Mining structural priors in data is a widely recognized technique for hyperspectral image (HSI) denoising tasks, whose typical ways include model-based methods and data-based methods. The model-based methods have good generalization ability, while the runtime can hardly meet the fast processing requirements of the practical situations due to the large size of an HSI X ϵ R ^MN×B . For the data-based methods, they perform relatively fast on new test data once they have been trained. However, their generalization ability is always insufficient. In this paper, we propose a fast model-based approach via a novel regularizer named the Representative Coefficient Total Variation (RCTV) to simultaneously characterize the low rank and local smooth properties. The RCTV regularizer is proposed based on the observation that the representative coefficient matrix U ϵ R ^MN×R ( R ≪ B ) obtained by orthogonally transforming the original HSI X can inherit the strong local-smooth prior of X. Since R/B is very small, the model based on the RCTV regularizer has lower time complexity. Additionally, we find that the representative coefficient matrix U is robust to noise, and thus the RCTV regularizer can somewhat promote the robustness of the HSI denoising model. Extensive experiments on mixed noise removal demonstrate that the proposed method realizes a perfect compromise between denoising performance and denoising speed as compared with other state-of-the-art methods. Remarkably, the denoising speed of our proposed method outperforms all competing model-based techniques and is comparable with the deep learning-based approaches. The code of our algorithm is released at https://github.com/andrew-pengjj/rctv.git.

Recovering low-rank matrices by nuclear norm minimization and local-smooth matrices by total variation seminorm minimization are two common methods in the context of compressive sensing. As a matter of fact, the two properties simultaneously exist in many real-world datasets, typically exampling hyperspectral images. The two methods may not perform well in this situation. To better address this issue, in this letter, we study the correlated total variation norm minimization problem both theoretically and numerically. We obtain an error bound for the robust recovery of our method in theory, which reflects that this model indeed benefits from low-rank and local-smooth properties of the matrix to be restored. Experiments on the recovery of hyperspectral images show that this model is superior to many other competing ones.

This paper considers the 1-bit compressed sensing (1-bit CS) of signals that are simultaneously sparse and gradient sparse. Since L1-norm and total variation (TV) penalties are beneficial for reconstructing element-sparse and gradient-sparse signals, respectively, we combine both to propose the L1-TV regularization model for 1-bit CS. We show that the proposed model has a closed-form solution so that one can easily calculate. Despite the apparent simplicity, our theoretical analysis reveals that the solution provides an approximate estimate for the underlying signal. Besides, in the case of introducing a dither vector, we develop an adaptive algorithm to accelerate the decay rate of recovery error. The key idea is that generating the dither for each iteration relying on the last estimate. In addition to theoretical analysis, we conduct a series of experiments on both synthetic and real-world data to show the superiority of our algorithms.

In this article, we bring forward a completely perturbed nonconvex Schatten $p$ -minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten- $p$ null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten- $p$ NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition $\delta_{2r}<1$ (Recht et al., 2010). In addition, we study the connection between RIP and Schatten- $p$ NSP and discern that Schatten- $p$ NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten $p$ -minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.