non-Euclidean Metrics Research Articles

Two approaches to the problem of defect correction in vector field tomography solving boundary value problems

The problem of vector field tomography consists of reconstructing a vector field from integrals of certain projections of the field over geodesic lines. In case of an homogeneous medium the underlying Riemannian metric is the Euclidean one and the geodesics are straight lines. Modeling inhomogeneous media we take also non-Euclidean metrics into account. It is well known that the corresponding integral transform has a nontrivial kernel consisting of potential fields with potentials being equal to a constant on the boundary of the domain. Thus, the reconstruction from these data may contain a potential part of this type. Since we consider only solenoidal fields, e.g.velocity fields of incompressible fluids, the accuracy of the solution can be improved detecting the undesirable potential part of the reconstruction and subtracting it. Depending on the boundary condition the sought potential is the solution of either a Dirichlet or a Neumann boundary value problem. In this article two approaches to recover the potential are suggested: One deals with the solution of the Dirichlet problem with a finite difference scheme, the other solves the Neumann problem with the help of boundary element methods.

Growth modeling of human mandibles using non-Euclidean metrics

From a set of 31 three-dimensional computed tomography (CT) scans we model the temporal shape and size of the human mandible for analysis, simulation, and prediction purposes. Each anatomical structure is represented using 14 851 semi-landmarks, and mapped into Procrustes tangent space. Exploratory subspace analyses are performed leading to linear models of mandible shape evolution in Procrustes space. The traditional variance analysis results in a one-dimensional growth model. However, working in a non-Euclidean metric results in a multimodal model with uncorrelated modes of biological variation related to independent component analysis. The applied non-Euclidean metric is governed by the correlation structure of the estimated noise in the data. The generative models are compared, and evaluated on the basis of a cross validation study. The new non-Euclidean analysis is completely data driven. It not only gives comparable results w.r.t. previous studies of the mean modeling error, but seems to better correlate to growth, and in addition provides the data analyst with alternative hypothesis of plausible shape evolution; hence aiding in the understanding of cranio-facial growth.

Modeling of morphogenesis of polyolefin particles: Catalyst fragmentation

AbstractThe growth of polyolefin particles in heterogeneous catalyzed polymerization reactors is simulated often by reaction‐transport models that introduce certain ad hoc effective spatial scales to the description of particle morphology and transport processes inside the particles. Because predictive capabilities of such models with respect to particle morphogenesis are limited, a multiscale model of a growing particle is developed. Maximum relevant information can be systematically extracted from particle microscopy images using the method of reconstructed porous media. The spatially 3‐D replicas of the particle pore space allow to calculate the effective transport and mechanical properties. Two new modeling techniques approximate catalyst particle fragmentation: Delaunay triangulation generalized into a 3‐D space with non‐Euclidean metrics and disconnection of the skeleton representation of the solid phase. Predictive mesoscopic modeling of particle morphogenesis is also discussed.

A generic geometric transformation that unifies a wide range of natural and abstract shapes

To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.

An efficient algorithm for 3D adaptive meshing

Many physical phenomena in science and engineering can be modeled by partial differential equations and solved by means of the finite element method. Such a method uses as computational spatial support a mesh of the domain where the equations are formulated. The ‘mesh quality’ is a key-point for the accuracy of the numerical simulation. One can show that this quality is related to the shape and the size of the mesh elements. In the case where the element sizes are not specified in advance, a quality mesh is a regular mesh (whose elements are almost equilateral). This problem is a particular case of a more general mesh generation problem whose purpose is to construct meshes conforming to a prespecified isotropic size field associated with the computational domain. Such meshes can be seen as ‘unit meshes’ (whose elements are of unit size) in an appropriate non-Euclidean metric. In this case, a quality mesh of the domain is a unit mesh as regular as possible. In this paper, we are concerned with the generation of such a mesh and we propose a method to achieve this goal. First, the boundary of the domain is meshed using an indirect scheme via parametric domains and then the mesh of the three-dimensional (3D) domain is generated. In both cases, an empty mesh is first constructed, then enriched by field points, and finally optimized. The field points are defined following an algebraic or an advancing-front approach and are connected using a generalized Delaunay type method. To show the overall meshing process, we give an example of a 3D domain encountered in a classical computational fluid dynamics problem.

New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity

Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy, that is, stochastic processes with variograms that have direction dependent correlation lengths and direction independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models can be separated into a product of one-dimensional functions oriented along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy, i.e., the ratio(s) of the correlation lengths. This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the covariance tensor. The procedure is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions. In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its implementation with numerical simulations.

Norm-dependent covariance permissibility of weakly homogeneous spatial random fields and its consequences in spatial statistics

Permissibility of a covariance function (in the sense of Bochner) depends on the norm (or metric) that determines spatial distance in several dimensions. A covariance function that is permissible for one norm may not be so for another. We prove that for a certain class of covariances of weakly homogeneous random fields, the spatial distance can be defined only in terms of the Euclidean norm. This class includes commonly used covariance functions. Functions that do not belong to this class may be permissible covariances for some non-Euclidean metric. Thus, a different class of covariances, for which non-Euclidean norms are valid spatial distances, is also discussed. The choice of a coordinate system and associated norm to describe a physical phenomenon depends on the nature of the properties being described. Norm-dependent permissibility analysis has important consequences in spatial statistics applications (e.g., spatial estimation or mapping), in which one is concerned about the validity of covariance functions associated with a physically meaningful norm (Euclidean or non-Euclidean).

Projective spaces of aC *-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection ε=2p−1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.

Subgradient Algorithm on Riemannian Manifolds

The subgradient method is generalized to the context of Riemannian manifolds. The motivation can be seen in non-Euclidean metrics that occur in interior-point methods. In that frame, the natural curves for local steps are the geodesies relative to the specific Riemannian manifold. In this paper, the influence of the sectional curvature of the manifold on the convergence of the method is discussed, as well as the proof of convergence if the sectional curvature is nonnegative.

Quantizing constrained systems

We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order hbar^2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order hbar^2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics, Feynman path integrals, and semiclassically in terms of adiabatic actions.

Task space decomposition for invariant control of constrained robots

An approach, motivated by analytical mechanics and linear algebra methods, is proposed for task space decomposition. The approach relies on the introduction of a new set of kinematic parameters describing the constrained motion of the end-effector, using the analytical forms of material and program constraints. These parameters define a new basis in the end-effector configuration space. A general inner product characterized by the unity matrix is introduced in this basis and in its dual, which gives rise to the definition of a new set of metrics for the end-effector configuration space. The vectors defined in these bases are considered as pseudo-orthogonal. Returning to the original bases of the configuration space, the symmetric matrices for the vanishing bilinear forms can be defined. In this way, the freedom and constraint subspaces can be defined in a rigorous, analytical way. The physical meaning of the resulting metrics is explained. It is shown that the task space decomposition is invariant, although non-Euclidean metrics are being used. To illustrate the application of the methodology and to explain further the properties of the task space decomposition, two examples are presented. In the first example, the robot end-effector tracks a planar surface. A particular definition of the program constraint gives rise to the introduction of skewed bases, which better explains the inherent features of the approach. In the second example, a task of operating a planar joystick, with constrained orientation of the end-effector, is considered. The relevant bases of the task space are translated in this case and it is possible to explain other features of the method.

Strong and weak convex hulls in non-Euclidean metric: theory and application

The notion of convexity is usually defined in the plane supplied with the Euclidean metric. This paper examines what remains if we equip the plane with a distance induced by a norm which is not necessarily the Euclidean one. The basic properties of the geodesic arcs according to these non-Euclidean metrics are stated. In some cases there exists more than one geodesic arc between two points. The two associated notions of convexity, both strong and weak, are the presented. The relationships between the notion of weak convex hull and the limit of closings of increasing size are stated. Finally an application in binary image pattern recognition is described.

Neighborhoods for distance transformations using ordered propagation

This paper presents a fast method for creating distance maps using ordered propagation techniques, where only the propagation front is processed in any stage of the operation. In this paper, we place the emphasis on the size of the neighborhoods used, the number of pixels that must be inspected for each pixel that is processed. We have developed algorithms for both non-Euclidean and Euclidean metrics where only two neighbors need to be inspected per pixel. For Euclidean distance maps, a version that is totally error-free is proposed.

Robust Estimation when More than One Variable per Equation of Condition has Error

It is well known that, in the errors-in-variables estimation problem, the results will have unnecessary asymptotic bias unless the algorithm is properly formulated. Similar difficulties can be expected with robust estimation techniques that are based on extending least squares to a noneuclidean metric. This paper presents an algorithm for robust estimation for the nonlinear model errors-in-variables case. The properties of the estimates produced by the algorithm are investigated, and a numerical example using data on galaxies is given.

Robust estimation when more than one variable per equation of condition has error

SUMMARY It is well known that, in the errors-in-variables estimation problem, the results will have unnecessary asymptotic bias unless the algorithm is properly formulated. Similar difficulties can be expected with robust estimation techniques that are based on extending least squares to a noneuclidean metric. This paper presents an algorithm for robust estimation for the nonlinear model errors-in-variables case. The properties of the estimates produced by the algorithm are investigated, and a numerical example using data on galaxies is given.

Ray Solutions on Surfaces of Revolution

Our concern is the development of techniques for the analysis of wave propagation and general transient response of thin shells to local loading. In general, ray methods are useful for such problems. In the present work, the ray solutions are obtained for a shell of revolution with high prestress, for which the bending stiffness of the shell wall can be neglected. The equation is that for an inhomogeneous, anisotropic medium with a non-Euclidean metric. Of special interest in the present shell problem is the global connectivity of ray paths. For an example the conical shell is considered. The rays for uniform pressure are shown to deviate substantially from geodesics. A ray emitted from a point on the cone may make several spirals around the small end of the cone before moving toward the large end in an almost straight line. The implication is that high focusing of energy can occur from a mild impact on the side of a conical shell.

Some Comments on Non-Euclidean Mental Maps

The Euclidean metric is perhaps the most commonly used and most convenient one for representing mapped phenomena. In this paper we examine the suitability of representing cognitive phenomena via the Euclidean metric. Some general properties of spaces are examined with particular emphasis on the properties of isotropy, incompleteness, and curvature, and a more detailed discussion is undertaken of the suitability of using curved spaces (particularly Reimannian spaces) for the representation of cognitive information. A final discussion is presented on the problems of handling manifolds with folds, warps, and tears; and speculations are made concerning the appropriateness of non-Euclidean metrics for the spatial representation of mental maps.

Computing the relative neighborhood graph in the L1 and L∞ metrics

The Relative Neighborhood Graph (RNG) of a set of n k-dimensional points connects “relatively close” neighbors: two points are connected by an edge if they are at least as close to each other as to any other point. Toussaint recently investigated the properties of the RNG in the Euclidean metric and proposed algorithms for its computation. This note examines one of the open problems listed by Toussaint: the extension of the analysis to non-Euclidean metrics. It is shown that Bentley's range query data structures may be used to improve the speed of the best known RNG algorithm in the L ∞ (for k ⩾ 2) and L 1 (for k = 2) metrics.

Distance measures over polyomino populations

This paper presents a statistical survey of distance measures over n-omino populations for n ≤ 9. Three metrics are considered and compared: one Euclidean and two nonEuclidean metrics—taxicab and graph. The measures are taken at three levels of aggregation: intercellular, aggregate, and total for each polyomino. Statistics are given for each population group, and some cross-metric comparisons are made between the population groups. The paper concludes with a comment on the relevance of this kind of study to architectural and urban design.

Le Progrès de l'Esprit

IN the introductory volume of his work on questions of structure and existence in mathematics, M. Lautman is mainly concerned with Prof. Weyl's distinction between the two types of modern mathematical systems in which the analysis of the infinite is contrasted with the synthetic methods of algebra and group theory. He shows that this distinction need not be conceived of as a fundamental opposition between two irreducible disciplines. On the contrary, among the modern theories of analysis, it is possible to discover points of view which are characteristic of algebra and group theory. For example, the theory of equations with an infinity of variables, the arithmetical theory of algebraic functions, the theory of non-Euclidean metrics, the theory of continuous groups, the analytic theory of numbers and the theory of Pfaffian forms, all occupy an equally intermediary position between algebra and analysis. For in all these, the methods are algebraic and the results extend to analysis; and although there is a certain analogy between contemporary physics and contemporary mathematics, in that both present a spectacle of facts which can be interpreted either by the calculus of the continuous or by that of the discrete, this duality of method, which is the principal source of the difficulties inherent in contemporary physics, is evidence of the profound unity of the mathematical sciences. Le Progres de l'Esprit Par Dr. Albert Lautman. 4: Essai sur l'unite des sciences mathematiques dans leur developpement actuel. Pp. 60. 15 francs. 5: Essai sur les notions de structure et d'existence en mathematiques, 1: Les schemas de structure. Pp. 82. 20 francs. 6: Essai sur les notions de structure et d'existence en mathematiques, 2: Les schemas de genese. Pp. 83–160. 20 francs. (Actualites scientifiques et industrielles, 589–591.) (Paris: Hermann et Cie., 1938.)