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

Noise smoothing for VR equipment in quaternions

Smooth motion generation is an important issue in the computer animation and virtual reality (VR) area. In general, the motion of a rigid body consists of translation and orientation. The former is described by a space curve in 3-dimensional Euclidean space, while the latter is represented by a curve in the unit quaternion space. Although there are well-known techniques for smoothing the translation curve in the Euclidean space, few results have been reported for smoothing motion as a whole. This paper improves the previous study and provides a more robust algorithm, which seeks to minimize the weighted sum of the strain-energy and the sum of the squared errors.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45883/1/10756_2004_Article_206666.pd

Non-uniform quaternion spline interpolation in vehicle kinematics

SGS-2022-00

AUTOMATIC FAIRING OF TWO-PARAMETER RATIONAL B-SPLINE MOTION

This paper deals with the problem of automatic fairing of two-parameter B

Uniform accelerated motions

An Affine matrix which maps an initial and final pose can be computed by solving a system of linear equations. Then there exists an interesting problem of finding a time varying affinity which maps the given set of poses and if it exists is always unique and should hold some interesting properites such as affine-invariant, reversible, preserve rigidity, similarities and volume. The Steady Affine Motions and Morphs (SAM) introduced by Jarek Rossignac and Alvar Vinacua solved this problem of time varying affinity and defines the quality of such affinity by the term steadiness. Until SAM, no mathematical definition of steadiness was available and intuitively SAM defined a steady animation to be continuous, to vary dimensions and angles monotonically and rather uniformly, and to move points along pleasing arcs that are free of unnecessary kinks or loops. The authors defined the term ”Steady” as a constant velocity motion in the local moving frame. SAM creates pleasing in-betweening motions that interpolates between an initial and final pose, B and C and the derived equation of beauty was At B with A = C B·-1. SAM is affine-invariant, reversible, preserves isometries (i.e., rigidity), similarities and volume. Previously proposed approaches came up with a solution for the time varying affinity problem, but there was no proper definition of how beautiful or how good the motion was. With the advent of SAM, the beauty of a motion can now be measured by the unsteadiness and Steady Affine motions and morphs is the one solution which comes to have a value of zero for the unsteadiness term. Uniform Accelerated Motions (UAM) carries forward the above definition of steadiness into a constant acceleration motion in the local moving frame. The time varying affinity At is computed using closed form expressions and some of its interesting properties are studied. The constant acceleration motion (in local frame) in UAM is then compared with the constant velocity motion (in local frame) of SAM and the resuls are discussed

Smooth Key-framing using the Image Plane

This paper demonstrates the use of image-space constraints for key frame interpolation. Interpolating in image-space results in sequences with predictable and controlable image trajectories and projected size for selected objects, particularly in cases where the desired center of rotation is not fixed or when the key frames contain perspective distortion changes. Additionally, we provide the user with direct image-space control over {\em how} the key frames are interpolated by allowing them to directly edit the object\u27s projected size and trajectory. Image-space key frame interpolation requires solving the inverse camera problem over a sequence of point constraints. This is a variation of the standard camera pose problem, with the additional constraint that the sequence be visually smooth. We use image-space camera interpolation to globally control the projection, and traditional camera interpolation locally to avoid smoothness problems. We compare and contrast three different constraint-solving systems in terms of accuracy, speed, and stability. The first approach was originally developed to solve this problem [Gleicher and Witken 1992]; we extend it to include internal camera parameter changes. The second approach uses a standard single-frame solver. The third approach is based on a novel camera formulation and we show that it is particularly suited to solving this problem