The comprehensive acquisition of electromagnetic spectrum data is crucial for applications, such as cognitive radio and electromagnetic mapping in electromagnetic environment monitoring. However, the inherent limitations of spectrum sensor sampling rates and environmental unreliability often result in data loss. Existing spectrum data recovery methods predominantly adopt generic algorithms from other domains, failing to adequately exploit the intrinsic characteristics of spectrum signals. This article introduces a novel low-rank autoregressive tensor completion algorithm leveraging the shift-invariant discrete wavelet transform (SIDWT) to recover and reconstruct incomplete electromagnetic spectrum data. The approach focuses on minimizing the truncated nuclear norm (T-TNN) of the SIDWT coefficient tensor, which exhibits stronger low-rank properties. Additionally, the AR(p) model is incorporated to impose temporal constraints, ensuring the reconstructed data closely approximates the original in both global and local characteristics. Additionally, to address the issue of high computational complexity, we have developed a fast algorithm. Experimental validation using real-world data demonstrates the efficacy of the proposed algorithm. It outperforms state-of-the-art algorithms on 78.1% of electromagnetic spectrum datasets and all audio signal spectrum datasets, achieving at least a 2% improvement in Relative Squared Error (root square error (RSE)) compared to the best-performing baseline. Notably, under high missing rates and on time missing (TM) datasets, the improvement exceeds 30%. The proposed method provides a novel solution for the completion of incomplete electromagnetic spectrum data. The proposed fast algorithm achieves approximately 40% improvement in computational speed without significant performance degradation.

In residential environment, electromagnetic power density exceeding a certain value will affect people's livelihood and health. In the monitoring of electromagnetic environmental quality of residential buildings, the grid method is generally used to measure the data value of electromagnetic radiation sources, and the visualization technology is used to display the data of electromagnetic radiation sources in the region. In this paper, we use the method of randomly deploying sensor nodes to sample grid electromagnetic data, which greatly saves the deployment cost of sensor nodes. However, it will lead to data loss and pulse noise interference. Giving that the general electromagnetic data visualization diagram are local smoothing and sparse in transformation domain, we propose to use the tensor form of electromagnetic data to completion/restoration or predict the area grid that cannot be monitored based on the completion theory. The prediction model based on tensor train and algorithm are given. Experimental results show that the method can make the data smoother visually and within a certain accuracy.

Camera failure or loss of storage components in imaging equipment may result in the loss of important image information or random pulse noise interference. The low-rank prior is one of the most important priors in image optimization processing. This paper reviews and compares some low-rank constraint models for image matrices. Firstly, an overview of image-inpainting models based on nuclear norm, truncated nuclear norm, weighted nuclear norm, and matrix-factorization-based F norm is presented, and corresponding optimization iterative algorithms are provided. Then, we use different image matrix low-order constraint models to recover satellite images from three types of pulse interference and provide our experimental visual and numerical results. Finally, it can be concluded that the method based on the weighted nuclear norm can achieve the best image restoration effect. The F norm method based on matrix factorization has the shortest computational time and can be used for large-scale low-rank matrix calculations. Compared with nuclear norm-based methods, weighted nuclear norm-based methods and truncated nuclear norm-based methods can significantly improve repair performance.

Camera malfunctions or loss of storage elements in imaging devices may lead to loss of important image information or random impulse noise interference. Low rank is one of the important prior information in image optimization processing. This paper uses different low-rank constraint models of the image matrices to recover the impulse interference satellite images. Firstly, an overview of image inpainting models based on nuclear norm, truncated nuclear norm, weighted nuclear norm, and matrix-factorization-based F-norm is presented, and corresponding optimiza-tion iterative algorithms are provided. Then, we conducted experiments under three types of pulse interference and provided visual and numerical results. Finally, it is concluded that the WSVT_ADMM method based on the weighted nuclear norm can obtain the best image inpainting results; The UV_ADMM method based on F norm of matrix factorization has the least computa-tion time and can be used for large-scale low-rank matrix computation; The WSVT_ADMM method based on weighted nuclear norm and the TSVT_ ADMM method based on truncated nuclear norms can significantly improve the repair effect compared to the nuclear norm-based methods such as SVT, SVP, and n_ADMM.

Employing low tensor rank decompositions in image inpainting has attracted increasing attention. This paper exploits a novel tensor-augmentation schemes to transform an image (a low-order tensor) to a higher-order tensor without changing the total number of pixels. The developed augmentation schemes enhance the low-rankness of an image under three tensor decompositions: matrix SVD, tensor train (TT) decomposition, and tensor singular value decomposition (t-SVD). By exploiting the schemes, we solve the image inpainting problem with three low-rank con-strained models which use the matrix rank, TT rank, and tubal rank as constrained priors re-spectively. The tensor tubal rank and tensor train multi-rank are developed from t-SVD and TT decomposition respectively. We exploit efficient ADMM algorithms for solving the three models. Experimental results demonstrate that our methods are effective for image inpainting and supe-rior to numerous close methods.