Parameter estimation and hypothesis testing in linear models
Parameter estimation and hypothesis testing in linear models
Model acquisition using stochastic projective geometry
Model acquisition using stochastic projective geometry
Statistical Optimization for Geometric Computation: Theory and Practice
Statistical Optimization for Geometric Computation: Theory and Practice
Tracking People with Twists and Exponential Maps
CVPR '98 Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
Nonlinear Estimation of the Fundamental Matrix with Minimal Parameters
IEEE Transactions on Pattern Analysis and Machine Intelligence
Uncertain Projective Geometry: Statistical Reasoning For Polyhedral Object Reconstruction (Lecture Notes in Computer Science)
How to Put Probabilities on Homographies
IEEE Transactions on Pattern Analysis and Machine Intelligence
Structure-from-motion using lines: representation, triangulation, and bundle adjustment
Computer Vision and Image Understanding
Conic fitting using the geometric distance
ACCV'07 Proceedings of the 8th Asian conference on Computer vision - Volume Part II
Efficient video mosaicking by multiple loop closing
PIA'11 Proceedings of the 2011 ISPRS conference on Photogrammetric image analysis
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Estimation using homogeneous entities has to cope with obstacles such as singularities of covariance matrices and redundant parametrizations which do not allow an immediate definition of maximum likelihood estimation and lead to estimation problems with more parameters than necessary. The paper proposes a representation of the uncertainty of all types of geometric entities and estimation procedures for geometric entities and transformations which (1) only require the minimum number of parameters, (2) are free of singularities, (3) allow for a consistent update within an iterative procedure, (4) enable to exploit the simplicity of homogeneous coordinates to represent geometric constraints and (5) allow to handle geometric entities which are at infinity or at least very far, avoiding the usage of concepts like the inverse depth. Such representations are already available for transformations such as rotations, motions (Rosenhahn 2002), homographies (Begelfor 2005), or the projective correlation with fundamental matrix (Bartoli 2004) all being elements of some Lie group. The uncertainty is represented in the tangent space of the manifold, namely the corresponding Lie algebra. However, to our knowledge no such representations are developed for the basic geometric entities such as points, lines and planes, as in addition to use the tangent space of the manifolds we need transformation of the entities such that they stay on their specific manifold during the estimation process. We develop the concept, discuss its usefulness for bundle adjustment and demonstrate (a) its superiority compared to more simple methods for vanishing point estimation, (b) its rigour when estimating 3D lines from 3D points and (c) its applicability for determining 3D lines from observed image line segments in a multi view setup.