Optimal Parameter Estimation with Homogeneous Entities and Arbitrary Constraints

Abstract

Well known estimation techniques in computational geometry usually deal only with single geometric entities as unknown parameters and do not account for constrained observations within the estimation.

The estimation model proposed in this paper is much more general, as it can handle multiple homogeneous vectors as well as multiple constraints. Furthermore, it allows the consistent handling of arbitrary covariance matrices for the observed and the estimated entities. The major novelty is the proper handling of singular observation covariance matrices made possible by additional constraints within the estimation. These properties are of special interest for instance in the calculus of algebraic projective geometry, where singular covariance matrices arise naturally from the non-minimal parameterizations of the entities.

The validity of the proposed adjustment model will be demonstrated by the estimation of a fundamental matrix from synthetic data and compared to heteroscedastic regression [1], which is considered as state-of-the-art estimator for this task. As the latter is unable to simultaneously estimate multiple entities, we will also demonstrate the usefulness and the feasibility of our approach by the constrained estimation of three vanishing points from observed uncertain image line segments.