Pose Estimation of 3D Free-Form Contours

In this article we discuss the 2D-3D pose estimation problem of 3D free-form contours. 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 rotationRand translation T ) of the 3D object to the reference camera system. The fusion of modeling free-form contours within the pose estimation problem is achieved by using the conformal geometric algebra. The conformal geometric algebra is a geometric algebra which models entities as stereographically 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. To model free-form contours in the conformal framework we use twists to model cycloidal curves as twist-depending functions and interpret n-times nested twist generated curves as functions generated by 3D 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 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.