基于监控视频背景建模的一类非凸矩阵优化算法研究
Research on a Class of Nonconvex Matrix Optimization Algorithm Based on Surveillance Video Background Modeling
摘要: 为了实现稳健的低秩–稀疏矩阵分解，本文考虑非凸非光滑优化模型，其中矩阵的秩函数采用奇异值的Capped-l1松弛，矩阵的l0范数采用矩阵的l1范数松弛。首先，给出了适用于矩阵问题的Capped-l1阈值算子。其次，提出了交替方向乘子法求解我们的非凸非光滑矩阵优化模型，并分析了算法的收敛性。最后，通过大量数值实验表明：在有噪声和无噪声的情况下，所提出的算法都能有效、稳健地分解出低秩–稀疏矩阵。并将提出的算法应用于监控视频的前景和背景分离问题，发现所提出的算法对于该问题有良好的性能，这说明该算法能够解决相关实际问题。
Abstract: In order to achieve robust low-rank sparse matrix decomposition, this paper considers a non-convex and nonsmooth optimization model, in which the ranking function of the matrix is relaxed by Capped-l1 regularization of the singular values, and l0 norm of the matrix is relaxed with the l1 norm of the matrix. First, we provide the closed-form thresholding operator for the matrixCapped-l1 function. Secondly, the alternating direction method of multipliers is proposed to solve our nonconvex and nonsmooth optimization model, and the convergence analysis is provided. Finally, a large number of numerical experiments show that the proposed algorithm can effectively and robustly decompose the low-rank and sparse matrices in the case of noise and noise. The proposed algorithm is applied to the background separation problem of surveillance video, and it is indicated that the proposed algorithm has good performance for this problem, which shows that the algorithm can solve the relevant practical problems.
文章引用：贾凯贤, 陈小英, 孟书羽, 程越, 张弦, 彭定涛. 基于监控视频背景建模的一类非凸矩阵优化算法研究[J]. 运筹与模糊学, 2023, 13(4): 2817-2830. https://doi.org/10.12677/ORF.2023.134282

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