Related papers: The Gibbs Representation of 3D Rotations
It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the…
Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection…
We resort to the concept of turns to provide a geometrical representation of the action of any lossless multilayer, which can be considered as the analogous in the unit disk to the sliding vectors in Euclidean geometry. This construction…
We propose a methodology to design Wigner representations in phase spaces with nontrivial topology having evolution equations with desired mathematical properties. As an illustration, two representations of molecular rotations are developed…
We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a…
A ``Wick rotation'' is applied to the noncommutative sphere to produce a noncommutative version of the hyperboloids. A harmonic basis of the associated algebra is given. It is noted that, for the one sheeted hyperboloid, the vector space…
The attitude of a rigid-body in the three dimensional space has a unique and global definition on the Special Orthogonal Group SO (3). This paper gives an overview of the rotation matrix, attitude kinematics and parameterization. The four…
This short note introduces a geometric representation for binary (or ternary) sequences. The proposed representation is linked to multivariate data plotting according to the radar chart. As an illustrative example, the binary Hamming…
3D data is a valuable asset the computer vision filed as it provides rich information about the full geometry of sensed objects and scenes. Recently, with the availability of both large 3D datasets and computational power, it is today…
We demonstrate an object tracking method for 3D images with fixed computational cost and state-of-the-art performance. Previous methods predicted transformation parameters from convolutional layers. We instead propose an architecture that…
Trajectory segmentation refers to dividing a trajectory into meaningful consecutive sub-trajectories. This paper focuses on trajectory segmentation for 3D rigid-body motions. Most segmentation approaches in the literature represent the…
A sequential application of the Grover algorithm to solve the iterated search problem has been improved by Ozhigov by parallelizing the application of the oracle. In this work a representation of the parallel Grover as dynamic system of…
3D Gaussian Splatting (3DGS) has recently emerged as a pioneering approach in explicit scene rendering and computer graphics. Unlike traditional neural radiance field (NeRF) methods, which typically rely on implicit, coordinate-based models…
It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex,…
Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix…
We propose a novel quaternion product unit (QPU) to represent data on 3D rotation groups. The QPU leverages quaternion algebra and the law of 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted…
This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC),…
Three-Dimensional Gaussian Splatting (3DGS) has shown substantial promise in the field of computer vision, but remains unexplored in the field of magnetic resonance imaging (MRI). This study explores its potential for the reconstruction of…
Recent 4D Gaussian Splatting (4DGS) methods achieve impressive dynamic scene reconstruction but often rely on piecewise linear velocity approximations and short temporal windows. This disjointed modeling leads to severe temporal…
Omnidirectional images and spherical representations of $3D$ shapes cannot be processed with conventional 2D convolutional neural networks (CNNs) as the unwrapping leads to large distortion. Using fast implementations of spherical and…