Euclidean Transformation Research Articles

Anisotropic generalized Procrustes analysis

Generalized Procrustes analysis is a popular method for matching several configurations by translations, rotations/reflections and scaling constants. It aims at producing a group average from these Euclidean similarity transformations followed by bi-linear approximation of this group average for graphical inspection. An extension that allows for anisotropic scaling (i.e. different scaling of different dimensions) is proposed. This method is illustrated using two real data sets from sensory analysis.

$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics Part II: Accelerated Observers and Fictitious Forces

We investigate a possible form of Schrodinger’s equation as it appears to moving observers. It is shown that, in this framework, accelerated motion requires fictitious potentials to be added to the original equation. The gauge invariance of the formulation is established. The example of accelerated Euclidean transformations is treated explicitly, which contain Galilean transformations as special cases. The relationship between an acceleration and a gravitational field is found to be compatible with the picture of the ‘Einstein elevator’. The physical effects of an acceleration are illustrated by the problem of the uniformly-accelerated harmonic oscillator.

Feature vector field and feature matching

In this paper, we propose a feature vector field for images, which is built by the inner products and exterior products of image gradients. The feature vector field effectively represents image edges and feature points including corners and edge points with big curvature. Using the feature vector field, some novel descriptors are constructed for point matching and curve matching. The descriptors are all invariant to image Euclidean transformations and linear intensity changes. The experiments show that the descriptors also have a good adaptability to small image affine transformation, JPEG compression and nonlinear changes of intensity.

Creating and classifying measures of linear association by optimization techniques

The idea of measures of linear association, such as Pearson's correlation coefficient, can be put in a general framework by axiomization. Groups of linear transformations on $\mathsf{R}^n$ can be exploited to create new and classify existing measures according to their invariance properties. Thus invariance under the Euclidean transformation group leads to the class of so-called geometric measures. Similarly, a measure is called algebraic if it is invariant under scalings. Pearson's coefficient is an example of an algebraic measure; it is not geometric. It is proved that, generally, a measure of linear association cannot possibly be both geometric and algebraic. A procedure is developed to convert a geometric measure into an algebraic and vice versa. Thus a kind of a duality between algebraic and geometric measures arises. In this duality measures can be reflexive or not.

Coordinate independent update formulas for versor Clifford neurons

We study the optimization of neural networks with Clifford geometric algebra versor and spinor nodes. For that purpose important multivector calculus results are introduced. Such nodes are generalizations of real, complex and quaternion spinor nodes. In particular we consider nodes that can learn all proper and improper Euclidean transformations with so-called conformal versors. Thus a single node can correctly compute full 3D screws and rotoinversions with off-origin axis and off- origin points of inversion. The latter is a unique property of our proposed versor neuron. Computing inversions by ordinary real-valued networks is not easily possible due to its nonli near nature. Simulation on learning inversions illustrating these facts are provided in the paper. theory is a priori coordinate independence. We consider a vari- ety of new multivector node models, their error functions and (optimal) weight update rules: Clifford group versor neurons, unitary conformal versor neurons, homogeneous conformal versor neurons, and weight unitarity by Lagrange multipliers. The a priori invariance gives deep geometric insight and allows for easy systematic generalization to even higher dimensional spaces. Based on these results it is also straightforward to devise versor (and spinor) MLP back-propagation algorithms.

Robust 3D object registration without explicit correspondence using geometric integration

3D vision-guided manipulation of components is a key problem of industrial machine vision. In this paper, we focus on the localization and pose estimation of known industrial objects from 3D measurements delivered by a scanning sensor. Since local information extracted from these measurements is unreliable due to noise, spatially unstructured measurements and missing detections, we present a novel objective function for robust registration without using correspondence information, based on the likelihood of model points. Furthermore, by extending Runge–Kutta-type integration directly to the group of Euclidean transformation, we infer object pose by computing the gradient flow directly on the related manifold. Comparison of our approach to existing state of the art methods shows that our method is more robust against poor initializations while having comparable run-time performance.

A survey on evaluation methods for image interpolation

Image interpolation is applied to Euclidean, affine and projective transformations in numerous imaging applications. However, due to the unique characteristics and wide applications of image interpolation, a separate study of their evaluation methods is crucial. The paper studies different existing methods for the evaluation of image interpolation techniques. Furthermore, an evaluation method utilizing ground truth images for the comparisons is proposed. Two main classes of analysis are proposed as the basis for the assessments: performance evaluation and cost evaluation. The presented methods are briefly described, followed by comparative discussions. This survey provides information for the appropriate use of the existing evaluation methods and their improvement, assisting also in the designing of new evaluation methods and techniques.

The deformation of flat connections and affine manifolds

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption is dropped, the manifold is not necessarily obtained as the quotient of the Euclidean space through a properly discontinuous group of affine transformations. In fact the universal cover may no longer be the Euclidean space. The main result of this paper states that if a flat connection of a bundle can be properly deformed into a metric connection then its Euler class vanishes. This is a partial result toward an old question of Chern.

Shape Metrics Based on Elastic Deformations

Deformations of shapes and distances between shapes are an active research topic in computer vision. We propose an energy of infinitesimal deformations of continuous 1- and 2-dimensional shapes that is based on the elastic energy of deformed objects. This energy defines a shape metric which is inherently invariant with respect to Euclidean transformations and yields very natural deformations which preserve details. We compute shortest paths between planar shapes based on elastic deformations and apply our approach to the modeling of 2-dimensional shapes.

Branch-and-Bound Methods for Euclidean Registration Problems

In this paper, we propose a practical and efficient method for finding the globally optimal solution to the problem of determining the pose of an object. We present a framework that allows us to use point-to-point, point-to-line, and point-to-plane correspondences for solving various types of pose and registration problems involving euclidean (or similarity) transformations. Traditional methods such as the iterative closest point algorithm or bundle adjustment methods for camera pose may get trapped in local minima due to the nonconvexity of the corresponding optimization problem. Our approach of solving the mathematical optimization problems guarantees global optimality. The optimization scheme is based on ideas from global optimization theory, in particular convex underestimators in combination with branch-and-bound methods. We provide a provably optimal algorithm and demonstrate good performance on both synthetic and real data. We also give examples of where traditional methods fail due to the local minima problem.

On the Topology and Geometry of Spaces of Affine Shapes

We define the space of affine shapes of k points in R n to be the topological quotient of (R n ) k modulo the natural action of the affine group of R n . These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ n k in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh n k in detail. Each Sh n k is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for k>n, expressing the metric purely in terms of "pre-shape" data, i.e. configurations of k points in R n .

A unified localization approach for machining allowance optimization of complex curved surfaces

The goal of workpiece localization is of interest to find the optimal Euclidean transformation that aligns the sampled points to the nominal CAD model to ensure sufficient stock allowance during the machining process. In this paper, a unified localization technique is developed for sculptured surface machining. This technique concerns an alignment process to satisfy a user-defined set of constraints for some specific surfaces where the machining allowance is preferentially guaranteed. The mathematical model of the constrained optimization alignment is firstly established, and is efficiently solved by a combination of the multipliers method and the BFGS algorithm to handle the large number of constraints in allowance optimization. To efficiently calculate the Euclidean oriented distance, a novel approach, which combines the robust arithmetic for multivariate Bernstein-form polynomials and Bezier surface segmentation algorithm, is presented based on recursive quadtree decomposition. Two typical sculptured surfaces are used to test the developed algorithm and comparisons between the proposed algorithm and the existing algorithms are given. Experiment results show that the proposed method is appropriate and feasible to distribute the stock allowance for proper sculptured surface machining.

A Fast Registration Method Using IP and Its Application to Ultrasound Image Registration

This paper presents a fast registration method based on solving an energy minimization problem derived by implicit polynomials (IPs). Once a target object is encoded by an IP, it will be driven fast towards a corresponding source object along the IP’s gradient flow without using point-wise correspondences. This registration process is accelerated by a new IP transformation method. Instead of the time-consuming transformation to a large discrete data set, the new method can transform the polynomial coefficients to maintain the same Euclidean transformation. Its computational efficiency enables us to improve a new application for real-time Ultrasound (US) pose estimation. The reported experimental results demonstrate the capabilities of our method in overcoming the limitations of a noisy, unconstrained, and freehand US image, resulting in fast and robust registration.

On the weak-equilibrium condition for derivation of algebraic heat flux model

Analogous to an algebraic Reynolds stress model, the algebraic heat flux model (AHFM) is derived from a second-moment closure by invoking the weak-equilibrium condition. The present study investigates this condition in detail as it applies to the advection and diffusive-transport terms. For the advection term, the correct form of this condition in non-inertial frames is obtained by means of an invariant Euclidean transformation. The validity of the diffusive-transport condition is examined through an a priori test using a DNS database for rotating turbulent channel flow with heat transfer. It is shown that the weak-equilibrium condition applied to diffusive-transport term tends to fail in the near-wall region. An alternative form is proposed that is based on an asymptotic analysis of the transport equation budget in the near-wall region. An evaluation of proposed form shows that it has the potential to improve the predictive ability of an ARSM for flows involving system rotation and/or streamline curvature.

Machining localization and quality evaluation of parts with sculptured surfaces using SQP method

Using automatic localization to ensure sufficient machining allowance or derive reliable evaluation results of machining quality, it is possible to cope with the case where the coordinate system in a part design specification is different from that for machining or measuring the part. To achieve workpiece localization and machining quality evaluation, it is essential to find the optimum Euclidean transformation that aligns the nominal mode to the sample points from a workpiece. This paper describes a unified algorithm for workpiece localization and quality evaluation. The optimum alignment model is firstly established with minimax criteria, and then sequential quadratic programming (SQP) is proposed to solve the optimization problem. A computational algorithm of point-to-surface distance and a linear differential motion model of the objective function with respect to alignment parameters are subsequently provided. The results show that the SQP-based workpiece localization and quality evaluation method is computationally more efficient than those based on direct search algorithms. It is found that the proposed method is effective in machining localization and quality evaluation even with a large set of sample points.

Fourier-based geometric shape prior for snakes

A novel method of snakes with shape prior is presented in this paper. We propose to add a new force which makes the curve evolve to particular shape corresponding to a template to overcome some well-known problems of snakes. The template is an instance or a sketch of the researched contour without knowing its exact geometric pose in the image. The prior information is introduced through a set of complete and locally stable invariants to Euclidean transformations (translation, rotation and scale factor) computed using Fourier transform on contours. The method is evaluated with the segmentation of myocardial scintigraphy slices and the tracking of an object in a video sequence.

Rigidity of the canonical isometric imbedding of the Hermitian symmetric space $Sp(n)/U(n)$

In this paper we discuss the rigidity of the canonical isometric imbedding $\pmb{f}_0$ of the Hermitian symmetric space $Sp(n)/U(n)$ into the Lie algebra $\mathfrak{sp}(n)$. We will show that if $n \ge 2$, then $\pmb{f}_0$ is strongly rigid, i.e., for any isometric immersion $\pmb{f}_1$ of a connected open set $U$ of $Sp(n)/U(n)$ into $\mathfrak{sp}(n)$ there is a euclidean transformation $a$ of $\mathfrak{sp}(n)$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.

The Linear Elliptic - Surfaces of Revolution

The paper presents a category of revolution surfaces which may be generated by the metrical Euclidean transformation compounded by revolutions and displacements. These surfaces are created by a straight line revolution around a revolution axis moving on an ellipse. This surface group is classified according to the relative positions of a revolution axis moving in contact with an ellipse, and according to the relative positions of a straight line and a revolution axis . The straight line revolution and axis motion can be for equal or different orientation and both can have equal or different radian frequencies. The transformation matrices for the vector equations of the surface and also some properties of these surfaces are presented in this paper.

Statistical inferences for two-dimensional workpiece localisation

In this paper, we study Workpiece Localisation (WL) which can be described as follows. Before machining, a rigid workpiece is fixed to a work table approximately at its nominal location after minimal calibration. With data measured on the fixed workpiece using a Coordinate Measuring Machine (CMM), we aim to determine the Euclidean transformation of the workpiece Computer-Aided Design (CAD) model frame relative to some known world reference frame. Recently the WL problems have been formulated as Least Squares Problems (LSP). Although various algorithms have been proposed to solve the LSP, they involve an iterative process to update the Euclidian transformation and the home points of the measured data. In this paper we propose a new formulation of the WL problems, which can be solved using standard nonlinear least squares method where updating the home points is unnecessary. Additionally, the new formulation enables constructing confidence intervals for the Euclidean transformation parameters.

Groups in Mind

Both Henri Poincaré and (more recently) Roger Shepard have argued that the geometry and topology of physical space are internalized by the mind in the form, not (or not only) of a Euclidean manifold, but in terms of the group of rigid Euclidean transformations. Since this issue can have bearing on various metaphysical and epistemological questions, we explore the different reasons they offer for holding this view. In this context, we show how most commentators misunderstand Poincaré's ‘heated sphere/plate’ model and introduce Shepard's ideas to the philosophy of science community.