Computing largest circles separating two sets of segments

A circle $C$ separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect $\Omega(n^2)$ times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on Computational Geometry, 199

Convex Tours of Bounded Curvature

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with $n$ vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible.Comment: 11 pages, 5 figures, abstract presented at European Symposium on Algorithms 199

Quantitative comparisons of forward problems in MEEG.

This document gives comparisons between several methods that solve the forward problem in MEEG by comparing their precision on a 3-layer spherical model. These methods are based on finite elements which either use surfacic meshes with triangles, volumic meshes with tetrahedra, or implicit elements deduced from levelsets

A Complete Analysis of Clarkson's Algorithm for Safe Determinant Evaluation

In this paper, we give a complete and self-contained analysis of Clarkson's algorithm that performs safe and efficient determinant evaluation of an $n\times n$ matrix with integer coefficients that can be expressed with $b$ bits. Clarkson's original paper is generally felt difficult to read. We complete the gaps in his exposition, simplifying the analysis where we can. The number of extra bits needed by this analysis is roughly equivalent to the number of bits needed by Clarkson's analysis. We show that the algorithm performs sign evaluation correctly if $b+O(n)$ bits are available. We give a table of the maximum numbers $b$ of bits available for the entries as a function of $n$, when the arithmetic is performed using a floating point processor complying with the IEEE 754 standard for double precision arithmetic (with $53$ bits available for the mantissa). We also gain some insight into the practical behavior of the algorithm by experimenting. In particular, we provide experimental evidence that the algorithm evaluates correctly the sign of determinants of order up to 15 with $48$-bit coefficients

A generic software design for Delaunay refinement meshing

This paper describes a generic software designed to implement meshing algorithms based on the Delaunay refinement paradigm. Such a meshing algorithm is generally described through a set of rules guiding the refinement of mesh elements. The central item of the software design is a generic class, called a mesher level, able to handle one of the rules guiding the refinement process. Several instantiations of the mesher level class can be stacked and tied together to implement the whole refinement process. As shown in this paper, the design is flexible enough to implement all currently known mesh generation algorithms based on Delaunay refinement. In particular it can be used to generate meshes approximating smooth or piecewise smooth surfaces, as well as to mesh three dimensional domains bounded by such surfaces. It also adapts to algorithms handling small input angles and various refinement criteria. This design highly simplifies the task of implementing Delaunay refinement meshing algorithms. It has been used to implemented several meshing algorithms in the CGAL library

A generic software design for Delaunay refinement meshing

This paper describes a generic software designed to implement meshing algorithms based on the Delaunay refinement paradigm. Such a meshing algorithm is generally described through a set of rules guiding the refinement of mesh elements. The central item of the software design is a generic class, called a mesher level, able to handle one of the rules guiding the refinement process. Several instantiations of the mesher level class can be stacked and tied together to implement the whole refinement process. As shown in this paper, the design is flexible enough to implement all currently known mesh generation algorithms based on Delaunay refinement. In particular it can be used to generate meshes approximating smooth or piecewise smooth surfaces, as well as to mesh three dimensional domains bounded by such surfaces. It also adapts to algorithms handling small input angles and various refinement criteria. This design highly simplifies the task of implementing Delaunay refinement meshing algorithms. It has been used to implemented several meshing algorithms in the CGAL library

High Resolution Surface Reconstruction from Overlapping Multiple-Views

International audienceExtracting a computer model of a real scene from a sequence of views, is one of the most challenging and fundamental problems in computer vision. Stereo vision algorithms allow us to extract from the images a sparse 3D point cloud on the scene surfaces. However, computing an accurate mesh of the scene based on such poor quality data points (noise, sparsity) is very difficult. Here we describe a simple yet original approach that uses both the stereo vision extracted point cloud and the calibrated images. Our method is a three-stage process in which the first stage merges, filters and smoothes the input 3D points. The second stage builds for each calibrated image a triangular depth-map and fuses the set of depth-maps into a triangle soup that minimize violations of size and visibility constraints. Finally, a mesh is computed from the triangle soup using a reconstruction method that combines restricted Delaunay triangulation and Delaunay refinement

Meshing 3D domains bounded by piecewise smooth surfaces

35 pagesThis paper proposes an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm handle multivolume domains defined by surfaces that may be non connected or non manifold. The boundary and subdivision surfaces are assumed to be described by a complex formed by surface patches stitched together along curve segments. The meshing algorithm is a Delaunay refinement and it uses the notion of restricted Delaunay triangulation to approximate the input curve segments and surface patches. The algorithm yields a mesh with good quality tetrahedra and offers a user control on the size of the tetrahedra. The vertices in the final mesh have a restricted Delaunay triangulation to any input feature which is a homeomorphic and accurate approximation of this feature. The algorithm also provides guarantee on the size and shape of the facets approximating the input surface patches. In its current state the algorithm suffers from a severe angular restriction on input constraints. It basically assumes that two linear subspaces that are tangent to non incident and non disjoint input features on a common point form an angle measuring at least 90 degrees