Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Exploiting quaternions to support expressive interactive character motion

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2003.Includes bibliographical references (p. 261-266).A real-time motion engine for interactive synthetic characters, either virtual or physical, needs to allow expressivity and interactivity of motion in order to maintain the illusion of life. Canned animation examples from an animator or motion capture device are expressive, but not very interactive, often leading to repetition. Conversely, numerical procedural techniques such as Inverse Kinematics (IK) tend to be very interactive, but often appear "robotic" and require parameter tweaking by hand. We argue for the use of hybrid example-based learning techniques to incorporate expert knowledge of character motion in the form of animations into an interactive procedural engine. Example-based techniques require appropriate distance metrics, statistical analysis and synthesis primitives, along with the ability to blend examples; furthermore, many machine learning techniques are sensitive to the choice of representation. We show that a quaternion representation of the orientation of a joint affords us computational efficiency along with mathematical robustness, such as avoiding gimbal lock in the Euler angle representation. We show how to use quaternions and their exponential mappings to create distance metrics on character poses, perform simple statistical analysis of joint motion limits and blend multiple poses together. We demonstrate these joint primitives on three techniques which we consider useful for combining animation knowledge with procedural algorithms: 1) pose blending, 2) joint motion statistics and 3) expressive IK. We discuss several projects designed using these primitives and offer insights for programmers building real-time motion engines for expressive interactive characters.by Michael Patrick Johnson.Ph.D

Abstract Fast Construction of Accurate Quaternion Splines

In 1992, Barr et al. proposed a method for interpolating orientations with unit quaternion curves by minimizing covariant acceleration. This paper presents a simple improved method which uses cubic basis functions to achieve a speedup of up to three orders of magnitude. A new criterion for automatic refinement based on the Euler-Lagrange error functional is also introduced