# Robust L/sub 1/ norm factorization in the presence of outliers and missing data by alternative convex programming

@article{Ke2005RobustL1, title={Robust L/sub 1/ norm factorization in the presence of outliers and missing data by alternative convex programming}, author={Q. Ke and T. Kanade}, journal={2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05)}, year={2005}, volume={1}, pages={739-746 vol. 1} }

Matrix factorization has many applications in computer vision. Singular value decomposition (SVD) is the standard algorithm for factorization. When there are outliers and missing data, which often happen in real measurements, SVD is no longer applicable. For robustness iteratively re-weighted least squares (IRLS) is often used for factorization by assigning a weight to each element in the measurements. Because it uses L/sub 2/ norm, good initialization in IRLS is critical for success, but is… Expand

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#### References

SHOWING 1-10 OF 27 REFERENCES

Factorization with Uncertainty

- Computer Science, Mathematics
- International Journal of Computer Vision
- 2004

A new approach to covariance-weighted factorization, which can factor noisy feature correspondences with high degree of directional uncertainty into structure and motion and provides a unified approach for treating corner-like points together with points along linear structures in the image. Expand

Factorization with Uncertainty and Missing Data: Exploiting Temporal Coherence

- Computer Science, Mathematics
- NIPS
- 2003

This paper uses the well known EM algorithm for factor analysis to perform factorization and places a prior on the temporal trajectory of the latent variables (the camera position) and shows that incorporating this prior gives a significant improvement in performance in challenging image sequences. Expand

Robust Factorization

- Mathematics, Computer Science
- IEEE Trans. Pattern Anal. Mach. Intell.
- 2002

A new and computationally efficient algorithm for applying an arbitrary error function in the factorization scheme that enables the use of robust statistical techniques and arbitrary noise models for the individual features. Expand

A Framework for Robust Subspace Learning

- Computer Science
- International Journal of Computer Vision
- 2004

The theory of Robust Subspace Learning (RSL) for linear models within a continuous optimization framework based on robust M-estimation is developed and applies to a variety of linear learning problems in computer vision including eigen-analysis and structure from motion. Expand

Linear Fitting with Missing Data for Structure-from-Motion

- Mathematics, Computer Science
- Comput. Vis. Image Underst.
- 2001

This work proposes a novel method for fitting a low rank matrix to a matrix with missing elements and shows that this method has desirable theoretical properties compared to previously proposed methods, because it can find solutions when there is less data present. Expand

Outlier correction in image sequences for the affine camera

- Mathematics, Computer Science
- Proceedings Ninth IEEE International Conference on Computer Vision
- 2003

This work presents an outlier correction scheme that iteratively updates the elements of the image measurement matrix, with the result that outliers are corrected and retained, giving improved reconstruction and smaller reprojection errors. Expand

Shape and motion from image streams under orthography: a factorization method

- Mathematics, Computer Science
- International Journal of Computer Vision
- 2004

A factorization method is developed that can overcome the difficulty by recovering shape and motion under orthography without computing depth as an intermediate step, and gives accurate results. Expand

Learning the parts of objects by non-negative matrix factorization

- Computer Science, Medicine
- Nature
- 1999

An algorithm for non-negative matrix factorization is demonstrated that is able to learn parts of faces and semantic features of text and is in contrast to other methods that learn holistic, not parts-based, representations. Expand

Multi-frame optical flow estimation using subspace constraints

- Mathematics, Computer Science
- Proceedings of the Seventh IEEE International Conference on Computer Vision
- 1999

This paper develops a method for simultaneous estimation of optical flow across multiple frames, which uses multi-frame subspace constraints to constrain the 2D correspondence estimation process itself, and not for 3D recovery. Expand

On the unification of line processes, outlier rejection, and robust statistics with applications in early vision

- Mathematics, Computer Science
- International Journal of Computer Vision
- 2004

It is shown how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier-process formulation. Expand