The quaternion-based three-dimensional beam theory

This paper presents the equations for the implementation of rotational quaternions in the geometrically exact three-dimensional beam theory. A new finite-element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam. The formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form. A strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented. Several numerical examples demonstrate the validity, performance and accuracy of the proposed approach. (C) 2009 Elsevier B.V. All rights reserved

Low-latency compression of mocap data using learned spatial decorrelation transform

Due to the growing needs of human motion capture (mocap) in movie, video games, sports, etc., it is highly desired to compress mocap data for efficient storage and transmission. This paper presents two efficient frameworks for compressing human mocap data with low latency. The first framework processes the data in a frame-by-frame manner so that it is ideal for mocap data streaming and time critical applications. The second one is clip-based and provides a flexible tradeoff between latency and compression performance. Since mocap data exhibits some unique spatial characteristics, we propose a very effective transform, namely learned orthogonal transform (LOT), for reducing the spatial redundancy. The LOT problem is formulated as minimizing square error regularized by orthogonality and sparsity and solved via alternating iteration. We also adopt a predictive coding and temporal DCT for temporal decorrelation in the frame- and clip-based frameworks, respectively. Experimental results show that the proposed frameworks can produce higher compression performance at lower computational cost and latency than the state-of-the-art methods.Comment: 15 pages, 9 figure

Surface Networks

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs