Gauge Freedoms and Uncertainty Modeling for 3D Computer Vision

Abstract

Parameter indeterminacies are inherent in 3D computer vision. We show in this thesis that how they are treated can have significant impact on the accuracy of the estimated 3D structure. However, there has not been a general and convenient method available for representing and analyzing the indeterminacies and their effects on accuracy. Consequently, up to the present their effects are usually ignored in uncertainty modeling research. In this work we a develop gauge-based uncertainty representation for 3D estimation that includes indeterminacies. We represent indeterminacies with orbits in the parameter space and model local linearized parameter indeterminacies as gauge freedoms. Combining this formalism with first order perturbation theory, we are able to model uncertainties along with parameter indeterminacies. The key to our work is a geometric interpretation of the parameters and gauge freedoms. We solve the problem of how to compare parameter uncertainties despite indeterminacies and added constraints. This permits us to extend the Cramer-Rao lower bound to problems that include parameter indeterminacies. In 3D computer vision the basic quantities that often cannot be recovered include scale, rotation and translation. We use our method to analyze the local effects of these indeterminacies on the estimated shape, and find all the local gauge freedoms. This enables us to express the uncertainties when additional information is available from measurements that constrain the gauge freedoms. Through analytical and empirical means we gain intuition into the effects of constraining the gauge freedoms, for both general Structure from Motion and stereo shape estimation. We include, in our uncertainty model, measurement errors and feature localization errors. These results along with our theory allow us to find optimal constraints on the gauge freedoms that maximize the accuracy of the part of the object we seek to estimate.