Showing papers by "Herbert Edelsbrunner published in 2003"

Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds

Abstract: . We present algorithms for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse—Smale complexes by canceling pairs of critical points in order of increasing persistence.

Morse-smale complexes for piecewise linear 3-manifolds

Abstract: We define the Morse-Smale complex of a Morse function over a 3-manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3-dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.

Loops in reeb graphs of 2-manifolds

Abstract: Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable We also give an algorithm that constructs the Reeb graph in time O(nlogn), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function

Surface Reconstruction by Wrapping Finite Sets in Space

Abstract: Given a finite point set in ℝ3, the surface reconstruction problem asks for a surface that passes through many but not necessarily all points. We describe an unambiguous definition of such a surface in geometric and topological terms,and sketch a fast algorithm for constructing it. Our solution overcomes past limitations to special point distributions and heuristic design decisions.

A multi-resolution data structure for two-dimensional Morse-Smale functions

Abstract: We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multi-resolution representations.

The weighted-volume derivative of a space-filling diagram

Abstract: Computing the volume occupied by individual atoms in macromolecular structures has been the subject of research for several decades. This interest has grown in the recent years, because weighted volumes are widely used in implicit solvent models. Applications of the latter in molecular mechanics simulations require that the derivatives of these weighted volumes be known. In this article, we give a formula for the volume derivative of a molecule modeled as a space-filling diagram made up of balls in motion. The formula is given in terms of the weights, radii, and distances between the centers as well as the sizes of the facets of the power diagram restricted to the space-filling diagram. Special attention is given to the detection and treatment of singularities as well as discontinuities of the derivative.

Morse Complexes for Piecewise Linear 3-Manifolds

Abstract: We define the Morse complex of a Morse function over a 3manifold as the overlay of the stable and unstable manifolds of all critical points. In the generic case, its 3-dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.

Relaxed scheduling in dynamic skin triangulation

Abstract: We introduce relaxed scheduling as a paradigm for mesh maintenance and demonstrate its applicability to triangulating a skin surface in R 3 .

Area and perimeter derivatives of a union of disks

Abstract: We give analytic inclusion-exclusion formulas for the area and perimeter derivatives of a union of finitely many disks in the plane.

Computing linking numbers of a filtration

Abstract: We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of non-trivial tangling in biomolecules, modeled as alpha complexes.

Relaxed Scheduling in Dynamic Skin Triangulation

Abstract: We introduce relaxed scheduling as a paradigm for mesh maintenance and demonstrate its applicability to triangulating a skin surface in $\Rspace^3$.

Area, perimeter and derivatives of a skin curve

Abstract: The body defined by a finite collection of disks is a subset of the plane bounded by a tangent continuous curve, which we call the skin. We give analytic formulas for the area, the perimeter, the area derivative, and the perimeter derivative of the body. Given the filtrations of the Delaunay triangulation and the Voronoi diagram of the disks, all formulas can be evaluated in time proportional to the number of disks.