2D Manifolds Research Articles

Ebola-challenge and revival of theoretical epidemiology: Why Extrapolations from early phases of epidemics are problematic.

Ebola-challenge and revival of theoretical epidemiology: Why Extrapolations from early phases of epidemics are problematic.

Data-Driven Color Manifolds

Color selection is required in many computer graphics applications, but can be tedious, as 1D or 2D user interfaces are employed to navigate in a 3D color space. Until now the problem was considered a question of designing general color spaces with meaningful (e.g., perceptual) parameters. In this work, we show how color selection usability improves by applying 1D or 2D color manifolds that predict the most likely change of color in a specific context. A typical use-case is manipulating the color of a banana; instead of presenting a 2D+1D RGB, CIE Lab, or HSV widget, our approach presents a simple 1D slider that captures the most likely change for this context. Technically, for each context, we learn a lower-dimensional manifold with varying density from labeled Internet examples. We demonstrate the increase in task performance of color selection in a user study.

Robust multi-modal medical image fusion via anisotropic heat diffusion guided low-rank structural analysis

This paper proposes a novel and robust multi-modal medical image fusion method, which is built upon a novel framework comprising multi-scale image decomposition based on anisotropic heat kernel design, scale-aware salient information extraction based on low-rank analysis, and scale-specific fusion rules. Our framework respects multi-scale structure features, while being robust to complex noise perturbation. First, anisotropic heat kernel is computed by constructing an image pyramid and embedding multi-level image properties into 2D manifolds in a divide-and-conquer way, consequently, multi-scale structure-preserving image decomposition can be accommodated. Second, to extract meaningfully scale-aware salient information, we conduct low-rank analysis over the image layer groups obtained in the first step, and employ the low-rank components to form the scale space of the salient features, wherein the underlying noise can be synchronously decoupled in a natural way. Third, to better fuse the complementary salient information extracted from multi-modal images, we design an S-shaped weighting function to fuse the large-scale layers, and employ the maximum selection principle to handle the small-scale layers. Moreover, we have conducted extensive experiments on MRI and PET/SPECT images. The comprehensive and quantitative comparisons with state-of-the-art methods demonstrate the informativeness, accuracy, robustness, and versatility of our novel approach.

Notes on magnetic curves in 3D semi-Riemannian manifolds

A magnetic field is defined by the property that its divergence is zero in three-dimensional semi-Riemannian manifolds. Each magnetic field generates a magnetic flow whose trajectories are curves $\gamma $, called magnetic curves. In this paper, we investigate the effect of magnetic fields on the moving particle trajectories by variational approach to the magnetic flow associated with the Killing magnetic field on three-dimensional semi-Riemannian manifolds. We then investigate the trajectories of these magnetic fields and give some characterizations and examples of these curves.

Spin-orbit-free topological insulators without time-reversal symmetry.

We explore the 32 crystallographic point groups and identify topological phases of matter with robust surface modes. For n=3,4, and 6 of the C_{nv} groups, we find the first-known 3D topological insulators without spin-orbit coupling, and with surface modes that are protected only by point groups; i.e., the relevant symmetries are purely crystalline and do not include time reversal. To describe these C_{nv} systems, we introduce the notions of (a)a halved mirror chirality, an integer invariant which characterizes half-mirror-planes in the 3D Brillouin zone, and (b)a bent Chern number, the traditional Thouless-Kohmoto-Nightingale-den Nijs invariant generalized to bent 2D manifolds. We find that a Weyl semimetallic phase intermediates two gapped phases with distinct halved chiralities. In addition to electronic systems without spin-orbit coupling, our findings also apply to intrinsically spinless systems such as photonic crystals and ultracold atoms.

Minimal time trajectories for two-level quantum systems with two bounded controls

In this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we treat the time-optimal control problem with techniques of optimal synthesis on 2D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the initial condition.

A new approach for magnetic curves in 3D Riemannian manifolds

A magnetic field is defined by the property that its divergence is zero in a three-dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a three-dimensional oriented Riemann manifold, (M3, g). And then, we investigate the trajectories of the magnetic fields called as N-magnetic and B-magnetic curves.

Efficient decomposition of line drawings of connected manifolds without face identification

This paper presents an algorithm for decomposing complex line drawings which depict connected 3D manifolds into multiple simpler drawings of individual manifolds. The decomposition process has three stages: decomposition at non-manifold vertices, along non-manifold edges and across internal faces. Once non-manifold vertices and/or edges are found, the decomposition can be performed straightforwardly. Thus the major task in this paper is decomposition across internal faces. This has two steps: basic decomposition and repair of incomplete parts. The decomposition process is performed before face identification which is computationally expensive. After decomposition, the time for face finding is much reduced and this, in turn, greatly improves the process for 3D reconstruction, which is the ultimate goal.

STL file format MeshIO class for ITK

This document describes the implementation of an ITK class to support the reading and writing of Meshes in STL file format. The Meshes are assumed to contain 2D manifolds embedded in a 3D space. In practice, it would be desirable to use this class mostly to read and write QuadEdgeMeshes.This paper is accompanied with the source code, input data, parameters and output data that the authors used for validating the algorithm described in this paper. This adheres to the fundamental principle that scientific publications must facilitate reproducibility of the reported results.

Generalized extrinsic distortion and applications

Curvature is a key feature in shape analysis and its estimation on discrete simplicial complexes benefits many geometry processing applications. However, its study has mostly remained focused on 2D manifolds and computationally practical extensions to higher dimensions remain an active area of computer science research. We examine the existing notions of distortion, an analog of curvature in the discrete setting, and classify them into two categories: intrinsic and extrinsic, depending on whether they use the interior or the dihedral angles of the tessellation. We then propose a generalization of extrinsic distortion to ce:italic> D /ce:italic> D and derive a weighting that can be used to compute mean curvature on tessellated hypersurfaces. We analyze the behavior of the operator on 3-manifolds in 4D and compare it to the well-known Laplace–Beltrami operator using ground truth hypersurfaces defined by functions of three variables, and a segmentation application, showing it to behave as well or better while being intuitively simple and easy to implement.

Structures of Periodical Knots and Links as Geometric Models of Complex Surfaces for Designing

Practical modeling of spatial surfaces is more convenient by means of transformation of their flat developments made as topologically connected kinetic structures. Topologically, any surface in 3D space consists of three types of elements: planar facets, linear edges and point vertexes. Planar facets and linear edges can be identifed respectively with the structural units of folding structures and kinematical nets. Here we consider a third possible type of flat transformable structures with vertexes as form-generative units, in which flat developments of surfaces are formed by arranged point sets given by contacting crossing points of the periodic knots and links made of elastic-flexible materials, so that their crossing points have real physical contacts. This new form-generative method can be applied to modeling of both oriented and non-oriented differentiable topological 2D manifolds. The method of form generation based upon the developing properties of periodic structures of knots and links may be applied to many practical fields, including art, design and architecture.

A quantum Mermin–Wagner theorem for quantum rotators on two-dimensional graphs

This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin–Wagner theorem [N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966)]10.1103/PhysRevLett.17.1133. In the model considered here (quantum rotators), the phase space of a single spin is a d-dimensional torus M, and spins (or particles) are attached to sites of a graph \documentclass[12pt]{minimal}\begin{document}$(\Gamma ,\mathcal {E})$\end{document}(Γ,E) satisfying a special bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus a half of the Laplace operator −Δ/2 on M. We assume that the interaction potential is C2-smooth and invariant under the action of a connected Lie group \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G (i.e., a Euclidean space \documentclass[12pt]{minimal}\begin{document}${\mathbb {R}}^{d^{\prime }}$\end{document}Rd′ or a torus M′ of dimension d′ ⩽ d) on M preserving the flat Riemannian metric. A part of our approach is to give a definition (and a construction) of a class of infinite-volume Gibbs states for the systems under consideration (the class \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G). This class contains the so-called limit Gibbs states, with or without boundary conditions. We use ideas and techniques originated from papers [R. L. Dobrushin and S. B. Shlosman, “Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics,” Commun. Math. Phys. 42, 31–40 (1975)10.1007/BF01609432; C.-E. Pfister, “On the symmetry of the Gibbs states in two-dimensional lattice systems,” Commun. Math. Phys. 79, 181–188 (1981)10.1007/BF01942060; J. Fröhlich and C. Pfister, “On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems,” Commun. Math. Phys. 81, 277–298 (1981)10.1007/BF01208901; B. Simon and A. Sokal, “Rigorous entropy-energy arguments,” J. Stat. Phys. 25, 679–694 (1981)10.1007/BF01022362; D. Ioffe, S. Shlosman and Y. Velenik, “2D models of statistical physics with continuous symmetry: The case of singular interactions,” Commun. Math. Phys. 226, 433–454 (2002)]10.1007/s002200200627 in combination with the Feynman–Kac representation, to prove that any state lying in the class \documentclass[12pt]{minimal}\begin{document}$\mathfrak {G}$\end{document}G (defined in the text) is \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not \documentclass[12pt]{minimal}\begin{document}${\tt G}$\end{document}G-invariant. In the next paper, under the same title we establish a similar result for a bosonic model where particles can jump from a vertex i ∈ Γ to one of its neighbors (a generalized Hubbard model).

On a multidimensional integral equation with data supported by low-dimensional analytic manifolds

Abstract. This paper is concerned with a 3D linear integral equation of the first kind which arises when solving nonlinear coefficient inverse problems for second order hyperbolic PDEs. We prove that the integral equation has at most one solution when its right part is given on appropriate analytic 2D and 1D manifolds. As compared to known uniqueness theorems, the dimension of a smooth manifold supporting problem data is decreased here from four to one. As auxiliary results, we give examples of 1D analytic manifolds of uniqueness and examples of minimal countable sets of uniqueness for harmonic functions in 3D domains.

Reconstructing shape boundaries with multimodal constraints

Shape primitives are a valuable input for reconstructing 3D models from point clouds. In this paper we present a method for clipping simple shape primitives at reasonable boundaries. The shape primitives, e.g. planes or cylinders, are 2D manifolds which are automatically detected in unstructured point clouds. Shape boundaries are necessary for generating valid 3D models from multiple shape primitives, because shape primitives possibly have dimensions of infinite extent or they are only partially present in the scene. Hints for reasonable boundaries of shape primitives are indicated by different input sources and constraints. Point clouds and range images provide information where shape primitives coincide with measured surface points. Edge detectors offer cues for surface boundaries in color images. The set of shape primitives is analyzed for constraints such as intersections. Due to an iterative approach, intermediate results provide additional constraints such as coplanar boundary points over multiple shape primitives. We present a framework for extracting and optimizing shape boundaries based on the given input data and multiple constraints. Further, we provide a simple user interface for manually adding constraints in order to improve the results. Our approach generates structurally simple 3D models from shape primitives and point clouds. It is useful for reconstructing scenes containing man-made objects, such as buildings, interior scenes, or engineering objects. The application of multiple constraints enables the reconstruction of proper 3D models despite noisy or incomplete point clouds.

BEHAVIOR PATTERNS IN MULTIPARAMETRIC DYNAMICAL SYSTEMS: LORENZ MODEL

In experimental and theoretical studies of Dynamical Systems, there are usually several parameters that govern the models. Thus, a detailed study of the global parametric phase space is not easy and normally unachievable. In this paper, we show that a careful selection of one straight line (or other 1D manifold) permits us to obtain a global idea of the evolution of the system in some circumstances. We illustrate this fact with the paradigmatic example of the Lorenz model, based on a global study, changing all three parameters. Besides, searching in other regions, for all the detected behavior patterns in one straight line, we have been able to see that missing topological structures of the chaotic attractors may be found on the chaotic-saddles.

Keypoints and Local Descriptors of Scalar Functions on 2D Manifolds

This paper addresses the problem of describing surfaces using local features and descriptors. While methods for the detection of interest points in images and their description based on local image features are very well understood, their extension to discrete manifolds has not been well investigated. We provide a methodological framework for analyzing real-valued functions defined over a 2D manifold, embedded in the 3D Euclidean space, e.g., photometric information, local curvature, etc. Our work is motivated by recent advancements in multiple-camera reconstruction and image-based rendering of 3D objects: there is a growing need for describing object surfaces, matching two surfaces, or tracking them over time. Considering polygonal meshes, we propose a new methodological framework for the scale-space representations of scalar functions defined over such meshes. We propose a local feature detector (MeshDOG) and region descriptor (MeshHOG). Unlike the standard image features, the proposed surface features capture both the local geometry of the underlying manifold and the scale-space differential properties of the real-valued function itself. We provide a thorough experimental evaluation. The repeatability of the feature detector and the robustness of feature descriptor are tested, by applying a large number of deformations to the manifold or to the scalar function.

Geometrically exact theory for contact interactions of 1D manifolds. Algorithmic implementation with various finite element models

The intuitive understanding of contact between bodies is based on the geometry of adjoining bodies. A more sophisticated approach of an advanced analysis including the application of various numerical methods is to take advantage of the geometry of an analyzed object and describe the problem in the best coordinate system. The best coordinate system to describe contact interaction in all its geometrical details is a coordinate system attached to the geometrical features of contacting bodies. This leads to a systematical analysis of geometrical situations leading to contact pairs – surface-to-surface, line-to-surface, point-to-surface, line-to-line, point-to-line. Each contact pair is inherited with a special coordinate system based on its geometrical properties. The current contribution is concentrating on contact between 1D manifolds in 3D space – this is the majority of edge-to-edge, beam-to-beam, cable-to-edge etc. contact cases. The geometrically exact curve-to-curve contact approach is then systematically combined together with various finite element approaches – classical finite elements, isogeometric beam finite elements and also with a new developed solid-beam approach. Examples illustrating the diversity of various finite element combinations e.g. within knot mechanics are shown.

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

In this work we consider the problem of shape reconstruction from an unor- ganized data set which has many important applications in medical imaging, scientific computing, reverse engineering and geometric modelling. The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differen- tial Equation (PDE)-based diffusion model derived by a minimal volume-like variational formulation. The evolution is driven both by the distance from the data set and by the curvature analytically computed by it. The distance function is computed by implicit local interpolants defined in terms of radial basis functions. Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale. The use of a level set method for the numerical computation of the sur- face reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction. Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions. Moreover, we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets. AMS subject classifications: 65D18, 65F10, 65N08

Further classification of 2D integrable mechanical systems with quadratic invariants

Four new integrable classes of mechanical systems on Riemannian 2D manifolds admitting a complementary quadratic invariant are introduced. Those systems have quite rich structure. They involve 11–12 arbitrary parameters that determine the metric of the configuration space and forces with scalar and vector potentials. Interpretations of special versions of them are pointed out as problems of motions of rigid body in a liquid or under action of potential and gyroscopic forces and as motions of a particle on the plane, sphere, ellipsoid, pseudo-sphere and other surfaces.

Symmetry-decomposed reduced transition operators for degenerate atomic states-Their generation and their utility

The p-particle reduced transition operator between the (ML = L, Ms = S) member of one degenerate (L, S) manifold and the (ML′ = V, MS′ = S′) member of a (possibly different) (L′, S′) manifold of eigenstates of a nonrelativistic atomic Hamiltonian contains sufficient information to construct the p-particle reduced transition operator between any member of the first manifold and any member of the second manifold. The procedure for such a construction has been given, and applied computationally to generate the symmetry-decomposed one-particle reduced transition operators among all the ML, MS states of a degenerate (L, S) manifold, from the nondecomposed one-particle reduced density operator of the (ML = L, MS = S) member of this manifold; thereby expediting the calculation of the one-particle reduced density operator of an arbitrary linear superposition of these degenerate states. To illustrate their applicability, these results were used to obtain the densities of the cubic-symmetry-adapted states of an atomic 1D manifold, (in a configuration-interaction approximation) which densities are shown graphically (using the PLOT79 graphics program). The computer program uses the coding of Carlos Bunge's atomic CI program, and can be appended directly to it or to its wavefunction or density-matrix output.