Pose Estimation 3D Free-form Contours

In this report 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 rotation $R$ and 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 stereographic projected entities in an 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 cycloidal 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 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

Monocular Pose Estimation Based on Global and Local Features

The presented thesis work deals with several mathematical and practical aspects of the monocular pose estimation problem. Pose estimation means to estimate the position and orientation of a model object with respect to a camera used as a sensor element. Three main aspects of the pose estimation problem are considered. These are the model representations, correspondence search and pose computation. Free-form contours and surfaces are considered for the approaches presented in this work. The pose estimation problem and the global representation of free-form contours and surfaces are defined in the mathematical framework of the conformal geometric algebra (CGA), which allows a compact and linear modeling of the monocular pose estimation scenario. Additionally, a new local representation of these entities is presented which is also defined in CGA. Furthermore, it allows the extraction of local feature information of these models in 3D space and in the image plane. This local information is combined with the global contour information obtained from the global representations in order to improve the pose estimation algorithms. The main contribution of this work is the introduction of new variants of the iterative closest point (ICP) algorithm based on the combination of local and global features. Sets of compatible model and image features are obtained from the proposed local model representation of free-form contours. This allows to translate the correspondence search problem onto the image plane and to use the feature information to develop new correspondence search criteria. The structural ICP algorithm is defined as a variant of the classical ICP algorithm with additional model and image structural constraints. Initially, this new variant is applied to planar 3D free-form contours. Then, the feature extraction process is adapted to the case of free-form surfaces. This allows to define the correlation ICP algorithm for free-form surfaces. In this case, the minimal Euclidean distance criterion is replaced by a feature correlation measure. The addition of structural information in the search process results in better conditioned correspondences and therefore in a better computed pose. Furthermore, global information (position and orientation) is used in combination with the correlation ICP to simplify and improve the pre-alignment approaches for the monocular pose estimation. Finally, all the presented approaches are combined to handle the pose estimation of surfaces when partial occlusions are present in the image. Experiments made on synthetic and real data are presented to demonstrate the robustness and behavior of the new ICP variants in comparison with standard approaches

The Twist Representation of Shape

We give a contribution to the representation problem of free-form curves and surfaces. Our proposal is an operational or kinematic approach based on the Lie group SE(3). While in Euclidean space the modelling of shape as orbit of a point under the action of SE(3) is limited, we are embedding our problem into the conformal geometric algebra R_4,1 of the Euclidean space R^3. This embedding results in a number of advantages which makes the proposed method a universal and flexible one with respect to applications. Especially advantagous is the equivalence of the proposed shape model to that of the Fourier representations

New directions in Nielsen-Reidemeister theory

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.Comment: 50 pages, surve

Pose Estimation of Free-form Objects: Theory and Experiments

In this report we present geometric foundations and an algorithmic approach to deal with the 2D-3D pose estimation problem for free-form surface models. This work is an extension to earlier studies presented in [29]. The discussion of 1D contour models in [29] is extended to 2D free-form surface models. We use a parametric representation of surfaces and apply Fourier transformations to gain low-pass descriptions of objects. We present an algorithm for pose estimation, which uses the silhouette of the object as pictorial information and recovers the 3D pose of the object even for changing aspects of the object during image sequences. We further present extensions to couple surface and contour information on objects and show the potential of our chosen approach for complex objects and scenes

Colour based rigid body tracking using three-dimensional graphics models

This paper introduces the first stage of a new model-based approach to three-dimensional (3D) human movement tracking. A ‘generate-and-test’ matching procedure was adopted by matching rendered images of a 3D computer graphics model of the human body to target images of rigid body motion. The set of pixels to be compared were just those corresponding to the model of the body in the rendered images. The matching criterion to optimise model position and orientation was based on the minimisation of the RGB colour difference between generated model images and associated target images. The method was able to track synthetic image sequences of a half twisting somersault accurately with root mean square errors of less than 5 mm and 0.3° for position and orientation estimates respectively. The suitability of the proposed approach for rigid body motion tracking was supported by additional tracking experiments on video image sequences of ‘wooden cross’ trajectories. Comparisons of tracked estimates against manual digitising estimates returned relatively small rms difference values on both side somersault and twisting somersault movements. The proposed approach has the potential to track video images of a human torso using a rigid body model and hence to track articulated movements by successively adding segments to the model in a hierarchical manner