Shape and motion from image streams under orthography: a factorization method
International Journal of Computer Vision
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
A Paraperspective Factorization Method for Shape and Motion Recovery
IEEE Transactions on Pattern Analysis and Machine Intelligence
Matrix computations (3rd ed.)
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Affine Structure and Motion from Points, Lines and Conics
International Journal of Computer Vision
Linear fitting with missing data for structure-from-motion
Computer Vision and Image Understanding
Factorization with Uncertainty
International Journal of Computer Vision
Principal Component Analysis with Missing Data and Its Application to Polyhedral Object Modeling
IEEE Transactions on Pattern Analysis and Machine Intelligence
Structure and Motion from Points, Lines and Conics with Affine Cameras
ECCV '98 Proceedings of the 5th European Conference on Computer Vision-Volume I - Volume I
Incremental Singular Value Decomposition of Uncertain Data with Missing Values
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Structure from Many Perspective Images with Occlusions
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part II
A Framework for Robust Subspace Learning
International Journal of Computer Vision - Special Issue on Computational Vision at Brown University
CVPR '97 Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition (CVPR '97)
Reconstruction from Affine Cameras Using Closure Constraints
ICPR '98 Proceedings of the 14th International Conference on Pattern Recognition-Volume 1 - Volume 1
Multiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision
A Unified Factorization Algorithm for Points, Line Segments and Planes with Uncertainty Models
ICCV '98 Proceedings of the Sixth International Conference on Computer Vision
Rank 1 Weighted Factorization for 3D Structure Recovery: Algorithms and Performance Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
3D Reconstruction by Fitting Low-Rank Matrices with Missing Data
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 1 - Volume 01
Damped Newton Algorithms for Matrix Factorization with Missing Data
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
On the Wiberg Algorithm for Matrix Factorization in the Presence of Missing Components
International Journal of Computer Vision
Journal of Mathematical Imaging and Vision
Motion segmentation with missing data using power factorization and GPCA
CVPR'04 Proceedings of the 2004 IEEE computer society conference on Computer vision and pattern recognition
The geometry of weighted low-rank approximations
IEEE Transactions on Signal Processing
Heteroscedastic Low-Rank Matrix Approximation by the Wiberg Algorithm
IEEE Transactions on Signal Processing
Optimal reduced-rank estimation and filtering
IEEE Transactions on Signal Processing
Recovering the missing components in a large noisy low-rank matrix: application to SFM
IEEE Transactions on Pattern Analysis and Machine Intelligence
International Journal of Computer Vision
Successively alternate least square for low-rank matrix factorization with bounded missing data
Computer Vision and Image Understanding
ACM Transactions on Graphics (TOG)
5D motion subspaces for planar motions
ECCV'10 Proceedings of the 11th European conference on computer vision conference on Computer vision: Part III
Hessian Matrix vs. Gauss-Newton Hessian Matrix
SIAM Journal on Numerical Analysis
Low-Rank Matrix Approximation with Weights or Missing Data Is NP-Hard
SIAM Journal on Matrix Analysis and Applications
Low-rank matrix decomposition in L1-norm by dynamic systems
Image and Vision Computing
Multilinear Factorizations for Multi-Camera Rigid Structure from Motion Problems
International Journal of Computer Vision
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Low-rank matrix approximation has applications in many fields, such as 3D reconstruction from an image sequence and 2D filter design. In this paper, one issue with low-rank matrix approximation is re-investigated: the missing data problem. Much effort was devoted to this problem, and the Wiberg algorithm or the damped Newton algorithm were recommended in previous studies. However, the Wiberg or damped Newton algorithms do not suit for large (especially "long") matrices, because one needs to solve a large linear system in every iteration. In this paper, we revitalize the usage of the Levenberg-Marquardt algorithm for solving the missing data problem, by utilizing the property that low-rank approximation is a minimization problem on subspaces. In two proposed implementations of the Levenberg-Marquardt algorithm, one only needs to solve a much smaller linear system in every iteration, especially for "long" matrices. Simulations and experiments on real data show the superiority of the proposed algorithms. Though the proposed algorithms achieve a high success rate in estimating the optimal solution by random initialization, as illustrated by real examples; it still remains an open issue how to properly do the initialization in a severe situation (that is, a large amount of data is missing and with high-level noise).