Automatic Differentiation and Optimization of Multivectors: Estimating Motors in Conformal Geometric Algebra

Abstract

This thesis is on the estimation of rigid body motions from observations of points, lines, and planes in conformal geometric algebra using nonlinear least-squares optimization. Distance measures based on the ratio and difference of points, lines, and planes are formulated and analyzed. Then, the properties of these distance measures, such as translation invariance and scale dependence are analyzed and discussed. A novel approach to motor estimation using the retraction-based framework of Riemannian optimization is presented. This approach is based on computing the update at each iteration in the tangent space of the motor manifold at the current iterate, and then mapping the solution to the motor manifold using a retraction map. Two retraction maps are presented. The first retraction is based on the exponential map, while the other is based on orthogonal projection onto the manifold. Cost functions and Jacobian matrices based on the distance measures for points, lines, and planes are presented, and it is shown how Jacobian matrices can be derived and computed efficiently. This includes a new approach for computing gradients and Jacobian matrices of multivector valued functions in Euclidean and conformal geometric algebra. The novelty of this approach is the use of automatic differentiation in conformal geometric algebra, which computes derivative values up to machine precision. Implementation details and experimental results of automatic differentiation of multivectors and motor estimation are presented. Performance evaluations of combinations of geometric algebra and automatic differentiation libraries show that there are significant performance differences, and recommendations on how to select the best solutions are made. Experimental results from motor estimation show that the proposed method has similar performance as state-of-the-art methods when estimating motors from observations of points, lines, and planes. The problem related to distance measures that are not translation invariant or scale invariant is discussed, and it is shown that the results are useful from an engineering standpoint although translation invariance is certainly a drawback of the distance measures that are used. Finally, the proposed method can be used for a wide range of optimization problems in conformal geometric algebra, including different distance measures, and data from different geometric objects.