# Recovering shape and motion by a dynamic system for low-rank matrix approximation in *L* _{1} norm

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## Abstract

To recover motion and shape matrices from a matrix of tracking feature points on a rigid object under orthography, we can do low-rank matrix approximation of the tracking matrix with its each column minus the row mean vector of the matrix. To obtain the row mean vector, usually 4-rank matrix approximation is used to recover the missing entries. Then, 3-rank matrix approximation is used to recover the shape and motion. Obviously, the procedure is not convenient. In this paper, we build a cost function which calculates the shape matrix, motion matrix as well as the row mean vector at the same time. The function is in *L* _{1} norm, and is not smooth everywhere. To optimize the function, a continuous-time dynamic system is newly proposed. With time going on, the product of the shape and rotation matrices becomes closer and closer, in *L* _{1}-norm sense, to the tracking matrix with each its column minus the mean vector. A parameter is implanted into the system for improving the calculating efficiency, and the influence of the parameter on approximation accuracy and computational efficiency are theoretically studied and experimentally confirmed. The experimental results on a large number of synthetic data and a real application of structure from motion demonstrate the effectiveness and efficiency of the proposed method. The proposed system is also applicable to general low-rank matrix approximation in *L* _{1} norm, and this is also experimentally demonstrated.

## Keywords

Structure from motion Low-rank matrix approximation Dynamic system*L*

_{1}norm Convergence

## Notes

### Acknowledgements

We thank the Editor and Reviewers for time and effort going in reviewing this paper. This work was supported by NSFC under Grants 61173182 and 61179071, and the Applied Basic Research Project (2011JY0124) and the International Cooperation and Exchange Project (2012HH0004) of Sichuan Province.

## References

- 1.Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography: a factorization method. Int. J. Comput. Vis.
**9**, 137–154 (1992) CrossRefGoogle Scholar - 2.Wang, G., Wu, Q.M.J.: Stratification approach for 3-D Euclidean reconstruction of nonrigid objects from uncalibrated image sequences. IEEE Trans. Syst. Man Cybern., Part B, Cybern.
**38**, 90–101 (2008) CrossRefGoogle Scholar - 3.Torresani, L., Hertzmann, A., Bregler, C.: Nonrigid structure-from-motion: estimating shape and motion with hierarchical priors. IEEE Trans. Pattern Anal. Mach. Intell.
**30**, 878–892 (2008) CrossRefGoogle Scholar - 4.Peng, Y., Ganesh, A., Wright, J., Ma, Y.: RASL: robust alignment by sparse and low-rank decomposition for linearly correlated images. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 700–763 (2010) Google Scholar
- 5.Wright, J., Ganesh, A., Rao, S., Peng, Y., Ma, Y.: Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. In: Bengio, Y., Schuurmans, D., Lafferty, J., Williams, C.K.I., Culotta, A. (eds.) Advances in Neural Information Processing Systems, pp. 2080–2088 (2009) Google Scholar
- 6.Liu, Y., You, Z., Cao, L.: A novel and quick SVM-based multi-class classifier. Pattern Recognit.
**39**, 2258–2264 (2006) MATHCrossRefGoogle Scholar - 7.Liu, Y., Sam Ge, S., Li, C., You, Z.: k-NS: a classifier by the distance to the nearest subspace. IEEE Trans. Neural Netw.
**22**(8), 1256–1268 (2011) CrossRefGoogle Scholar - 8.Morita, T., Kanade, T.: A sequential factorization method for recovering shape and motion from image streams. IEEE Trans. Pattern Anal. Mach. Intell.
**19**, 858–867 (1997) CrossRefGoogle Scholar - 9.Hartley, R., Schaffalitzky, F.: Power factorization: 3D reconstruction with missing or uncertain data. In: Australia-Japan Advanced Workshop on Computer Vision, pp. 1–9 (2003) Google Scholar
- 10.Buchanan, A.M., Fitzgibbon, A.W.: Damped newton algorithms for matrix factorization with missing data. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), vol. 2, pp. 316–322 (2005) Google Scholar
- 11.Morgan, A.B.: Investigation into matrix factorization when elements are unknown. Technical report, Visual Geometry Group, Department of Engineering Science, University of Oxford (2004) Google Scholar
- 12.Ye, J.: Generalized low rank approximations of matrices. Mach. Learn.
**61**, 167–191 (2005) MATHCrossRefGoogle Scholar - 13.Liu, J., Chen, S., Zhou, Z.-H., Tan, X.: Generalized low-rank approximations of matrices revisited. IEEE Trans. Neural Netw.
**21**, 621–632 (2010) CrossRefGoogle Scholar - 14.Wiberg, T.: Computation of principal components when data are missing. In: Proc. Second Symp. Computational Statistics, Berlin, pp. 229–236 (1976) Google Scholar
- 15.Okatani, T., Deguchi, K.: On the Wiberg algorithm for matrix factorization in the presence of missing components. Int. J. Comput. Vis.
**72**, 329–337 (2007) CrossRefGoogle Scholar - 16.Chen, P.: Heteroscedastic low-rank matrix approximation by the Wiberg algorithm. IEEE Trans. Signal Process.
**56**, 1429–1439 (2008) MathSciNetCrossRefGoogle Scholar - 17.Chen, P.: Optimization algorithms on subspaces: revisiting missing data problem in low-rank matrix. Int. J. Comput. Vis.
**80**(1), 125–142 (2008) CrossRefGoogle Scholar - 18.Lin, Z., Chen, M., Wu, L., Ma, Y.: The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices, October (2009) Google Scholar
- 19.Cai, J.-F., Candés, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim.
**20**, 1956–1982 (2010) MathSciNetMATHCrossRefGoogle Scholar - 20.Ke, Q., Kanade, T.: Robust
*L*_{1}norm factorization in the presence of outliers and missing data by alternative convex programming. In: 2005 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), vol. 1, pp. 739–746 (2005) CrossRefGoogle Scholar - 21.Eriksson, A., van den Hengel, A.: Efficient computation of robust low-rank matrix approximations in the presence of missing data using the
*L*_{1}norm. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 771–778 (2010) CrossRefGoogle Scholar - 22.Li, S.Z.: Markov Random Field Modeling in Image Analysis. Springer, Berlin (2001) MATHCrossRefGoogle Scholar
- 23.Zucker, S.: Differential geometry from the Frenet point of view: boundary detection, stereo, texture and color. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, pp. 359–373. Springer, Berlin (2006) Google Scholar
- 24.Liu, Y.: Automatic range image registration in the Markov chain. IEEE Trans. Pattern Anal. Mach. Intell.
**32**, 12–29 (2010) MATHCrossRefGoogle Scholar - 25.Weickert, J., SteidI, G., Mrazek, P., Welk, M., Brox, T.: Diffusion filters and wavelets: what can they learn from each other? In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, pp. 3–17. Springer, Berlin (2006) Google Scholar
- 26.Stoykova, E., Alatan, A.A., Benzie, P., Grammalidis, N., Malassiotis, S., Ostermann, J., Piekh, S., Sainov, V., Theobalt, C., Thevar, T., Zabulis, X.: 3-D time-varying scene capture technologiesła survey. IEEE Trans. Circuits Syst. Video Technol.
**17**, 1568–1586 (2007) CrossRefGoogle Scholar - 27.Liu, Y., You, Z., Cao, L.: A functional neural network computing some eigenvalues and eigenvectors of a special real matrix. Neural Netw.
**18**, 1293–1300 (2005) MATHCrossRefGoogle Scholar - 28.Liu, Y., You, Z., Cao, L.: A concise functional neural network computing the largest modulus eigenvalues and their corresponding eigenvectors of a real skew matrix. Theor. Comput. Sci.
**367**, 273–285 (2006) MathSciNetMATHCrossRefGoogle Scholar - 29.Orban, G.A., Janssen, P., Vogels, R.: Extracting 3D structure from disparity. Trends Neurosci.
**29**(8), 466–473 (2006) CrossRefGoogle Scholar - 30.la Torre, F.D., Blackt, M.J.: Robust principal component analysis for computer vision. In: Proceedings of Eighth IEEE International Conference on Computer Vision (ICCV 2001), vol. 1, pp. 362–369 (2001) CrossRefGoogle Scholar
- 31.El-Melegy, M.T., Al-Ashwal, N.H.: A variational technique for 3D reconstruction from multiple views. In: International Conference on Computer Engineering & Systems (ICCES 07), pp. 38–43 (2007) Google Scholar
- 32.Zhong, H., Hung, Y.: Multi-stage 3D reconstruction under circular motion. Image Vis. Comput.
**25**, 1814–1823 (2007) CrossRefGoogle Scholar - 33.Martinec, D., Pajdla, T.: 3D reconstruction by fitting low-rank matrices with missing data. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR 2005, vol. 1, pp. 198–205 (2005) Google Scholar
- 34.Keshavan, R.H., Montanari, A., Oh, S.: Matrix completion from a few entries. IEEE Trans. Inf. Theory
**56**, 2980–2998 (2010) MathSciNetCrossRefGoogle Scholar - 35.Toh, K.-C., Yun, S.: An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pac. J. Optim.
**6**, 65–640 (2010) MathSciNetGoogle Scholar - 36.Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New York (2002) MATHGoogle Scholar