A General Method for Errors-in-Variables
Problems in Computer Vision
Bogdan Matei and Peter Meer
Department of Electrical and Computer Engineering
Rutgers University, Piscataway, NJ 08854-8058, USA
The Errors-in-Variables (EIV) model from statistics is often
employed in computer vision though only rarely under this name.
In an EIV model all the measurements are corrupted by noise while
the a priori information is captured with a nonlinear constraint
among the true (unknown)
values of these measurements. To estimate the model parameters and
the uncorrupted data, the constraint can be linearized,
i.e., embedded in a higher dimensional space. We show that
linearization introduces data-dependent (heteroscedastic) noise
and propose an iterative procedure,
the heteroscedastic EIV (HEIV) estimator to obtain consistent
estimates in the most general, multivariate case.
Analytical expressions for the
covariances of the parameter estimates and corrected data points,
a generic method for the enforcement of ancillary constraints
arising from the underlying geometry are also given.
The HEIV estimator minimizes the first order approximation of the
geometric distances between the measurements and the true data
points, and thus can be a substitute for the widely used
Levenberg-Marquardt based direct solution of the original,
nonlinear problem. The HEIV
estimator has however the advantage of a weaker dependence on the
initial solution and a faster convergence. In comparison to
renormalization paradigm (an earlier solution of the same problem)
the HEIV estimator has more solid theoretical foundations which
translate into better numerical behavior. We show
that the HEIV estimator can provide an accurate solution to most
3D vision estimation tasks, and illustrate its performance
through two case studies:
calibration and the estimation of the fundamental matrix.
2000 Computer Vision and Pattern Recognition
Conference , June 2000, Hilton Head Island, SC, vol.II, 18-25.
Return to Research: Estimation under heteroscedasticity