Euclidean Motion Research Articles

High-Resolution Radar Imaging of Space Targets Based on HRRP Series

When wide or ultrawideband, low pulse repetition frequency (PRF) radar is applied to the imaging of space targets; it is highly possible that motion through range cell and azimuth under-sampling occurs, which will result in image smearing. To figure out this problem, this paper proposes a novel, three-step imaging method using the high-resolution range profile (HRRP) series. In the first step, high-quality HRRP series are obtained based on the theory of sparse signal representation. Then, based on the Kalman predictor and the minimum Euclidean distance criterion, motion and amplitude feature-based scatterer trajectory association is carried out to form the scatterer trajectory matrix, from which the scatterer locations are conveniently solved in the last step. Compared to the traditional imaging techniques based on Doppler analysis, the proposed method is able to mitigate the influence of azimuth under-sampling, and may provide a new solution to high-resolution imaging of targets moving nonuniformly in low PRF scenarios. Finally, simulations have proved the effectiveness of the proposed method.

A Multi-Orientation Analysis Approach to Retinal Vessel Tracking

This paper presents a method for retinal vasculature extraction based on biologically inspired multi-orientation analysis. We apply multi-orientation analysis via so-called invertible orientation scores, modeling the cortical columns in the visual system of higher mammals. This allows us to generically deal with many hitherto complex problems inherent to vessel tracking, such as crossings, bifurcations, parallel vessels, vessels of varying widths and vessels with high curvature. Our approach applies tracking in invertible orientation scores via a novel geometrical principle for curve optimization in the Euclidean motion group SE(2). The method runs fully automatically and provides a detailed model of the retinal vasculature, which is crucial as a sound basis for further quantitative analysis of the retina, especially in screening applications.

Applications of the Exponential Function of the Euclidean Motion Group to the Theory of Gearing

The present paper provides a first step to a new approach to the theory of gearing, which uses modern differential geometry in order to ensure a strict and coordinate-independent formulation. Here, we are mainly concerned with a basic equation, namely, the equation of meshing, of two rotating surfaces in mesh. Since we are able to solve this equation by the time parameter, we derive parameterizations of the mating pinion from a bevel gear as well as a parameterization for gears produced by special machine tools.

Unitarising measures for the representations of affine group and associated invariant operators

Let G denote the group of affine transformations on Cn, or one of its subgroups such as the translation group or the Euclidean motion group. In this paper, we study unitarisability of holomorphic representations T of G into the space of square integrable C-valued functions with respect to a positive real measure μ with support in a domain D in Cn. Moreover, we investigate the infinitesimal representation approach of a unitary representation (Tg,μ) on the Lie algebra G of G in order to determine a class of second-order differential operators (called of Ornstein–Uhlenbeck type) invariant with respect to the unitarising measures μ.

On the inverse kinematics of an a priori unknown general 6R-Robot

SUMMARYThis paper looks at the inverse kinematics problem of an a priori unknown 6R-Robot from the representation point of view. It describes a new representation of the Euclidean motion group. With this representation, the inverse kinematics problem can be treated entirely numerical. No symbolical methods are required.

Chasles’ fixed point theorem for Euclidean motions

Chasles’ theorem, a classic and important result of kinematics, states that every orientation-preserving isometry of \({\mathbb{R}^3}\) is a screw motion. We show that this is equivalent to the assertion that each proper Euclidean motion that is not a pure translation, acting on the space of oriented lines, has a unique fixed point (the axis of the screw motion). We use that formulation to derive a simple and novel constructive proof of Chasles’ theorem.

Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS)

We introduce an algorithm for diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both voxel space and diffusion-gradient space. The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models. It works by embedding the measurement space into a space with defined metric, in this case the Lie group of three-dimensional Euclidean motion SE(3). Subsequently, pairwise comparisons of the values of the diffusion weighted signal are used for adaptation. POAS preserves the edges of the observed fine and anisotropic structures. It is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and variability of quantities derived from the data for specific models. We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and hence acquisition time while achieving a similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.

A Note on Confidence Momentum and Term Structure of Confidence with Applications to Financial Markets

This note is based on a recent confidence index introduced in the context of compensating probability factors for deviations of subjective probability measures from equivalent martingale measures. The index is adjusted for loss gain probability spreads, and it explains momentum in confidence. We use the index to introduce a confidence matrix operator which shows how a subject transforms gain domain into fear of loss. So she is loss averse or risk averse. By contrast, the adjoint confidence matrix operator is an Euclidean motion which rotates and reverses loss domain into hope of gain. Thus, signifying risk seeking over loss domains in hope of gain. Simulation of the model shows that the distribution of prior loss [gain] probabilities is a predictor of confidence momentum and fields of confidence. Moreover, our field theory of confidence mimics a sample of Gallup Monthly Economic Confidence Index, and depicts a term structure of confidence for hope and fear. It plainly shows that the growth rate of Gallup Economic Confidence Index -- which is highly correlated with popular confidence indexes like UBS/Gallup Investor Optimism Index; Michigan Consumer Confidence Index; and Yale Investor Confidence Index -- predict bubbles and crashes.

On the norm of the Lp -Fourier transform on compact extensions of ℝ n

Abstract This paper aims to prove that the norm of the Lp -Fourier transform of the semidirect product ℝ n ⋊ K $\mathbb {R}^n\rtimes K$ is A p n , where 1 < p ≤ 2 $1 < p \le 2$ , q = p / ( p - 1 ) $q=p/(p-1)$ , A p = p 1 2 p q - 1 2 q $A_p=p^{\frac{1}{2p}}q^{\frac{-1}{2q}}$ , and K stands for a compact subgroup of automorphisms of ℝ n . An extremal function is given by an extension of a Gaussian function. Besides, as an example of non-compact extension, the universal covering group of the Euclidean motion group of the plane is also treated and an estimate of the norm is obtained.

Neutral wrenches of 3-parametric robot-manipulators of the spherical rank 1

Let SE(3) be the Lie group of all Euclidean motions in the Euclidean space E 3, let se(3) be its Lie algebra and se*(3) the space dual to se(3). This paper deals with structures of the subspaces of se*(3) which are formed by all the forces whose power exerted on the robot effector is zero.

Non-rigid isometric ICP: A practical registration method for the analysis and compensation of form errors in production engineering

The unprecedented success of the iterative closest point (ICP) method for registration in geometry processing and related fields can be attributed to its efficiency, robustness, and wide spectrum of applications. Its use is however quite limited as soon as the objects to be registered arise from each other by a transformation significantly different from a Euclidean motion. We present a novel variant of ICP, tailored for the specific needs of production engineering, which registers a triangle mesh with a second surface model of arbitrary digital representation. Our method inherits most of ICP’s practical advantages but is capable of detecting medium-strength bendings i.e. isometric deformations. Initially, the algorithm assigns to all vertices in the source their closest point on the target mesh and then iteratively establishes isometry, a process which, very similar to ICP, requires intermediate re-projections. A NURBS-based technique for applying the resulting deformation to arbitrary instances of the source geometry, other than the very mesh used for correspondence estimation, is described before we present numerical results on synthetic and real data to underline the viability of our approach in comparison with others.

Rigidity in the class of orientable compact surfaces of minimal total absolute curvature

Consider an orientable compact surface in three-dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion.

Regularity of minimal intrinsic graphs in 3-dimensional sub-Riemannian structures of step 2

This work provides a characterization of the regularity of noncharacteristic intrinsic minimal graphs for a class of vector fields that includes non-nilpotent Lie algebras as the one given by Euclidean motions of the plane. The main result extends a previous one on the Heisenberg group, using similar techniques to deal with nonlinearities. This wider setting provides a better understanding of geometric constraints, together with an extension of the potentialities of specific tools as the lifting–freezing procedure and interpolation inequalities. As a consequence of the regularity, a foliation result for minimal graphs is obtained.

Accurate three-dimensional registration of magnetic resonance images for detecting local changes in cartilage thickness

The purpose of this study is to develop a three-dimensional registration method for monitoring knee joint disease from magnetic resonance (MR) image data sets. A global optimization technique was used for identifying anatomically corresponding points of knee femur surfaces (bone cartilage interfaces). In a first pre-registration step, we used the principal axes transformation to correct for different knee joint positions and orientations in the MR scanner. In a second step, we presented a global search algorithm based on Lipschitz optimization theory. This technique can simultaneously determine the translation and rotation parameters through searching a six-dimensional space of Euclidean motion metrics (translation and rotation) after calculating the point correspondences. The point correspondences were calculated by using the Hungarian algorithm. The accuracy of registration was evaluated using 20 porcine knees. There were 300 corresponding landmark points over the 20 pig knees. We evaluated the registration accuracy by measuring the root-mean-square distance (RMSD) error of corresponding landmark points between two femur surfaces (two time-points). The results show that the average RMSD was 1.22 ± 0.10 mm (SD) by the iterative closest point (ICP) method, 1.17 ± 0.10 mm the by expectation-maximization-ICP method, 1.02 ± 0.06 mm by the genetic method, and 0.93 ± 0.04 mm by the proposed method. Compared with the other three registration approaches, the proposed method achieved the highest registration accuracy.

On Two Superintegrable Nonlinear Oscillators in N Dimensions

We consider the classical superintegrable Hamiltonian system given by $$\mathcal{H}_{\lambda}=\mathcal{T}+\mathcal{U}=\frac{\mathbf{p}^2}{2(1+{\lambda}\mathbf{q}^2)} + \frac{ \omega^2\mathbf{q}^2}{2(1+{\lambda}\mathbf{q}^2)},$$ where $\mathcal{U}$ is known to be the “intrinsic” oscillator potential on the Darboux spaces of nonconstant curvature determined by the kinetic energy term $\mathcal{T}$ and parametrized by λ. We show that $\mathcal{H}_{\lambda}$ is Stäckel equivalent to the free Euclidean motion, a fact that directly provides a curved Fradkin tensor of constants of motion for $\mathcal{H}_{\lambda}$ . Furthermore, we analyze in terms of λ the three different underlying manifolds whose geodesic motion is provided by $\mathcal{T}$ . As a consequence, we find that $\mathcal{H}_{\lambda }$ comprises three different nonlinear physical models that, by constructing their radial effective potentials, are shown to be two different nonlinear oscillators and an infinite barrier potential. The quantization of these two oscillators and its connection with spherical confinement models is briefly discussed.

A stochastic kinematic model of class averaging in single-particle electron microscopy

Single-particle electron microscopy is an experimental technique that is used to determine the three-dimensional (3D) structure of biological macromolecules and the complexes that they form. In general, image processing techniques and reconstruction algorithms are applied to micrographs, which are 2D images taken by electron microscopes. Each of these planar images can be thought of as a projection of the macromolecular structure of interest from an a priori unknown direction. A class is defined as a collection of projection images with a high degree of similarity, presumably resulting from taking projections along similar directions. In practice, micrographs are very noisy and those in each class are aligned and averaged in order to reduce the background noise. Errors in the alignment process are inevitable due to noise in the electron micrographs. This error results in blurry averaged images. In this paper, we investigate how blurring parameters are related to the properties of the background noise in the case when the alignment is achieved by matching the mass centers and the principal axes of the experimental images. We observe that the background noise in micrographs can be treated as Gaussian. Using the mean and variance of the background Gaussian noise, we derive equations for the mean and variance of translational and rotational misalignments in the class averaging process. This defines a Gaussian probability density on the Euclidean motion group of the plane. Our formulation is validated by convolving the derived blurring function representing the stochasticity of the image alignments with the underlying noiseless projection and comparing with the original blurry image.

Deconvolution on the Euclidean motion group

Deconvolution is an important class of inverse problems. The goal of this paper is to develop the optimality properties of deconvolution density estimators on the noncommutative and noncompact six-dimensional Euclidean motion group. The idea is that if the underlying random object is a composition of several independent objects, then the underlying density is a convolution of densities of those independent objects. Therefore, deconvolution allows one to recover the main components of the mixture. We illustrate this with some numerical work.

Automatic 3D MR Image Registration and Its Evaluation for Precise Monitoring of Knee Joint Disease

We describe a technique for the registration of three dimensional (3D) knee femur surface points from MR image data sets; it is a technique that can track local cartilage thickness changes over time. In the first coarse registration step, we use the direction vectors of the volume given by the cloud of points of the MR image to correct for different knee joint positions and orientations in the MR scanner. In the second fine registration step, we propose a global search algorithm that simultaneously determines the optimal transformation parameters and point correspondences through searching a six dimensional space of Euclidean motion vectors (translation and rotation). The present algorithm is grounded on a mathematical theory - Lipschitz optimization. Compared with the other three registration approaches (ICP, EM-ICP, and genetic algorithms), the proposed method achieved the highest registration accuracy on both animal and clinical data.

Determination of rotation vectors from multi‐point data and practical applications

AbstractThe present work aims to present a different set of parameters for Euclidean motions which have a definite physical interpretation. The orientation here is obtained via vector‐parameters which are elements of a Lie group with a nice and clear composition law. The method is quite convenient for describing of rigid body and manipulators motions such as scene and image processing and recognition. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Novel 3D Model Retrieval Approach Based on Model-Partitioning and Fuzzy Relevance Feedback

3D models play an important role in many applications, so there is an urgent need for an effective content based 3D model retrieval system. A variety of 3D model retrieval methods have been proposed in recent years. Shape distributions show superiority over other methods due to ease of computation and invariance to Euclidean motion, but there is poor retrieval performance for the loss in information. This paper introduces two model-partitioning methods to improve shape distributions, in which the two enhanced descriptors are combined with a fuzzy feedback method. Experimental results show that the proposed methods can achieve better retrieval performance.