Euclidean Motion Research Articles

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

A nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker-Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

Følner Nets for Semidirect Products of Amenable Groups

AbstractFor unimodular semidirect products of locally compact amenable groups N and H, we show that one can always construct a Følner net of the form (Aα × Bβ) for G, where (Aα) is a strong form of Følner net for N and (Bβ) is any Følner net for H. Applications to the Heisenberg and Euclidean motion groups are provided.

Integration of combinatorial decompositions in the presence of collinearities

Many results in Combinatorial Integral Geometry are derived by integration of the combinatorial decompositions associated with finite point sets {P i } given in the plane ℝ2. However, most previous cases of integration of the decompositions in question were carried out for the point sets {P i } containing no triads of collinear points, where the familiar algorithm sometimes called the “Four indicator formula” can be used. The present paper is to demonstrate that the complete combinatorial algorithm valid for sets {P i } not subject to the mentioned restriction opens the path to various results, including the field of Stochastic Geometry. In the paper the complete algorithm is applied first in an integration procedure in a study of the perforated convex domains, i.e convex domains containing a finite array of non-overlapping convex holes. The second application is in the study of random colorings of the plane that are Euclidean motions invariant in distribution, basing on the theory of random polygonal windows from the so-called Independent Angles (IA) class. The method is a direct averaging of the complete combinatorial decompositions written for colorings observed in polygonal windows from the IA class. The approach seems to be quite general, but promises to be especially effective for the random coloring generated by random Poisson polygon process governed by the Haar measure on the group of Euclidean motions of the plane, assuming that a point P ∈ ℝ2 is colored J if P is covered by exactly J polygons of the Poisson process. A general theorem clearing the way for Laplace transform treatment of the random colorings induced on line segments is formulated.

Combinatorial principles in stochastic geometry in the plane: A review

The paper contains a review of the main results of Yerevan research group in planar stochastic geometry, in particular the second order random geometrical processes using the methods of integration of combinatorial decompositions and invariant imbedding. For the present review we have chosen the results concerning tomography (or stereology) of random processes of lines, random tessellations (mosaics) and Boolean models (not necessarily Poisson) in the plane. The results are obtained either under the assumption of invariance with respect to the group T of translations, or the group M of the Euclidean motions of ℝ2. In each case the marked point processes {P i ,Ψ i } of intersections induced by the random pattern in question on the test lines are studied, where {P i } is the point process of intersections induced on a test line of direction α, the mark Ψ i is the angle at which the intersection with the test line of direction α occurs at point P i . In the case of random line processes, these approaches lead to differential relations between the joint probabilities for the numbers of intersections that occur in disjoint intervals on a test line having direction α, and the first and the second order Palm probabilities of the similar events. By an analysis of these relations, conditions of Poissonity of n-dimensional distributions for any n ≥ 1 are derived. In the case of random tessellations the distribution of the length of so-called typical edge of direction α is obtained in terms of the probability distribution of the corresponding {P i ,Ψ i }. As regards random M-invariant Boolean models, the present review includes a description of the method based on summation of Pleijel-type identities for the part of the realization within a finite disc and subsequent calculation of the limit when the radius of the disc tends to infinity. An alternating process of marked intervals on a line is called black-recurrent if the black marked intervals are independent and the lengths of the white intervals constitute an independent sequence of independent random variables. One of the earliest results in the field states that black-recurrence implies an exponential distribution for the length of the typical white interval on a test line.

The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green’s functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations.

Algebraic methods in mechanism analysis and synthesis

SUMMARYAlgebraic methods in connection with classical multidimensional geometry have proven to be very efficient in the computation of direct and inverse kinematics of mechanisms as well as the explanation of strange, pathological behavior. In this paper, we give an overview of the results achieved within the last few years using the algebraic geometric method, geometric preprocessing, and numerical analysis. We provide the mathematical and geometrical background, like Study's parametrization of the Euclidean motion group, the ideals belonging to mechanism constraints, and methods to solve polynomial equations. The methods are explained with different examples from mechanism analysis and synthesis.

Affine congruence by dissection of intervals

Tarski’s circle squaring problem (see [8]) has motivated the following question: Can a circular disc be dissected into finitely many topological discs such that images of these pieces under suitable Euclidean motions form a dissection of a square? Dubins, Hirsch, and Karush give a negative answer in [1]. However, one can get positive results if the group of Euclidean motions is replaced by suitable other groups of affine maps of the plane (see [3, 5, 6, 7]). The general concept behind these phenomena is the congruence by dissection of discs with respect to some fixed group of affine transformations of R2. Let d denote the Euclidean distance in the plane R2. We recall that a topological disc D is the image of the closed unit disc {x ∈ R2 : d(x, 0) ≤ 1} under a homeomorphism of the plane onto itself. We say that D is dissected into the discs D1, . . . , Dn if D = D1∪. . .∪Dn and int(Di ) ∩ int(D j ) = ∅ for 1 ≤ i < j ≤ n, int(Di ) denoting the interior of Di . Given a group G of affine transformations of R2, two topological discs D, E are called congruent by dissection with respect to G if and only if there exist dissections of D and E into the same finite number n ≥ 1 of subdiscs D1, . . . , Dn and E1, . . . , En , respectively,

Invertible Orientation Scores as an Application of Generalized Wavelet Theory

Inspired by the visual system of many mammals, we consider the construction of—and reconstruction from—an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in \(\mathbb{L}_2 \)(ℝ2) and oriented wavelet φ ∈ \(\mathbb{L}_2 \)(ℝ2). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet φ and deal with the question of which oriented wavelet φ is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores.

The Cartan model of the canonical vector bundles over the Grassmannian manifolds

We give a representation of the canonical vector bundles over the Grassmannian manifolds G(n, p) as noncompact symmetric affine spaces together with their Cartan model in the group of the Euclidean motions SE(n).

Chord calculus and stochastic geometry

Chord calculus is a collection of integration procedures applied to to the combinatorial decompositions that give the solution of the Buffon-Sylvester problem for n needles in a plane or the similar problem in IR 3. It is a source of various integral geometry identities, some of which find their application in Stochastic geometry. In the present paper these applications are focused on random convex polygons and polyhedrons, where we define certain classes where rather simple tomography analysis is possible. The choice of these classes (the Independent Angles class and the Independent Orientations class) is due to the nature of the results of the Chord calculus. The last section points at an application of the convex polygons from the Independent Angles class to Boolean sets in the plane (Boolean models) whose probability distibutions are invariant with respect to the group of Euclidean motions of the plane.

A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group

This paper presents a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane. The proposed inversion method is based on the observation that the exponential Radon transform can be modified to obtain a new transform, defined as the modified exponential Radon transform, that can be expressed as a convolution on the Euclidean motion group. The convolution representation of the modified exponential Radon transform is block diagonalized in the Euclidean motion group Fourier domain. Further analysis of the block diagonal representation provides a class of relationships between the spherical harmonic decompositions of the Fourier transforms of the function and its exponential Radon transform. These relationships and the block diagonalization lead to three new reconstruction algorithms. The proposed algorithms are implemented using the fast implementation of the Euclidean motion group Fourier transform and their performances are demonstrated in numerical simulations. Our study shows that convolution representation and harmonic analysis over groups motivates novel solutions for the inversion of the exponential Radon transform.

AnLp-Lq-Version of Morgan's Theorem for then-Dimensional Euclidean Motion Group

We establish anLp-Lq-version of Morgan's theorem for the group Fourier transform on then-dimensional Euclidean motion groupM(n).

Euclidean motion group representations and the singular value decomposition of the Radon transform

The matrix elements of the unitary irreducible representations of the Euclidean motion group are closely related to the Bessel and Gegenbauer functions. These special functions also arise in the singular functions of the singular value decomposition (SVD) of the Radon transform. In this paper, our objective is to study the Radon transform using the harmonic analysis over the Euclidean motion group and explain the origin of the special functions present in the SVD of the Radon transform from the perspective of group representation theory. Starting with a convolution representation of the Radon transform over the Euclidean motion group, we derive a method of inversion for the Radon transform using the harmonic analysis over the Euclidean motion group. We show that this inversion formula leads to an alternative derivation of the SVD of the Radon transform. This derivation reveals the origin of the special functions present in the SVD of the Radon transform. The derivation of the SVD of the Radon transform is a special case of a general result developed in this paper. This result shows that an integral transform with a convolution kernel is separable if the matrix elements of the irreducible unitary representations of the underlying group are separable.

Error propagation on the Euclidean group with applications to manipulator kinematics

Error propagation on the Euclidean motion group arises in a number of areas such as errors that accumulate from the base to the distal end of manipulators. We address error propagation in rigid-body poses in a coordinate-free way, and explain how this differs from other approaches proposed in the literature. In this paper, we show that errors propagate by convolution on the Euclidean motion group, SE(3). When local errors are small, they can be described well as distributions on the Lie algebra se(3). We show how the concept of a highly concentrated Gaussian distribution on SE(3) is equivalent to one on se(3). We also develop closure relations for these distributions under convolution on SE(3). Numerical examples illustrate how convolution is a valuable tool for computing the propagation of both small and large errors

Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

Inspired by the early visual system of many mammalians we consider the construction of-and reconstruction from- an orientation score $${\it U_f}:\mathbb{R}^2 \times S^{1} \to \mathbb{C}$$ as a local orientation representation of an image, $$f:\mathbb{R}^2 \to \mathbb{R}$$ . The mapping $$f\mapsto {\it U_f}$$ is a wavelet transform $$\mathcal{W}_{\psi}$$ corresponding to a reducible representation of the Euclidean motion group onto $$\mathbb{L}_{2}(\mathbb{R}^2)$$ and oriented wavelet $$\psi \in \mathbb{L}_{2}(\mathbb{R}^2)$$ . This wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis (constructed by irreducible representations of the similitude group) that it allows a stable reconstruction from one (single scale) orientation score. Since our wavelet transform is a unitary mapping with stable inverse, we directly relate operations on orientation scores to operations on images in a robust manner. Furthermore, by geometrical examination of the Euclidean motion group $$G=\mathbb{R}^2 \mathbb{R}\times \mathbb{T}$$ , which is the domain of our orientation scores, we deduce that an operator ? on orientation scores must be left invariant to ensure that the corresponding operator $$\mathcal{W}_{\psi}^{-1}\Phi \mathcal{W}_{\psi}$$ on images is Euclidean invariant. As an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on G. As an application we detect elongated structures in (medical) images and automatically close the gaps between them. Finally, we consider robust orientation estimates by means of channel representations, where we combine robust orientation estimation and learning of wavelets resulting in an auto-associative processing of orientation features. Here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto-associative behavior of the system.

On Inductive Limit Type Actions of the Euclidean Motion Group on Stable UHF Algebras

AbstractAn invariant is presented which classifies, up to equivariant isomorphism, C*-dynamical systems arising as limits from inductive systems of elementary C*-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles.

Weak-perspective structure from motion for strongly contaminated data

It is widely known that, for the affine camera model, both shape and motion data can be factorised directly from the measurement matrix containing the image coordinates of the tracked feature points. However, classical algorithms for structure from motion (SfM) are not robust: measurement outliers, that is, incorrectly detected or matched feature points can destroy the result. A few methods to robustify SfM have already been proposed. Different outlier detection schemes have been used. We examine an efficient algorithm by Trajković and Hedley [Trajković, M., Hedley, M., 1997. Robust recursive structure and motion recovery under affine projection. In: Proc. British Machine Vision Conference. Available from: <http://www.bmva.ac.uk/bmvc/1997/papers/082/082.html>] who use the affine camera model and the least median of squares (LMedS) method to separate inliers from outliers. LMedS is only applicable when the ratio of inliers exceeds 50%. We show that the least trimmed squares (LTS) method is more efficient in robust SfM than LMedS. In particular, we demonstrate that LTS can handle inlier ratios below 50%. We also show that using the real (Euclidean) motion data results in more precise SfM than using the affine motion data. Based on these observations, we propose a novel robust SfM algorithm and discuss its advantages and limits. Furthermore, we introduce a RANSAC based outlier detector that also provides robust results. The proposed methods and the Trajković procedure are quantitatively compared on synthetic data in different simulated situations. The methods are also tested on synthesised and real video sequences.

Intrinsic subdivision with smooth limits for graphics and animation

This article demonstrates the definition of subdivision processes in nonlinear geometries such that smoothness of limits can be proved. We deal with curve subdivision in the presence of obstacles, in surfaces, in Riemannian manifolds, and in the Euclidean motion group. We show how to model kinematic surfaces and motions in the presence of obstacles via subdivision. As to numerics, we consider the sensitivity of the limit's smoothness to sloppy computing.

Second order 3D shape features: An exhaustive study

In recent years many techniques for 3D shape retrieval and classification were proposed. Most of them follow the feature vector paradigm, i.e. the shape is extended with some compact feature representation, on which basis the objects are compared and similarity measures are computed. A main demand of such similarity measures is their invariance to Euclidean motion. There are three main direction to obtain such invariances: Matched Filter Approaches, Pose Normalization or Transformation Group Integration. Among those, registration approaches perform best in retrieval accuracy, while their computational expense however is rather high. In contrast, representation obtained by group integration are fast to compute and compare, but show bad retrieval performance due to its loss in information. In this article we try to close this gap and show that it is also possible to obtain meaningful representations of surface models by group integration approaches.

"Shape Activity": a continuous-state HMM for moving/deforming shapes with application to abnormal activity detection

The aim is to model "activity" performed by a group of moving and interacting objects (which can be people, cars, or different rigid components of the human body) and use the models for abnormal activity detection. Previous approaches to modeling group activity include co-occurrence statistics (individual and joint histograms) and dynamic Bayesian networks, neither of which is applicable when the number of interacting objects is large. We treat the objects as point objects (referred to as "landmarks") and propose to model their changing configuration as a moving and deforming "shape" (using Kendall's shape theory for discrete landmarks). A continuous-state hidden Markov model is defined for landmark shape dynamics in an activity. The configuration of landmarks at a given time forms the observation vector, and the corresponding shape and the scaled Euclidean motion parameters form the hidden-state vector. An abnormal activity is then defined as a change in the shape activity model, which could be slow or drastic and whose parameters are unknown. Results are shown on a real abnormal activity-detection problem involving multiple moving objects.