pre-defined transform basis, the advantages of the proposed model are that (i) the dictionary can be learned based on the given data observations so that the basis can be more adaptively and accurately constructed, and (ii) the low-rankness of the coding coefficients can allow the linear combination of dictionary features more effectively. Also we develop a multi-block proximal alternating minimization algorithm for solving such tensor learning and coding model, and show that the sequence generated by the algorithm can globally converge to a critical point. Extensive experimental results for real data sets such as videos, hyperspectral images, and traffic data are reported to demonstrate these advantages and show the performance of the proposed tensor learning and coding method is significantly better than the other tensor completion methods in terms of several evaluation metrics.

In this paper, we develop a new alternating projection method to compute nonnegative low rank matrix approximation for nonnegative matrices. In the nonnegative low rank matrix approximation method, the projection onto the manifold of fixed rank matrices can be expensive as the singular value decomposition is required. We propose to use the tangent space of the point in the manifold to approximate the projection onto the manifold in order to reduce the computational cost. We show that the sequence generated by the alternating projections onto the tangent spaces of the fixed rank matrices manifold and the nonnegative matrix manifold, converge linearly to a point in the intersection of the two manifolds where the convergent point is sufficiently close to optimal solutions. This convergence result based inexact projection onto the manifold is new and is not studied in the literature. Numerical examples in data clustering, pattern recognition and hyperspectral data analysis are given to demonstrate that the performance of the proposed method is better than that of nonnegative matrix factorization methods in terms of computational time and accuracy.

Recently, convolutional neural network (CNN)-based methods are proposed for hyperspectral images (HSIs) denoising. Among them, unsupervised methods such as the deep image prior (DIP) have received much attention because these methods do not require any training data. However, DIP suffers from the semi-convergence behavior, i.e., the iteration of DIP needs to terminate by referring to the ground-truth image at the optimal iteration point. In this paper, we propose the spatial-spectral constrained deep image prior (S2DIP) for HSI mixed noise removal. Specifically, we incorporate DIP with a spatial-spectral total variation (SSTV) term to fully preserve the spatial-spectral local smoothness of the HSI and an $\ell_1$-norm term to capture the complex sparse noise. The proposed S2DIP jointly leverages the expressive power brought from the deep CNN without any training data and exploits the HSI and noise structures via hand-crafted priors. Thus, our method avoids the semi-convergence behavior, showing higher stabilities than DIP. Meanwhile, our method largely enhances the HSI denoising ability of DIP. To tackle the proposed denoising model, we develop an alternating direction multiplier method algorithm. Extensive experiments demonstrate that the proposed S2DIP outperforms optimization-based and supervised CNN-based state-of-the-art HSI denoising methods.

The main aim of this paper is to develop a new algorithm for computing nonnegative low rank tensor approximation for nonnegative tensors that arise in many multi-dimensional imaging applications. Nonnegativity is one of the important property as each pixel value refers to nonzero light intensity in image data acquisition. Our approach is different from classical nonnegative tensor factorization (NTF) which requires each factorized matrix and/or tensor to be nonnegative. In this paper, we determine a nonnegative low Tucker rank tensor to approximate a given nonnegative tensor. We propose an alternating projections algorithm for computing such nonnegative low rank tensor approximation, which is referred to as NLRT. The convergence of the proposed manifold projection method is established. Experimental results for synthetic data and multi-dimensional images are presented to demonstrate the performance of NLRT is better than state-of-the-art NTF methods.

Hyperspectral images are of crucial importance in order to better understand features of different materials. To reach this goal, they leverage on a high number of spectral bands. However, this interesting characteristic is often paid by a reduced spatial resolution compared with traditional multispectral image systems. In order to alleviate this issue, in this work, we propose a simple and efficient architecture for deep convolutional neural networks to fuse a low-resolution hyperspectral image (LR-HSI) and a high-resolution multispectral image (HR-MSI), yielding a high-resolution hyperspectral image (HR-HSI). The network is designed to preserve both spatial and spectral information thanks to an architecture from two folds: one is to utilize the HR-HSI at a different scale to get an output with a satisfied spectral preservation; another one is to apply concepts of multi-resolution analysis to extract high-frequency information, aiming to output high quality spatial details. Finally, a plain mean squared error loss function is used to measure the performance during the training. Extensive experiments demonstrate that the proposed network architecture achieves best performance (both qualitatively and quantitatively) compared with recent state-of-the-art hyperspectral image super-resolution approaches. Moreover, other significant advantages can be pointed out by the use of the proposed approach, such as, a better network generalization ability, a limited computational burden, and a robustness with respect to the number of training samples.

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.

In this paper, we propose a novel low-tubal-rank tensor recovery model, which directly constrains the tubal rank prior for effectively removing the mixed Gaussian and sparse noise in hyperspectral images. The constraints of tubal-rank and sparsity can govern the solution of the denoised tensor in the recovery procedure. To solve the constrained low-tubal-rank model, we develop an iterative algorithm based on bilateral random projections to efficiently solve the proposed model. The advantage of random projections is that the approximation of the low-tubal-rank tensor can be obtained quite accurately in an inexpensive manner. Experimental examples for hyperspectral image denoising are presented to demonstrate the effectiveness and efficiency of the proposed method., Comment: Accepted by IGARSS 2019

Multi-dimensional images, such as color images and multi-spectral images, are highly correlated and contain abundant spatial and spectral information. However, real-world multi-dimensional images are usually corrupted by missing entries. By integrating deterministic low-rankness prior to the data-driven deep prior, we suggest a novel regularized tensor completion model for multi-dimensional image completion. In the objective function, we adopt the newly emerged tensor nuclear norm (TNN) to characterize the global low-rankness prior of the multi-dimensional images. We also formulate an implicit regularizer by plugging into a denoising neural network (termed as deep denoiser), which is convinced to express the deep image prior learned from a large number of natural images. The resulting model can be solved by the alternating directional method of multipliers algorithm under the plug-and-play (PnP) framework. Experimental results on color images, videos, and multi-spectral images demonstrate that the proposed method can recover both the global structure and fine details very well and achieve superior performance over competing methods in terms of quality metrics and visual effects., Comment: Accepted by Neurocoputing. Code availiable at https://github.com/TaiXiangJiang/Deep-Plug-and-Play-Prior-for-Low-Rank-Tensor-Completion

As low-rank modeling has achieved great success in tensor recovery, many research efforts devote to defining the tensor rank. Among them, the recent popular tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), obtains promising results. However, the framework of the t-SVD and the tensor tubal rank are applicable only to three-way tensors and lack of flexibility to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-$k_1k_2$ tensor unfolding, as the process of lexicographically stacking the mode-$k_1k_2$ slices of an $N$-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-$k$ tensor matricization. Based on it, we define a novel tensor rank, the tensor $N$-tubal rank, as a vector whose elements contain the tubal rank of all mode-$k_1k_2$ unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed $N$-tubal rank, we establish its convex relaxation: the weighted sum of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.

Rain streak removal is an important issue and has recently been investigated extensively. Existing methods, especially the newly emerged deep learning methods, could remove the rain streaks well in many cases. However the essential factor in the generative procedure of the rain streaks, i.e., the motion blur, which leads to the line pattern appearances, were neglected by the deep learning rain streaks approaches and this resulted in over-derain or under-derain results. In this paper, we propose a novel rain streak removal approach using a kernel guided convolutional neural network (KGCNN), achieving the state-of-the-art performance with simple network architectures. We first model the rain streak interference with its motion blur mechanism. Then, our framework starts with learning the motion blur kernel, which is determined by two factors including angle and length, by a plain neural network, denoted as parameter net, from a patch of the texture component. Then, after a dimensionality stretching operation, the learned motion blur kernel is stretched into a degradation map with the same spatial size as the rainy patch. The stretched degradation map together with the texture patch is subsequently input into a derain convolutional network, which is a typical ResNet architecture and trained to output the rain streaks with the guidance of the learned motion blur kernel. Experiments conducted on extensive synthetic and real data demonstrate the effectiveness of the proposed method, which preserves the texture and the contrast while removing the rain streaks.

Rain streak removal is an important issue in outdoor vision systems and has recently been investigated extensively. In this paper, we propose a novel video rain streak removal approach FastDeRain, which fully considers the discriminative characteristics of rain streaks and the clean video in the gradient domain. Specifically, on the one hand, rain streaks are sparse and smooth along the direction of the raindrops, whereas on the other hand, clean videos exhibit piecewise smoothness along the rain-perpendicular direction and continuity along the temporal direction. Theses smoothness and continuity results in the sparse distribution in the different directional gradient domain, respectively. Thus, we minimize 1) the $\ell_1$ norm to enhance the sparsity of the underlying rain streaks, 2) two $\ell_1$ norm of unidirectional Total Variation (TV) regularizers to guarantee the anisotropic spatial smoothness, and 3) an $\ell_1$ norm of the time-directional difference operator to characterize the temporal continuity. A split augmented Lagrangian shrinkage algorithm (SALSA) based algorithm is designed to solve the proposed minimization model. Experiments conducted on synthetic and real data demonstrate the effectiveness and efficiency of the proposed method. According to comprehensive quantitative performance measures, our approach outperforms other state-of-the-art methods especially on account of the running time. The code of FastDeRain can be downloaded at https://github.com/TaiXiangJiang/FastDeRain., Comment: codes available at https://github.com/TaiXiangJiang/FastDeRain

In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank. We build two PSTNN-based minimization models for two typical tensor recovery problems, i.e., the tensor completion and the tensor principal component analysis. We give two algorithms based on the alternating direction method of multipliers (ADMM) to solve proposed PSTNN-based tensor recovery models. Experimental results on the synthetic data and real-world data reveal the superior of the proposed PSTNN.