Hyperspectral images (HSIs) are often corrupted by a mixture of several types of noise during the acquisition process, e.g., Gaussian noise, impulse noise, dead lines, stripes, etc. Such complex noise could degrade the quality of the acquired HSIs, limiting the precision of the subsequent processing. In this paper, we present a novel tensor-based HSI restoration approach by fully identifying the intrinsic structures of the clean HSI part and the mixed noise part. Specifically, for the clean HSI part, we use tensor Tucker decomposition to describe the global correlation among all bands, and an anisotropic spatial–spectral total variation regularization to characterize the piecewise smooth structure in both spatial and spectral domains. For the mixed noise part, we adopt the $\ell _1$ norm regularization to detect the sparse noise, including stripes, impulse noise, and dead pixels. Despite that TV regularization has the ability of removing Gaussian noise, the Frobenius norm term is further used to model heavy Gaussian noise for some real-world scenarios. Then, we develop an efficient algorithm for solving the resulting optimization problem by using the augmented Lagrange multiplier method. Finally, extensive experiments on simulated and real-world noisy HSIs are carried out to demonstrate the superiority of the proposed method over the existing state-of-the-art ones.

Hyperspectral image (HSI) denoising is a fundamental task in remote sensing image processing, which is helpful for HSI subsequent applications, such as unmixing and classification. Thanks to the powerful representation ability of untrained deep neural networks, deep image prior (DIP)-based methods achieve tremendous successes in image processing (e.g., denoising and inpainting). However, DIP-based methods neglect the tensor low-rank prior of the underlying HSI which will be beneficial to capturing the global structure of the underlying HSI. To address this issue, we propose a novel model for HSI denoising, which can simultaneously take respective advantages of the tensor low-rank prior and the deep spatial-spectral prior. The tensor low-rank prior leads to a better global structure and the deep spatial-spectral prior is complementary to preserve better local details. On the one hand, we adopt low-rank tensor ring (TR) decomposition to characterize the tensor low-rank prior and capture the global structure of the underlying HSI. On the other hand, we employ untrained deep neural networks to flexibly represent the deep spatial-spectral prior and capture the local details of the underlying HSI. To solve the proposed model, we develop an efficient alternating minimization algorithm. Experimental results on simulated and real data validate the advantages of the proposed model in HSI denoising. Compared with state-of-the-art HSI denoising methods, the proposed method preserves better local details and the global structure of the underlying HSI.

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor recovery. 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.

Hyperspectral imaging, providing abundant spatial and spectral information simultaneously, has attracted a lot of interest in recent years. Unfortunately, due to the hardware limitations, the hyperspectral image (HSI) is vulnerable to various degradations, such as noises (random noise), blurs (Gaussian and uniform blur), and downsampled (both spectral and spatial downsample), each corresponding to the HSI denoising, deblurring, and super-resolution tasks, respectively. Previous HSI restoration methods are designed for one specific task only. Besides, most of them start from the 1-D vector or 2-D matrix models and cannot fully exploit the structurally spectral-spatial correlation in 3-D HSI. To overcome these limitations, in this article, we propose a unified low-rank tensor recovery model for comprehensive HSI restoration tasks, in which nonlocal similarity within spectral-spatial cubic and spectral correlation are simultaneously captured by third-order tensors. Furthermore, to improve the capability and flexibility, we formulate it as a weighted low-rank tensor recovery (WLRTR) model by treating the singular values differently. We demonstrate the reweighed strategy, which has been extensively studied in the matrix, also greatly benefits the tensor modeling. We also consider the stripe noise in HSI as the sparse error by extending WLRTR to robust principal component analysis (WLRTR-RPCA). Extensive experiments demonstrate the proposed WLRTR models consistently outperform state-of-the-art methods in typical HSI low-level vision tasks, including denoising, destriping, deblurring, and super-resolution.

The main aim of this paper is to study total variation (TV) regularization in deblurring and sparse unmixing of hyperspectral images. In the model, we also incorporate blurring operators for dealing with blurring effects, particularly blurring operators for hyperspectral imaging whose point spread functions are generally system dependent and formed from axial optical aberrations in the acquisition system. An alternating direction method is developed to solve the resulting optimization problem efficiently. According to the structure of the TV regularization and sparse unmixing in the model, the convergence of the alternating direction method can be guaranteed. Experimental results are reported to demonstrate the effectiveness of the TV and sparsity model and the efficiency of the proposed numerical scheme, and the method is compared to the recent Sparse Unmixing via variable Splitting Augmented Lagrangian and TV method by Iordache et al.