# Nonconvex plus quadratic penalized low-rank and sparse decomposition for noisy image alignment

## Abstract

This paper proposes a general method for dealing with the problem of recovering the low-rank structure, in which the data can be deformed by some unknown transformations and corrupted by sparse or nonsparse noises. Nonconvex penalization method is used to remedy the drawbacks of existing convex penalization method and a quadratic penalty is further used to better tackle the nonsparse noises in the data. We exploits the local linear approximation (LLA) method for turning the resulting nonconvex penalization problem into a series of weighted convex penalization problems and these subproblems are efficiently solved via the augmented Lagrange multiplier (ALM). Besides comparing with the method of robust alignment by sparse and low-rank decomposition for linearly correlated images (RASL), we also propose a nonconvex penalized lowrank and sparse decomposition (NLSD) model as comparison. Numerical experiments are conducted on both controlled and uncontrolled data to demonstrate the outperformance of the proposed method over RASL and NLSD.

## 摘要

### 创新点

1. (1)

针对采用 L1 范数进行凸优化时带来的较大偏差问题, 本文提出采用非凸优化的方式来减小偏差并获得了较好的对准效果。

2. (2)

实际应用中的数据存在稀疏噪声的同时也经常存在一些非稀疏噪声, 本文提出增加一项二次惩罚函数来对非稀疏噪声部分进行估计。

3. (3)

本文提出了一种 LLA-ALM 算法, 来求解一系列非凸优化子问题的局部最优解。

This is a preview of subscription content, access via your institution.

## References

1. 1

Eckart C, Young G. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1: 211–218

2. 2

Jolliffe I. Principal Component Analysis. Berlin: Springer-Verlag, 1986

3. 3

Candès E J, Li X D, Ma Y, et al. Robust principal component analysis? J ACM, 2011, 58: 1–37

4. 4

Huang G B, Jain V, Learned-Miller E G. Unsupervised joint alignment of complex images. In: Proceedings of IEEE International Conference on Computer Vision (ICCV), Rio de Janeiro, 2007. 1–8

5. 5

Learned-Miller E G. Data driven image models through continuous joint alignment. IEEE Trans Patt Anal Mach Intell, 2006, 28: 236–250

6. 6

Cox M, Sridharan S, Lucey S, et al. Least squares congealing for unsupervised alignment of images. In: Proceedings of IEEE International Conference on Computer Vision (ICCV), Kyoto, 2009. 1949–1956

7. 7

Vedaldi A, Guidi G, Soatto S. Joint data alignment up to (lossy) transformations. In: Proceedings of IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage Alaska, 2008. 1–8

8. 8

Fan J Q, Li R Z. Variable selection via nonconcave penalized likelihood and its oracle properties. J Amer Stat Assoc, 2001, 96: 1348–1360

9. 9

Zou H. The adaptive lasso and its oracle properties. J Amer Stat Assoc, 2006, 101: 1418–1429

10. 10

Leng C L, Lin Y, Wahba G. A note on the lasso and related procedures in model selection. Stat Sinica, 2006, 16: 1273–1284

11. 11

Zhang C H. Nearly unbiased variable selection under minimax concave penalty. Ann Stat, 2010, 38: 894–942

12. 12

Zhang C H, Zhang T. A general theory of concave regularization for high-dimensional sparse estimation problems. Stat Sci, 2012, 27: 576–593

13. 13

Fan J Q, Xue L Z, Zou H. Strong oracle optimality of folded concave penalized estimation. Ann Stat, 2014, 42: 819–849

14. 14

Bunea F, She Y Y, Wegkamp M H. Optimal selection of reduced rank estimators of high-dimensional matrices. Ann Stat, 2011, 39: 1282–1309

15. 15

Wang S S, Liu D H, Zhang Z H. Nonconvex relaxation approaches to robust matrix recovery. In: Proceedings of 23rd International Joint Conference on Artificial Intelligence, Beijing, 2013. 1764–1770

16. 16

Deng Y, Dai Q H, Liu R S, et al. Low-rank structure learning via log-sum heuristic recovery. IEEE Trans Neural Netw Learn Syst, 2013, 24: 383–396

17. 17

Cao W F, Wang Y, Yang C, et al. Folded-concave penalization appoarches to tensor completion. Neurocoumputing, 2015, 152: 261–273

18. 18

Zhou Z H, Li X D, Wright J, et al. Stable principal component pursuit. In: Proceedings of IEEE International Symposium on Information Theory (ISIT), Austin, 2010. 1518–1522

19. 19

Peng Y G, Ganesh A, Wright J, et al. RASL: robust alignment by sparse and low-rank decomposition for linearly correalted images. IEEE Trans Patt Anal Mach Intell, 2012, 34: 2233–2246

20. 20

Basri R, Jacobs D W. Lambertian reflectance and linear subspaces. IEEE Trans Patt Anal Mach Intell, 2003, 25: 218–233

21. 21

Wang X G, Zhang Z D, Ma Y, et al. Robust subspace discovery via relaxed rank minimization. Neural Comput, 2014, 26: 611–635

22. 22

Guo X H, Zhao R Z, An G Y, et al. An algorithm of face alignment and recognition by sparse and low rank decomposition. In: Proceedings of International Conference on Signal Processing (ICSP), HangZhou, 2014. 1036–1040

23. 23

Wright J, Yang A Y, Ganesh A, et al. Robust face recognition via sparse representation. IEEE Trans Patt Anal Mach Intell, 2009, 31: 210–227

24. 24

Huang Z W, Zhao X W, Shan S G, et al. Coupling alignments with recognition for still-to-video face recognition. In: Proceedings of IEEE International Conference on Computer Vision (ICCV), Sydney, 2013. 3296–3303

25. 25

Zhang Z D, Liang X, Ganesh A, et al. TILT: transform invariant low-rank textures. In: Proceedings of the Asian Conference on Computer Vision (ACCV), Queenstown, 2011. 314–328

26. 26

Cai J F, Candès E J, Shen Z W. A singular value thresholding algorithm for matrix completion. SIAM J Optim, 2010, 20: 1956–1982

27. 27

Zhang X Q, Wang D, Zhou Z Y, et al. Simultaneous rectification and alignment via robust recovery of low-rank tensors. In: Proceedings of 27th Annual Conference on Neural Information Processing Systems, Lake Tahoe, 2013. 1637–1645

## Author information

Authors

### Corresponding author

Correspondence to Zhi Han.

## Rights and permissions

Reprints and Permissions

Chen, X., Han, Z., Wang, Y. et al. Nonconvex plus quadratic penalized low-rank and sparse decomposition for noisy image alignment. Sci. China Inf. Sci. 59, 052107 (2016). https://doi.org/10.1007/s11432-015-5419-2

• Accepted:

• Published:

### Keywords

• low-rank decomposition
• nonconvex relaxation