# 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.

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