This paper presents a method for coarse alignment of point clouds by introducing a new descriptor based on the local curvature. The method is developed for model fitting a CAD model for use in robotic assembly. The method is based on selecting keypoints depending on shape factors calculated from the local covariance matrix of the surface. A descriptor is then calculated for each keypoint by fitting two spheres that describe the curvature of the surface. The spheres are calculated using conformal geometric algebra, which gives a convenient and efficient description of the geometry. The keypoint descriptors for the model and the observed point cloud are then compared to estimate the corresponding keypoints, which are used to calculate the displacement. The method is tested in several experiments. One experiment is for robotic assembly, where objects are placed on a table and their position and orientation is estimated using a 3D CAD model.

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Abstract. This paper presents an approach for detecting primitive geometric objects in point clouds captured from 3D cameras. Primitive objects are objects that are well defined with parameters and mathematical relations, such as lines, spheres and ellipsoids. RANSAC, a robust parameter estimator that classifies and neglects outliers, is used for object detection. The primitives considered are modeled, filtered and fitted using the conformal model of geometric algebra. Methods for detecting planes, spheres and cylinders are suggested. Least squares fitting of spheres and planes to point data are done analytically with conformal geometric algebra, while a cylinder is fitted by defining a nonlinear cost function which is optimized using a nonlinear least squares solver. Furthermore, the suggested object detection scheme is combined with an octree sampling strategy that results in fast detection of multiple primitive objects in point clouds.

Abstract. In this work, we present a novel approach to nonlinear optimization of multivectors in the Euclidean and conformal model of geometric algebra by introducing automatic differentiation. This is used to compute gradients and Jacobian matrices of multivector valued functions for use in nonlinear optimization where the emphasis is on the estimation of rigid body motions.

Abstract. In this paper we present a novel method for nonlinear rigid body motion estimation from noisy data using heterogeneous sets of objects of the conformal model in geometric algebra. The rigid body motions are represented by motors. We employ state-of-the-art nonlinear optimization tools and compute gradients and Jacobian matrices using forward-mode automatic differentiation based on dual numbers. The use of automatic differentiation enables us to employ a wide range of cost functions in the estimation process. This includes cost functions for motor estimation using points, lines and planes. Moreover, we explain how these cost functions make it possible to use other geometric objects in the conformal model in the motor estimation process, e.g., spheres, circles and tangent vectors. Experimental results show that we are able to successfully estimate rigid body motions from synthetic datasets of heterogeneous sets of conformal objects including a combination of points, lines and planes.