Research Seminar - February 05, 2003

Pose Estimation of 3D Free-form Contours in Conformal Geometry

Abstract:

We will discuss the 2D-3D pose estimation problem of 3D free-form contours. Although a well-known and often considered problem, we will consider the pose estimation with a conceptually and algorithmically new approach, which results in an efficient and flexible framework for that key task. From a cognitive point of view it would be preferable to estimate the pose not only from points but from higher order geometric entities in a linear manner. This can only be realized in a classical mathematical framework.

In our scenario we observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation (containing a rotation R and a translation T) of the 3D object to a reference camera system. The fusion of modeling free-form contours within the pose estimation problem is achieved by using the embedding of the problem in the conformal geometric algebra. The conformal geometric algebra is a geometric algebra which models entities as stereographic projected entities in a homogeneous model. This leads to a linear description of kinematics on the one hand and projective geometry on the other hand. In that algebraic framework Faugeras' stratification of space can be easily realized.

To model free-form contours in the conformal framework we use coupled twists to model algebraic curves of any order as twist-depending functions and interpret n-times nested cycloidal curves as functions represented by 3D Fourier descriptors. Twists are generators of a general rotation in space and thus, enable the operational definition of algebraic curves by using their tangential space representation. That means, we interpret curves as orbits of Lie group actions on a point, generated by twists as elements of the corresponding Lie algebra. The transition parts of the twist transformations are related to the coefficients of a Fourier series development of a curve. For discrete curves, i.e. contours, these coefficients are well-known as Fourier descriptors. This means, we use the twist concept to apply a spectral domain representation of 3D contours within the pose estimation problem. We will show that twist representations of objects can be numerically efficient and easily be applied to the pose estimation problem. The pose problem itself is formalized as an implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion. Several experiments visualize the robustness and real-time performance of our algorithms.