Affine interpolation in a lie group framework

ACM Transactions on Graphics (TOG), 2011

We propose to measure the beauty of an affine motion by the inverse of its steadiness, which we define as its Average Relative Acceleration (ARA). Steady affine motions, for which ARA= 0, include translations, rotations, screws, and the golden spiral. To facilitate the design of beautiful in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C, we propose the Steady Affine Morph

IEEE Computer Graphics and Applications, 1997

M ost conventional media depend on engaging and appealing characters. Empty spaces and buildings would not fare well as television or movie programming, yet virtual reality usually offers up such spaces. The problem lies in the difficulty of creating computer-generated characters that display real-time, engaging interaction and realistic motion.

1992

Affine transfonnations of 2-D frame buffer images are a common computer graphics operation. Such transfonnations take a rectangular raster of image memory, perfonn some affine transformation (e.g. scale, shift, shear, rotate) upon it, and write the result into some other rectangular raster of image memory. If the source and destination share the same memory, the operation is termed in-place. Previous in-place affine transformation algorithms on an m by n region required O(max(m,n)) space for internal buffers. The algorithm presented here requires 0(1) (constant) space: this allows in-place affine transfonnations to be perfonned on large images on processors with small memory.

Computer-Aided Design, 1998

This paper investigates methods for computing a smooth motion that interpolates a given set of positions and orientations. The position and orientation of a rigid body can be described with an element of the group of spatial rigid body displacements, X(3). To find a smooth motion that interpolates a given set of positions and orientations is therefore the same as finding an interpolating curve between the corresponding elements of SE(3). To make the interpolation on SE(3) independent of the representation of the group, we use the coordinate-free framework of differential geometry. It is necessary to choose inertial and body-fixed reference frames to describe the position and orientation of the rigid body. We first show that trajectories that are independent of the choice of these frames can be obtained by using the exponential map on S&3). However, these trajectories may exhibit rapid changes in the velocity or higher derivatives. The second contribution of the paper is a method for finding the maximally smooth interpolating curve. By adapting the techniques of the calculus of variations to S&3), necessary conditions are derived for motions that are equivalent to cubic splines in the Euclidean space. These necessary conditions result in a boundary value problem with interior-point constraints. A simple and efficient numerical method for finding a solution is then described. Finally, we discuss the dependence of the computed trajectories on the metric on S.!?(3) and show that independence of the trajectories from the choice of the reference frames can be achieved by using a suitable metric.

International Journal of Digital …, 2010

This paper presents a new and compact 3D representation for nonrigid objects using the motion vectors between two consecutive frames. Our method relies on an Octree to recursively partition the object into smaller parts. Each part is then assigned a small number of motion parameters that can accurately represent that portion of the object. Finally, an adaptive thresholding, a singular value decomposition for dealing with singularities, and a quantization and arithmetic coding further enhance our proposed method by increasing the compression while maintaining very good signal-noise ratio. Compared to other methods that use tri-linear interpolation, Principle Component Analysis (PCA), or non-rigid partitioning (e.g., FAMC) our algorithm combines the best attributes in most of them. For example, it can be carried out on a frame-to-frame basis, rather than over long sequences, but it is also much easier to compute. In fact, we demonstrate a computation complexity of Θ(n 2) for our method, while some of these methods can reach complexities of O(n 3) and worse. Finally, as the result section demonstrates, the proposed improvements do not sacrifice performance since our method has a better or at least very similar performance in terms of compression ratio and PSNR.

Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2013

In this paper, we introduce a novel framework for computing a path of diffeomorphisms between a pair of input diffeomorphisms. Direct computation of a geodesic path on the space of diffeomorphisms Diff(Ω) is difficult, and it can be attributed mainly to the infinite dimensionality of Diff(Ω). Our proposed framework, to some degree, bypasses this difficulty using the quotient map of Diff(Ω) to the quotient space Diff(M)/Diff(M) μ obtained by quotienting out the subgroup of volume-preserving diffeomorphisms Diff(M) μ . This quotient space was recently identified as the unit sphere in a Hilbert space in mathematics literature, a space with well-known geometric properties. Our framework leverages this recent result by computing the diffeomorphic path in two stages. First, we project the given diffeomorphism pair onto this sphere and then compute the geodesic path between these projected points. Second, we lift the geodesic on the sphere back to the space of diffeomerphisms, by solving a...

Parameterizing three degree-of-freedom (DOF) rotations is difficult to do well. Many graphics applications demand that we be able to compute and differentiate positions and orientations of articulated figures with respect to their rotational (and other) parameters, as well as integrate differential equations, optimize functions of DOFs, and interpolate orientations. Widely used parameterizations such as Euler angles and quaternions are well suited to only a few of these operations.

Multibody System Dynamics, 2014

In Cosserat solids such as shear deformable beams and shells, the displacement and rotation fields are independent. The finite element implementation of these structural components within the framework of flexible multibody dynamics requires the interpolation of rotation and motion fields. In general, the interpolation process does not preserve fundamental properties of the interpolated field. For instance, interpolation of an orthogonal rotation tensor does not yield an orthogonal tensor, and furthermore, does not preserve the tensorial nature of the rotation field. Consequently, many researchers have been reluctant to apply the classical interpolation tools used in finite element procedures to interpolate these fields. This paper presents a systematic study of interpolation algorithms for rotation and motion. All the algorithms presented here preserve the fundamental properties of the interpolated rotation and motion fields, and furthermore, preserve their tensorial nature. It is also shown that the interpolation of rotation and motion is as accurate as the interpolation of displacement, a widely accepted tool in the finite element method. The algorithms presented in this paper provide interpolation tools for rotation and motion that are accurate, easy to implement, and physically meaningful.

We have not found in the computer graphics literature what we believe may be the simplest algorithm for implementing a 3D rotation with a mouse. We present this algorithm, and observe that it leads to new mathematical realizations of the 3D rotation group and its double-cover, the unit quaternions. We use this to obtain an algorithm for interpolating rotations of a 3D scene that is more efficient than the standard quaternion based methods, and also to derive the quaternion composition formula from the geometry of 3D rotations.

Computer graphics are widely improved in many kind of output according to the advancement of devices and technology. 2D and 3D graphic are commonly used to display the output in purpose of evaluation, enhancement and improvement in many transformations. In order to get deeply understanding of 2D and 3D transformation, the fundamental of transformations is importantly to highlight. This work discussed about basic of 2D and 3D transformation which are translations, rotation and scaling. Moreover, the composition of those transformations are also mentioned.