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

Movimientos simétrico lineales esféricos segmentados para interpolación de orientaciones en planificación de trayectorias de herramienta en CNC de 5 Ejes

RESUMEN: Este artículo emplea biarcos cuaterniónicos para interpolar un conjunto de orientaciones con restricciones de velocidad angular. La curva cuaterniónica resultante representa un movimiento simétrico lineal esférico segmentado con continuidad C1 . El propósito de este esfuerzo es poner en uso los movimientos simétrico lineales desde el punto de vista de aproximación e interpolación de movimiento y presentar su potencial aplicación en la simulación de mecanizado por Control Numérico Computarizado (CNC) y planeación de trayectorias de herramienta. Los biarcos cuaterniónicos pueden ser usados para aproximar curvas B-spline cuaterniónicas que representan movimientos esféricos racionales, los cuales tienen aplicaciones en planeación de trayectorias de robots, en CAD/CAM y en gráficas por computador.ABSTRACT: This paper employs quaternion biarcs to interpolate a set of orientations with angular velocity constraints. The resulting quaternion curve represents a piecewise line-symmetric spherical motion with C1 continuity. The purpose of this effort is to put line-symmetric motions into use from the viewpoint of motion approximation and interpolation, and to present their potential applications in Computerized Numerical Control (CNC) machining simulation and tool path planning. Quaternion biarcs may be used to approximate B-spline quaternion curves that represent rational spherical motions that have applications in robot path planning, CAD/CAM and computer graphics

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

Curve and surface framing for scientific visualization and domain dependent navigation

Thesis (Ph.D.) - Indiana University, Computer Science, 1996Curves and surfaces are two of the most fundamental types of objects in computer graphics. Most existing systems use only the 3D positions of the curves and surfaces, and the 3D normal directions of the surfaces, in the visualization process. In this dissertation, we attach moving coordinate frames to curves and surfaces, and explore several applications of these frames in computer graphics and scientific visualization. Curves in space are difficult to perceive and analyze, especially when they are densely clustered, as is typical in computational fluid dynamics and volume deformation applications. Coordinate frames are useful for exposing the similarities and differences between curves. They are also useful for constructing ribbons, tubes and smooth camera orientations along curves. In many 3D systems, users interactively move the camera around the objects with a mouse or other device. But all the camera control is done independently of the properties of the objects being viewed, as if the user is flying freely in space. This type of domain-independent navigation is frequently inappropriate in visualization applications and is sometimes quite difficult for the user to control. Another productive approach is to look at domain-specific constraints and thus to create a new class of navigation strategies. Based on attached frames on surfaces, we can constrain the camera gaze direction to be always parallel (or at a fixed angle) to the surface normal. Then users will get a feeling of driving on the object instead of flying through the space. The user's mental model of the environment being visualized can be greatly enhanced by the use of these constraints in the interactive interface. Many of our research ideas have been implemented in Mesh View, an interactive system for viewing and manipulating geometric objects. It contains a general purpose C++ library for nD geometry and supports a winged-edge based data structure. Dozens of examples of scientifically interesting surfaces have been constructed and included with the system

Motion enriching using humanoide captured motions

Animated humanoid characters are a delight to watch. Nowadays they are extensively used in simulators. In military applications animated characters are used for training soldiers, in medical they are used for studying to detect the problems in the joints of a patient, moreover they can be used for instructing people for an event(such as weather forecasts or giving a lecture in virtual environment). In addition to these environments computer games and 3D animation movies are taking the benefit of animated characters to be more realistic. For all of these mediums motion capture data has a great impact because of its speed and robustness and the ability to capture various motions. Motion capture method can be reused to blend various motion styles. Furthermore we can generate more motions from a single motion data by processing each joint data individually if a motion is cyclic. If the motion is cyclic it is highly probable that each joint is defined by combinations of different signals. On the other hand, irrespective of method selected, creating animation by hand is a time consuming and costly process for people who are working in the art side. For these reasons we can use the databases which are open to everyone such as Computer Graphics Laboratory of Carnegie Mellon University.Creating a new motion from scratch by hand by using some spatial tools (such as 3DS Max, Maya, Natural Motion Endorphin or Blender) or by reusing motion captured data has some difficulties. Irrespective of the motion type selected to be animated (cartoonish, caricaturist or very realistic) human beings are natural experts on any kind of motion. Since we are experienced with other peoples’ motions, and comparing each motion to the others, we can easily judge one individual’s mood from his/her body language. As being a natural master of human motions it is very difficult to convince people by a humanoid character’s animation since the recreated motions can include some unnatural artifacts (such as foot-skating, flickering of a joint)