Euclidean Motion Group Research Articles

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.

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.

On the geometry of the space of oriented lines of Euclidean space

We prove that the space of all oriented lines of the n-dimensional Euclidean space admits a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions, exactly when n=3 or n=7 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). Up to equivalence, there are two such metrics for each dimension, and they are of split type and complete. Besides, we prove that the given metrics are Kahler or nearly Kahler if n=3 or n=7, respectively.

On theorems of Beurling and Hardy for the Euclidean Motion group

We establish an analogue of Beurling's uncertainty principle for the group Fourier transform on the Euclidean motion group. We also prove the most general version of Hardy's theorem on it which characterises functions on the motion group that are controlled by the heat kernel associated to the Laplacian of the Euclidean space.

Curvatures and Tolerances in the Euclidean Motion Group

We investigate the action of imprecisely defined affine and Euclidean transformations and compute tolerance zones of points and subspaces. Tolerance zones in the Euclidean motion group are analyzed by means of linearization and bounding the linearization error via the curvatures of that group with respect to an appropriate metric.

An Uncertainty Principle Related to the Euclidean Motion Group

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/, g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), .-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms Christensen and Schlichtkrull Euclidean motion group 251 4. The uncertainty principle Since [X,YX] = Y21 [X,Y2]=-Yl the uncertainty principle gives 1(7*7, )/,/>| = '(n([X, Y2])fJ)' The first author would like to thank the people at NUI Galway for the help and friendliness he received while writing his thesis.

AN UNCERTAINTY PRINCIPLE RELATED TO THE EUCLIDEAN MOTION GROUP

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.

Gliding spline motions and applications

We consider the ICP (iterative closest point) algorithm, which may in general be used for moving ‘active’ elements such as curves and surfaces towards geometric objects whose distance field is computable. We show how it may be accelerated, and how it can be applied to the design of near-Euclidean near-contact spline motions. One particular application of this concept is the modeling of milling tool paths in five-axis milling. The method involves computing the distance from and footpoints in both the Euclidean motion group and the configuration space of surface-surface contact.

An operational calculus for the euclidean motion group with applications in robotics and polymer science

In this article we develop analytical and computational tools arising from harmonic analysis on the motion group of three-dimensional Euclidean space. We demonstrate these tools in the context of applications in robotics and polymer science. To this end, we review the theory of unitary representations of the motion group of three dimensional Euclidean space. The matrix elements of the irreducible unitary representations are calculated and the Fourier transform of functions on the motion group is defined. New symmetry and operational properties of the Fourier transform are derived. A technique for the solution of convolution equations arising in robotics is presented and the corresponding regularized problem is solved explicity for particular functions. A partial differential equation from polymer science is shown to be solvable using the operational properties of the Euclidean-group Fourier transform.

Synthesis of Binary Manipulators Using the Fourier Transform on the Euclidean Group

In this paper we apply the Fourier transform on the Euclidean motion group to solve problems in kinematic design of binary manipulators. In recent papers it has been shown that the workspace of a binary manipulator can be viewed as a function on the motion group, and it can be generated as a generalized convolution product. The new contribution of this paper is the numerical solution of mathematical inverse problems associated with the design of binary manipulators. We suggest an anzatz function which approximates the manipulator’s density in analytical form and has few free fitting parameters. Using the anzatz functions and Fourier methods on the motion group, linear and non-linear inverse problems (i.e., problems of finding the manipulator’s parameters which produce the total desired workspace density) are solved.

Line congruences as surfaces in the space of lines

The space of lines in R 3 can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.

Uncertainty principles like Hardy's theorem on some Lie groups

AbstractWe extend an uncertainty principle due to Cowling and Price to Euclidean spaces, Heisenberg groups and the Euclidean motion group of the plane. This uncertainty principle is a generalisation of a classical result due to Hardy. We also show that on the real line this uncertainty principle is almost equivalent to Hardy's theorem.

Noncommutative gauge theories in matrix theory

We present a general framework for Matrix theory compactified on a quotient space R^n/G, with G a discrete group of Euclidean motions in R^n. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of G associated with a projective regular representation. Also we show how to extend our treatments to incorporate orientifolds.

The Marcinkiewicz-Zygmund law of large numbers on the group of Euclidean motions and the diamond group

An analogue of the Marcinkiewicz-Zygmund strong law of large numbers is proved for the group of Euclidean motions and for the diamond group.

On the 6 × 6 cartesian stiffness matrix for three-dimensional motions

It has been recently observed by Pigoski et al. [1, 2], and by Ciblak and Lipkin [3] that the Cartesian stiffness matrix associated with a linear elastic coupling between two rigid bodies is, in general, asymmetric if the resulting forces and moments do not sum to zero. In this paper, methods of differential geometry and properties of Lie groups are used to show that in any conservative system subjected to a non-zero external load, if motions in SE(3), the special Euclidean group of rigid body motions in three dimensions, are considered, the resulting 6 × 6 Cartesian stiffness matrix is asymmetric in any inertial (fixed) or body-fixed reference frame. We prove the general result that in any subgroup of SE(3), if the system is subject to external forces and moments, the Cartesian stiffness matrix is symmetric if finite displacements along the basis twists are used to generate the stiffness matrix commute. Further, we derive several useful properties of stiffness matrices using ideas from Lie theory. In particular, we offer a simple proof to show that the stiffness matrix in the body-fixed reference frame is the transpose of the stiffness matrix in the inertial reference frame, a result also derived in [3]. Finally, we outline a method to construct a symmetric stiffness matrix by choosing an appropriate moving reference frame that is not fixed to any rigid body.

Hardy’s theorem for the 𝑛-dimensional Euclidean motion group

An uncertainty principle, due to Hardy, for Fourier transform pairs on R \Bbb R says that if the function f f is “very rapidly decreasing”, then the Fourier transform cannot also be “very rapidly decreasing” unless f f is identically zero. In this paper we state and prove an analogue of Hardy’s theorem for the n n -dimensional Euclidean motion group.

Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group

In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible unitary representations of the motion group and discuss the corresponding Fourier transform of functions on the motion group. The main reasons why exact solutions of this convolution equation do not exist in many cases are discussed. Techniques for regularization of the problem and numerical methods for finding approximate solutions are presented. Examples are considered and approximate solutions are found.

Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles.

An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups

A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result to other Lie groups. (For example, see [18].) In sharp contrast to Schwartz’s theorem, is a result due to Hardy [5] which says that f and f cannot both be “very rapidly decreasing”. More precisely, if |f(x)| ≤ Ae−α|x|2 and |f(y)| ≤ Be−β|y| 2 and αβ > 1 4 , then f ≡ 0. (See [2], pp. 155-157.) However, as far as we are aware, until very recently no systematic attempt was made to generalize Hardy’s theorem to other Lie groups. In [12], [13], and [15], this result has been generalized to the Heisenberg groups Hn, the Euclidean motion groups M(n) and for certain eigenfunction expansions. In this paper we establish an analogue of Hardy’s theorem for a class of noncompact semisimple Lie groups and all symmetric spaces of the noncompact type. Hardy’s theorem can also be viewed as a sort of ‘Uncertainty Principle’. The results in [12] and [13] are presented from this point of view. (In [1], Cowling and Price have proved an “L − L” version of Hardy’s theorem on R. The theorem of Beurling in [9] is similar in spirit to Hardy’s theorem, although far more general, and indeed Hardy’s theorem, as well as the result of Cowling and Price, can be deduced from it as special cases.)