Spatialtemporal traffic data are collected by a wide array of data collection devices deployed in intelligent transportation systems, which play crucial roles in data-driven intelligent transportation systems. However, hardware malfunctions or software system errors can lead to the inability to collect accurate data. Incomplete data in intelligent transportation systems can cause difficulties for subsequent applications, such as traffic flow anomaly detection and forecasting. Recently, low-rank tensor-based methods that characterize the global correlation of high-dimensional data have achieved superior performance. However, the majority of tensor-based approaches primarily account for the global correlation within the target traffic data, which is not sufficient to recover some challenging missing scenarios, such as fiber missing and slice missing that often occur in real situations. In this manuscript, we suggest a flexible method to explore the global correlation, local smoothness, and nonlocal self-similar redundancy in traffic data to improve recovery accuracy. Specifically, we utilize the multidimensional tensor nuclear norm to characterize the correlation structure, the multidimensional total variation to maintain smoothness detail, and the plug-and-play term to promote nonlocal self-similarity. The primary merits of this model lie in the fact that these priors characterize data features from different perspectives and complement each other. We develop an alternating direction method of multipliers to achieve efficient optimization for each variable. The proposed method is comprehensively evaluated through experiments across seven missing scenarios. Extensive experiments demonstrate that the suggested method with the aid of different priors outperforms many state-of-the-art approaches, especially structured missing scenarios.

In this paper, we extend the definition of tensor products from using an invertible matrix to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. Based on this novel definition, we define the concepts of tensor rank and tensor nuclear norm, ensuring consistency with their matrix counterparts, and derive a singular value thresholding (${\ast }_{L,R}$ SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is integral to our proposed tensor robust principal component analysis (${\ast }_{LR}$ TRPCA) algorithm. The computational complexity of the ${\ast }_{LR}$ TRPCA algorithm is $O(k·(n_1{n}_2{n}_3+p·\min (n_1^2{n}_2,n_1{n}_2^2)))$ for k iterations. According to this complexity analysis, by using a right-invertible matrix that selects p rows from the $n_3$ rows of the invertible matrix used in the tensor product with an invertible matrix, the computational load is approximately reduced to $p/n_3$ of what it would be with an invertible matrix, highlighting the efficiency gain in terms of computational resources. We apply this efficient algorithm to grayscale video denoising and motion detection problems, where it demonstrates significant improvements in processing speed while maintaining comparable quality levels to existing methods, thereby providing a promising approach for handling multi-linear data and offering valuable insights for advanced data analysis tasks.

In this paper, we extend the definition of tensor products from using invertible transformations to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. On the basis of this new definition, we define the concepts of tensor rank and tensor nuclear norm and investigate their properties, ensuring consistency with their matrix counterparts. We then derive a singular value thresholding (SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is a key component of our proposed tensor robust principal component analysis (TRPCA) algorithm. We conduct a complexity analysis of the proposed algorithm, demonstrating its computational efficiency, and apply it to grayscale video denoising and motion detection problems, where it shows significant improvements in efficiency while maintaining a similar level of quality. This work provides a promising approach for handling large-scale data, offering new insights and solutions for advanced data analysis tasks.