Minimal Representations for Uncertainty and Estimation in Projective Spaces

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.