背景消去是一種很常用在移動物件偵測的方法，它利用影片中沒有被前景所遮蔽的背景部分去建立背景模型以達到移動物件偵測的效果，然而當前景的面積相對較大或是前景的移動量過小時，偵測的效果會因為影片中部分背景長時間被前景所遮蔽而導致變差，因此我們提出了稀疏與低張量秩模型來克服背景遮蔽的問題。我們考慮了背景在空間上低矩陣秩的特性，將背景在時間和空間上的低矩陣秩特性結合，使之能夠更好的描述背景在時空上所具有的高線性關聯性，而在實驗結果的部分可以看見，我們的方法對於具有背景遮蔽的影片以及具有高結構性背景的影片之偵測效果都有很好的改善。

Background subtraction is a common method utilized to detect moving objects. The main idea is estimate the background model according to the non-occluded background. However, when the foreground is comparatively large or the moving displacement of foreground is negligible, the estimated result will be inaccurate because the background is occluded by foreground most of the time. In order to overcome the occluded background problem, we consider the spatial low-rank property of background, and propose to combine the spatial low-rank property and the temporal low-rank property to better characterize the strong correlation existing in spatio-temporal dimension of the background. The proposed method extends the low-rank matrix modeling to low-rank tensor modeling for the background. Experimental results show that the low-rank tensor modeling improves the result under occluded background or highly structured background.

中文摘要 I
Abstract II
List of contents III
1. Introduction 1
2. Related Work 5
2.1.Principle component pursuit.…………………………………………………5
2.2. Detecting contiguous outliers in the low-rank representation……...………..6
3. Proposed Method 12
3.1. Motivation………………………………………………………………….12
3.2. Low-rank tensor background model 12
3.3. Patch representation 14
3.4. Algorithm 15
4. Experimental Results 20
4.1 Experimental setting………………………………………………………..,20
4.2 Highly structured background case…………………………………………20
4.3 Occluded background case………………………………………………….21
4.4 Other cases………………………………………………………………….21
4.5 Discussion and limitation………………………………………………….22
5. Conclusion 32
6. References 33

[1] W. Yu, X. Zhou, C. Yang, “Moving object detection by detecting. contiguous outliers in the low-rank representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 35, no. 3, pp. 597-610, 2012.
[2] C. Stauffer and W. Grimson, “Adaptive Background Mixture Models for Real-Time Tracking,” in Proc. of IEEE Int. Conf. Compt. Vis. Pattern Recogn., 1999.
[3] D. Gutchess, M. Trajkovics, E. Cohen-Solal, D. Lyons, and A. Jain, “A background model initialization algorithm for video surveillance,” in Proc. of IEEE Int. Conf. Comput. Vis., 2001.
[4] Jingyu Yan and Marc Pollefeys, “A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and non-degenerate,” in European Conference on Computer Vision, 2006.
[5] E. Candes, X. Li, Y. Ma, and J. Wright, “Robust Principal Component Analysis?” Journal of the ACM, vol.58, no.3, 2011.
[6] E. Simoncelli and W. Freeman, “A flexible architecture for multi-scale derivative computation,” in Proc. of IEEE Int. Conf. Image Process., 1995, pp. 444–447.
[7] M. Fazel, “Matrix rank minimization with applications,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 2002.
[8] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma, “Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization,” in Proc. Adv. Neural Inf. Process. Syst., 2009.
[9] J. Liu, P. Musialski, P. Wonka, J. Ye, “Tensor completion for estimating missing values in visual data,” in Proc. of IEEE Int. Conf. Comput. Vis., 2009.
[10] R. Mazumder, T. Hastie, and R. Tibshirani, “Spectral Regularization Algorithms for Learning Large Incomplete Matrices,” J. Mach. Learn. Res, vol. 11, pp. 2287–2322, 2010.
[11] Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 23, no. 11, pp. 1222–1239, 2001.
[12] V. Kolmogorov and R. Zabih, “What Energy Functions Can Be Minimizedvia Graph Cuts?” IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 2, pp. 147–159, 2004.
[13] J. Odobez and P. Bouthemy, “Robust multiresolution estimation of parametric motion models,” J. Visual Commun. Image repres., vol. 6, no. 4, pp. 348–365, 1995.
[14] T. Brox and J. Malik, “Object segmentation by long term analysis of point trajectories,” in Proc. of Eur. Conf. Comput. Vis., 2010.
[15] H. Wang and D. Suter, “A novel robust statistical method for background initialization and visual surveillance,” in Proc.of Asian Conf. Comput. Vis., 2006.
[16] L. Li, W. Huang, I. Gu, and Q. Tian, “Statistical modeling of complex backgrounds for foreground object detection,” IEEE Trans. Image Processing, vol. 13, no. 11, pp. 1459–1472, 2004.
[17] D. J. Dailey, F. W. Cathey and S. Pumrin, “An algorithm to estimate mean traffic speed using uncalibrated cameras,” IEEE Trans. on Intelligent Transportation Systems, vol. 1, no. 2, pp. 98–107, 2000.