Normal Umbrella: A new primitive for triangulating parametric surfaces

Typical methods for the triangulation of parametric surfaces use a sampling of the parameter space, and the wrong choice of parameterization can spoil a triangulation or even cause the algorithm to fail. We present a new method that uses a local tessellation primitive for almost-uniformly sampling and triangulating a surface, so that its parameterization becomes irrelevant. If sampling density or triangle shape has to be adaptive, the uniform mesh can be used either as an initial coarse mesh for a refinement process, or as a fine mesh to be reduced

Generation of Myocardial Wall Surface Meshes from Segmented MRI

This paper presents a novel method for the generation of myocardial wall surface meshes from segmented 3D MR images, which typically have strongly anisotropic voxels. The method maps a premeshed sphere to the surface of the segmented object. The mapping is defined by the gradient field of the solution of the Laplace equation between the sphere and the surface of the object. The same algorithm is independently used to generate the surface meshes of the epicardium and endocardium of the four cardiac chambers. The generated meshes are smooth despite the strong voxel anisotropy, which is not the case for the marching cubes and related methods. While the proposed method generates more regular mesh triangles than the marching cubes and allows for a complete control of the number of triangles, the generated meshes are still close to the ones obtained by the marching cubes. The method was tested on 3D short-axis cardiac MR images with strongly anisotropic voxels in the long-axis direction. For the five tested subjects, the average in-slice distance between the meshes generated by the proposed method and by the marching cubes was 0.4 mm

Fast extraction of adaptive multiresolution meshes with guaranteed properties from volumetric data

We present a new algorithm for extracting adaptive multiresolution triangle meshes from volume datasets. The algorithm guarantees that the topological genus of the generated mesh is the same as the genus of the surface embedded in the volume dataset at all levels of detail. In addition to this "hard constraint" on the genus of the mesh, the user can choose to specify some number of soft geometric constraints, such as triangle aspect ratio, minimum or maximum total number of vertices, minimum and/or maximum triangle edge lengths, maximum magnitude of various error metrics per triangle or vertex, including maximum curvature (area) error, maximum distance to the surface, and others. The mesh extraction process is fully automatic and does not require manual adjusting of parameters to produce the desired results as long as the user does not specify incompatible constraints. The algorithm robustly handles special topological cases, such as trimmed surfaces (intersections of the surface with the volume boundary), and manifolds with multiple disconnected components (several closed surfaces embedded in the same volume dataset). The meshes may self-intersect at coarse resolutions. However, the self-intersections are corrected automatically as the resolution of the meshes increase. We show several examples of meshes extracted from complex volume datasets

ELISa: A new tool for fast modelling of eclipsing binaries

We present a new, fast, and easy to use tool for modelling light and radial velocity curves of close eclipsing binaries with built-in methods for solving an inverse problem. The main goal of ELISa (Eclipsing binary Learning and Interactive System) is to provide an acceptable compromise between computational speed and precision during the fitting of light curves and radial velocities of eclipsing binaries. The package is entirely written in the Python programming language in a modular fashion, making it easy to install, modify, and run on various operating systems. ELISa implements Roche geometry and the triangulation process to model a surface of the eclipsing binary components, where the surface parameters of each surface element are treated separately. Surface symmetries and approximations based on the similarity between surface geometries were used to reduce the runtime during light curve calculation significantly. ELISa implements the least square trust region reflective algorithm and Markov-chain Monte Carlo optimisation methods to provide the built-in capability to determine parameters of the binary system from photometric observations and radial velocities. The precision and speed of the light curve generator were evaluated using various benchmarks. We conclude that ELISa maintains an acceptable level of accuracy to analyse data from ground-based and space-based observations, and it provides a significant reduction in computational time compared to the current widely used tools for modelling eclipsing binaries.Comment: 15 pages, 18 figure

Regularity criteria for the topology of algebraic curves and surfaces

In this paper, we consider the problem of analysing the shape of an object defined by polynomial equations in a domain. We describe regularity criteria which allow us to determine the topology of the implicit object in a box from information on the boundary of this box. Such criteria are given for planar and space algebraic curves and for algebraic surfaces. These tests are used in subdivision methods in order to produce a polygonal approximation of the algebraic curves or surfaces, even if it contains singular points. We exploit the representation of polynomials in Bernstein basis to check these criteria and to compute the intersection of edges or facets of the box with these curves or surfaces. Our treatment of singularities exploits results from singularity theory such as an explicit Whitney stratification or the local conic structure around singularities. A few examples illustrate the behavior of the algorithms

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

AbstractIn this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface

Fast extraction of adaptive multiresolution meshes with guaranteed properties from volumetric data

We present a new algorithm for extracting adaptive multiresolution triangle meshes from volume datasets. The algorithm guarantees that the topological genus of the generated mesh is the same as the genus of the surface embedded in the volume dataset at all levels of detail. In addition to this "hard constraint" on the genus of the mesh, the user can choose to specify some number of soft geometric constraints, such as triangle aspect ratio, minimum or maximum total number of vertices, minimum and/or maximum triangle edge lengths, maximum magnitude of various error metrics per triangle or vertex, including maximum curvature (area) error, maximum distance to the surface, and others. The mesh extraction process is fully automatic and does not require manual adjusting of parameters to produce the desired results as long as the user does not specify incompatible constraints. The algorithm robustly handles special topological cases, such as trimmed surfaces (intersections of the surface with the volume boundary), and manifolds with multiple disconnected components (several closed surfaces embedded in the same volume dataset). The meshes may self-intersect at coarse resolutions. However, the self-intersections are corrected automatically as the resolution of the meshes increase. We show several examples of meshes extracted from complex volume datasets