Rotation Group SO Research Articles

Filtering from observations on Stiefel manifolds

This paper addresses the problem of filtering processes on the rotation group SO(n) when partial observations, i.e. processes on the Stiefel manifold, are only available. In particular, we consider the problem of estimating the angular velocity of an object when only partial observation of its orientation is available. Interpolation schemes are proposed to overcome the discrete nature of observations and a practical numerical solution (particle filtering) is developed for filtering on the Stiefel manifold. The proposed solution to the filtering problem is based on the anti-development signal concept, which allows to overcome the non-additivity nature of the noise present in the observed signal. The solutions proposed are general and can be transposed to many manifolds. However, the presented examples and simulations consider the case of partially observed signals taking values on the special orthogonal group when one or more components are missing. In this particular case, the observed process takes its values in the Stiefel manifold. A comparison of the proposed particle filter with the extended Kalman filter is presented. Results on synthetic and real (sport science) data show the behavior of the proposed algorithm.

A Novel Sampling Theorem on the Rotation Group

We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.

Computational design of twisty joints and puzzles

We present the first computational method that allows ordinary users to create complex twisty joints and puzzles inspired by the Rubik's Cube mechanism. Given a user-supplied 3D model and a small subset of rotation axes, our method automatically adjusts those rotation axes and adds others to construct a "non-blocking" twisty joint in the shape of the 3D model. Our method outputs the shapes of pieces which can be directly 3D printed and assembled into an interlocking puzzle. We develop a group-theoretic approach to representing a wide class of twisty puzzles by establishing a connection between non-blocking twisty joints and the finite subgroups of the rotation group SO(3). The theoretical foundation enables us to build an efficient system for automatically completing the set of rotation axes and fast collision detection between pieces. We also generalize the Rubik's Cube mechanism to a large family of twisty puzzles.

Quaternionic Version of Rotation Groups

<p>Quaternionic version of rotation group SO(3) has been constructed. We construct<br />a quatenionic version of rotation operation that act to a quaternionic version of a<br />space coordinate vector. The computation are done for every rotation about each<br />coordinate axes (x,y, and z). The rotated quaternionic space coordinate vector con-<br />tain some unknown constants which determine the quaternionic rotation operator.<br />By solving for that constants, we get the expression of the quaternionics version<br />of the rotation operator. Finally the generators of th</p>

Orthogonal Matrix Retrieval In Cryo-Electron Microscopy.

In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size (2l + 1) × (2l + 1) for each l = 0,1,2,…. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.

Numerical Determination of the Mean Value of an Elasticity Tensor by Integration over the Rotation Group with Haar Measure

Let A be a 3D symmetric elasticity tensor not necessary isotropic. If μ is an invariant measure on SO(3), then μ is a convex combinaison of the Haar measure. The nearest isotropic elasticity tensor is obtained by integrating the tensor A on the rotation group SO(3). For the numerical approach, we integrate the elasticity tensor on the unit ordinary ball B(0, 1).

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.

Independent bases on the spatial wavefunction of four-identical-particle systems

We construct the independent bases on the spatial wavefunction of four-identical-particle systems classified under the rotational group SO(3) and the permutation group S4 with the usage of transformation coefficients that relate wavefunctions described in one set of internal coordinates with those in another. The basis functions for N ⩽ 2 are presented in the explicit expressions based on the harmonic oscillator model. Such independent bases are supposed to play a key role in the construction of the wavefunctions of the five-quark states and the variation calculation of four-body systems. Our prescription avoids the spurious states and can be programmed for arbitrary N.

MICZ-Kepler: Dynamics on the cone over SO(n)

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of the “Kepler problem” on the cone over the rotation group SO(n). As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, |ϕ|2/r 2, agrees with the rotational part of the cone’s kinetic energy.

Exact mapping of the quantum states in arbitraryN-level systems to the positions of classical coupled oscillators

The quantum dynamics of arbitrary $N$-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary $N$-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension $is$ general for any $N$. The time evolution of an $N$-level system [generated by a complex element of the SU($N$) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in $n$ Euclidean dimensions [the rotation group SO($n$)] is then mapped directly to $n$ oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent $N$-level systems scales as ${N}^{2}$ for the density matrix formalism but increases only linearly with $N$ for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.

Quaternionic Attitude Estimation for Robotic and Human Motion Tracking Using Sequential Monte Carlo Methods With von Mises-Fisher and Nonuniform Densities Simulations

In recent years, wireless positioning and tracking devices based on semiconductor micro electro-mechanical system (MEMS) sensors have successfully integrated into the consumer electronics market. Information from the sensors is processed by an attitude estimation program. Many of these algorithms were developed primarily for aeronautical applications. The parameters affecting the accuracy and stability of the system vary with the intended application. The performance of these algorithms occasionally destabilize during human motion tracking activities, which does not satisfy the reliability and high accuracy demand in biomedical application. A previous study accessed the feasibility of using semiconductor based inertial measurement units (IMUs) for human motion tracking. IMU hardware has been redesigned and an attitude estimation algorithm using sequential Monte Carlo (SMC) methods, or particle filter, for quaternions was developed. The method presented in this paper uses von Mises-Fisher and a nonuniform simulation to provide density estimation of the rotation group SO(3). Synthetic signal simulation, robotics applications, and human applications have been investigated.

Kernel density estimation on the rotation group and its application to crystallographic texture analysis

We are concerned with kernel density estimation on the rotation group SO(3). We prove asymptotically optimal convergence rates for the minimax risk of the mean integrated squared error for different function classes including bandlimited functions, functions with bounded Sobolev norm and functions with polynomially decaying Fourier coefficients and give optimal kernel functions. Furthermore, we consider kernel density estimation with nonnegative kernel functions and prove analogous saturation behavior as it is known for the Euclidean case, i.e., the optimal minimax rate does not improve for smoothness classes of functions which are more than two times differentiable. We also benchmark several families of kernel functions with respect to their capability for kernel density estimation. To make our finding applicable, we give a fast algorithm for the computation of the kernel density estimator for large sampling sets and illustrate our theoretical findings by numerical experiments. Finally, we apply our results to answer a long standing question in crystallographic texture analysis on the number of orientation measurements needed to estimate the underlying orientation density function up to a given accuracy.

Properties and applications of Fisher distribution on the rotation group

We study properties of Fisher distribution (von Mises–Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

Stochastic description of geometric phase for polarized waves in random media

We present a stochastic description of multiple scattering of polarized waves in the regime of forward scattering. In this regime, if the source is polarized, polarization survives along a few transport mean free paths, making it possible to measure an outgoing polarization distribution. We consider thin scattering media illuminated by a polarized source and compute the probability distribution function of the polarization on the exit surface. We solve the direct problem using compound Poisson processes on the rotation group SO(3) and non-commutative harmonic analysis. We obtain an exact expression for the polarization distribution which generalizes previous works and design an algorithm solving the inverse problem of estimating the scattering properties of the medium from the measured polarization distribution. This technique applies to thin disordered layers, spatially fluctuating media and multiple scattering systems and is based on the polarization but not on the signal amplitude. We suggest that it can be used as a non-invasive testing method.

The Jordan Structure of Residual Dynamics Used to Solve Linear Inverse Problems

With a detailed investigation of n linear algebraic equations Bx = b, we find that the scaled residual dynamics for y2 S n 1 is equipped with four struc- tures: the Jordan dynamics, the rotation group SO(n), a generalized Hamiltonian formulation, as well as a metric bracket system. Therefore, it is the first time that we can compute the steplength used in the iterative method by a novel algorithm based on the Jordan structure. The algorithms preserving the length of y are developed as the structure preserving algorithms (SPAs), which can significantly accelerate the convergence speed and are robust enough against the noise in the numerical solutions of ill-posed linear inverse problems.

Geometry of Entanglement Witnesses Parametrized by SO(3) Group

We characterize a set of positive maps in matrix algebra of 4 × 4 complex matrices. Equivalently, we provide a subset of entanglement witnesses in ℂ4⊗ℂ4 parametrized by the rotation group SO (3). Interestingly, these maps/witnesses define two intersecting convex cones in the 3-dimensional parameter space. The existence of two cones is related to the topological structure of the underlying orthogonal group. We perform detailed analysis of the corresponding geometric structure.

Majorana–Oppenheimer Approach to Maxwell Electrodynamics. Part II. Curved Riemannian Space

The Riemann–Silberstein–Majorana–Oppenheimer approach to the Maxwell electrodynamics in the presence of electrical sources, arbitrary media and curved space-time is investigated within the matrix formalism and tetrad method. Symmetries of the matrix Maxwell equation under transformations of the local gauge complex rotation group SO(3,C) is demonstrated explicitly. Equivalence of the approach to general covariant Proca technique and spinor formalism is shown.

ϕ-Lagrangian and Hamiltonian mechanics on SO(2)

Purpose: In this paper, the canonical formalism for plane rotation group SO(2) is presented. Methods: The φ-Lagrangian method is applied in order to find a Lagrangian that the Euler-Lagrange equations would correspondingly coincide with the Lie equations. Results: The canonical formalism for plane rotation group SO(2) is presented. The Lagrangian and Hamiltonian are explicitly constructed. Conclusions: We introduce a family of Lagrange functions on the plane rotation group and verify that Euler-Lagrange and Hamilton equations associated to each of these functions must coincide with the Lie equations in the Lie transformation group.

Optimal Control on the Rotation Group SO (3)

A typical left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived in a simple and elegant manner. These equations are then explicitly integrated by Jacobi elliptic functions.

Fast oriented bounding box optimization on the rotation group SO (3,ℝ)

An exact algorithm to compute an optimal 3D oriented bounding box was published in 1985 by Joseph O'Rourke, but it is slow and extremely hard to implement. In this article we propose a new approach, where the computation of the minimal-volume OBB is formulated as an unconstrained optimization problem on the rotation group SO (3,ℝ). It is solved using a hybrid method combining the genetic and Nelder-Mead algorithms. This method is analyzed and then compared to the current state-of-the-art techniques. It is shown to be either faster or more reliable for any accuracy.