Shape and motion from image streams under orthography: a factorization method
International Journal of Computer Vision
Affine analysis of image sequences
Affine analysis of image sequences
A Paraperspective Factorization Method for Shape and Motion Recovery
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Factorization Based Algorithm for Multi-Image Projective Structure and Motion
ECCV '96 Proceedings of the 4th European Conference on Computer Vision-Volume II - Volume II
A Factorization Method for Affine Structure from Line Correspondences
CVPR '96 Proceedings of the 1996 Conference on Computer Vision and Pattern Recognition (CVPR '96)
A Unified Factorization Algorithm for Points, Line Segments and Planes with Uncertainty Models
ICCV '98 Proceedings of the Sixth International Conference on Computer Vision
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Incremental Singular Value Decomposition of Uncertain Data with Missing Values
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Plane+Parallax, Tensors and Factorization
ECCV '00 Proceedings of the 6th European Conference on Computer Vision-Part I
Machine Vision and Applications
Computer Vision and Image Understanding
Geometrical fitting of missing data for shape from motion under noise distribution
ICNC'06 Proceedings of the Second international conference on Advances in Natural Computation - Volume Part II
Uncertainty analysis using geometrical property between 2d-to-3d under affine projection
FSKD'06 Proceedings of the Third international conference on Fuzzy Systems and Knowledge Discovery
Incorporating non-motion cues into 3d motion segmentation
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part III
Optimal multi-frame correspondence with assignment tensors
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part III
A novel recovery algorithm of incomplete observation matrix for converting 2-d video to 3-d content
IWICPAS'06 Proceedings of the 2006 Advances in Machine Vision, Image Processing, and Pattern Analysis international conference on Intelligent Computing in Pattern Analysis/Synthesis
SLSFS'05 Proceedings of the 2005 international conference on Subspace, Latent Structure and Feature Selection
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Factorization using Singular Value Decomposition (SVD) is often used for recovering 3D shape and motion from feature correspondences across multiple views. SVD is powerful at finding the global solution to the associated least-square-error minimization problem. However, this is the correct error to minimize only when the x and y positional errors in the features are uncorrelated and identically distributed. But this is rarely the case in real data. Uncertainty in feature position depends on the underlying spatial intensity structure in the image, which has strong directionality to it. Hence, the proper measure to minimize is covariance-weighted squared-error (or the Mahalanobis distance). In this paper, we describe a new approach to covariance-weighted factorization, which can factor noisy feature correspondences with high degree of directional uncertainty into structure and motion. Our approach is based on transforming the raw-data into a covariance-weighted data space, where the components of noise in the different directions are uncorrelated and identically distributed. Applying SVD to the transformed data now minimizes a meaningful objective function. We empirically show that our new algorithm gives good results for varying degrees of directional uncertainty. In particular, we show that unlike other SVD-based factorization algorithms, our method does not degrade with increase in directionality of uncertainty, even in the extreme when only normal-flow data is available. It thus provides a unified approach for treating corner-like points together with points along linear structures in the image.