提取視訊前景是電腦視覺中一項任務，其目的是從包含動態前景物體與靜態背景的影像序列中，準確地提取出前景資訊。本文提出一種改良Candès 等人所發展的主成分追蹤方法，建構了一個高效秩1主成分追蹤模型，應用於視訊中的前景提取。該方法假設所有影格共享相同的背景，並將整段視訊表示為一個資料矩陣，將其分解為稀疏矩陣與秩1矩陣之和。 我們進一步基於增廣拉格朗日乘子法設計出一種高效演算法，其透過閉合形式解的更新實現快速收斂，並顯著降低計算成本。為了評估方法效能，我們將所提出的模型與現有的非凸主成分追蹤演算法–加速交替投影法–進行比較。數值實驗結果顯示，透過高效秩1主成分追蹤模型在某些情境下於精確度與執行速度具有相對優勢，結果驗證了所提方法在提取視訊前景上的有效性與穩健性，顯示其作為提取前景建模工具的實用潛力。

Video foreground extraction is a fundamental task in computer vision, aiming to accurately extract the foreground from a sequence of frames containing dynamic foreground objects and a static background. This thesis proposes an improved principal component pursuit (PCP) model developed by Candès et al., constructing an efficient rank-1 PCP model applied to foreground extraction modeling in video. The method assumes that all frames shares a common background, and formulates the video sequence as a data matrix to be decomposed into sparse and rank-1 components. We further develop an efficient algorithm based on the Augmented Lagrange Multiplier (ALM) method, which enables closed-form updates for fast convergence and significantly reduces computational cost. To evaluate performance, the proposed model is compared with the existing nonconvex PCP solver, Accelerated Alternating Projection (AccAltProj). Numerical experiments demonstrate that the rank-1 model exhibits relative advantages in both accuracy and runtime under certain scenarios. The results confirm the effectiveness and robustness of the proposed method in video foreground extraction, highlighting its practical potential as a foreground extraction modeling tool.

[1] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimiza-
tion and statistical learning via the alternating direction method of multipli-
ers, Foundations and Trends in Machine Learning, 3 (2010), pp. 1-122.
[2] T. Bouwmans, A. Sobral, S. Javed, S. K. Jung, and E. Zahzah, Decomposi-
tion into low-rank plus additive matrices for background/foreground sep-
aration: a review for a comparative evaluation with a large-scale dataset,
Computer Science Review, 23 (2017), pp. 1-71.
[3] H. Cai, J. Cai, and K. Wei, Accelerated alternating projections for robust prin-
cipal component analysis, arXiv:1711.05519v4 [cs.IT], 2019.
[4] H. Cai, K. Hamm, L. Huang, and D. Needell, Robust CUR decomposition:
theory and imaging applications, arXiv:2101.05231v2 [cs.CV], 2021.
[5] E. J. Candès, X. Li, Y. Ma, and J. Wright, Robust principal component analy-
sis? Journal of the ACM, 58 (2011), Article 11.
[6] V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky, Rank-
sparsity incoherence for matrix decomposition, SIAM Journal on Optimiza-
tion, 21 (2011), pp. 572–596.
[7] J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time back-
ground/foreground separation in video, arXiv:1404.7592v1 [cs.CV], 2014.
[8] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins
University Press, 2013.
[9] V. Hovhannisyan, Y. Panagakis, P. Parpas, and S. Zafeiriou, Fast multilevel
algorithms for compressive principal component pursuit, SIAM Journal on
Imaging Sciences, 12 (2019), pp. 624-649.
[10] V. Hovhannisyan, Y. Panagakis, S. Zafeiriou, and P. Parpas, Multilevel ap-
proximate robust principal component analysis, IEEE International Conference
on Computer Vision Workshops (ICCVW), 2017, pp. 536-544.
[11] Z. Lin, R. Lin, and Z. Su, Linearized alternating direction method with adap-
tive penalty for low-rank representation, Proceedings of the 24th International
Conference on Neural Information Processing Systems, 24 (2011), pp. 612-620.
[12] B. E. Moore, C. Gao, and R. R. Nadakuditi, Panoramic Robust PCA for
foreground-background separation on noisy, free-motion camera video,
IEEE Transactions on Computational Imaging, 5 (2019), pp. 195-211.
[13] P. Netrapalli, U. Niranjan, S. Sanghavi, A. Anandkumar, and P. Jain, Non-
convex robust PCA, In Advances in Neural Information Processing Systems, 27
(2014), pp. 1107–1115.
[14] X. Xiu1, Y. Yang1, W. Liu, L. Kong, and M. Shang, An improved total vari-
ation regularized RPCA for moving object detection with dynamic back-
ground, American Institute of Mathematical Sciences, 16 (2020), pp. 1685-1698.
[15] S. Zuo, X. Chen, R. Ansari, and Y. Lu, 21 cm signal recovery via robust prin-
cipal component analysis, The Astronomical Journal, 157 (2019), pp. 1-13.
[16] C. Zheng and H. Li, Small infrared target detection based on harmonic and
sparse matrix decomposition, Optical Engineering, 52 (2013), 066401.