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.)
A Sequential Factorization Method for Recovering Shape and Motion From Image Streams
IEEE Transactions on Pattern Analysis and Machine Intelligence
Heteroscedastic Regression in Computer Vision: Problems with Bilinear Constraint
International Journal of Computer Vision - Special issue on a special section on visual surveillance
Multiview Constraints on Homographies
IEEE Transactions on Pattern Analysis and Machine Intelligence
Factorization with Uncertainty
International Journal of Computer Vision
IEEE Transactions on Pattern Analysis and Machine Intelligence
Lambertian Reflectance and Linear Subspaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
The Rank 4 Constraint in Multiple (=3) View Geometry
ECCV '96 Proceedings of the 4th European Conference on Computer Vision-Volume II - Volume II
A Paraperspective Factorization Method for Shape and Motion Recovery
ECCV '94 Proceedings of the Third European Conference-Volume II on Computer Vision - Volume II
Incremental Singular Value Decomposition of Uncertain Data with Missing Values
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part I
Multi-View Subspace Constraints on Homographies
ICCV '99 Proceedings of the International Conference on Computer Vision-Volume 2 - Volume 2
Multiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision
Rank 1 Weighted Factorization for 3D Structure Recovery: Algorithms and Performance Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
An Analysis of Linear Subspace Approaches for Computer Vision and Pattern Recognition
International Journal of Computer Vision
Journal of Mathematical Imaging and Vision
Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique
Journal of Mathematical Imaging and Vision
The geometry of weighted low-rank approximations
IEEE Transactions on Signal Processing
Three-dimensional modeling from two-dimensional video
IEEE Transactions on Image Processing
Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique
Journal of Mathematical Imaging and Vision
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It is well known that one can collect the coefficients of five (or more) homographies between two views into a large, rank deficient matrix. In principle, this implies that one can refine the accuracy of the estimates of the homography coefficients by exploiting the rank constraint. However, the standard rank-projection approach is impractical for two different reasons: it requires many homographies to even score a modest gain; and, secondly, correlations between the errors in the coefficients will lead to poor estimates.In this paper we study these problems and provide solutions to each. Firstly, we show that the matrices of the homography coefficients can be recast into two parts, each consistent with ranks of only one. This immediately establishes the prospect of realistically (that is, with as few as only three or four homographies) exploiting the redundancies of the homographies over two views. We also tackle the remaining issue: correlated coefficients. We compare our approach with the "gold standard"; that is, non-linear bundle adjustment (initialized from the ground truth estimate--the ideal initialization). The results confirm our theory and show one can implement rank-constrained projection and come close to the gold standard in effectiveness. Indeed, our algorithm (by itself), or our algorithm further refined by a bundle adjustment stage; may be a practical algorithm: providing generally better results than the "standard" DLT (direct linear transformation) algorithm, and even better than the bundle adjustment result with the DLT result as the starting point. Our unoptimized version has roughly the same cost as bundle adjustment and yet can generally produce close to the "gold standard" estimate (as illustrated by comparison with bundle adjustment initialized from the ground truth).Independent of the merits or otherwise of our algorithm, we have illuminated why the naive approach of direct rank-projection is relatively doomed to failure. Moreover, in revealing that there are further rank constraints, not previously known; we have added to the understanding of these issues, and this may pave the way for further improvements.