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15
Topological inference via meshing
 IN SOCG: PROCEEDINGS OF THE 26TH ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
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Mesh Generation and Geometric Persistent Homology
, 2011
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interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S.
New Bounds on the Size of Optimal Meshes
"... The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents ..."
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The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing φ, then the number of vertices in an optimal mesh will be O(φ d n), where d is the input dimension. We give a new analysis of this integral showing that the output size is only Θ(n + nlogφ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).
A New Approach to OutputSensitive Voronoi Diagrams
"... We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest ..."
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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and nearlinear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures. 1
LinearTime Surface Reconstruction
, 2009
"... By now it is moderately well understood how to take as input a set of points that lie on an unknown manifold, and produce as output a piecewise linear approximation to the manifold. The question I explore here is: how fast can we do it? Previous solutions achieve a sequential runtime of O(n log n) i ..."
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By now it is moderately well understood how to take as input a set of points that lie on an unknown manifold, and produce as output a piecewise linear approximation to the manifold. The question I explore here is: how fast can we do it? Previous solutions achieve a sequential runtime of O(n log n) in low ambient dimensions. I show that if the points are specified in floating point coordinates, we in fact achieve linear work, and we can run in logarithmically many parallel rounds. 1
Donald Sheehy Geometry, Topology, and Data 1 Executive Summary
"... When I lived in Pittsburgh, the local classical music station had a slogan reminding listeners that “all music was once new”. The same holds in computer science where the basic concepts that any undergraduate might be expected to learn were once the cutting edge of theory. It is easy to forget this. ..."
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When I lived in Pittsburgh, the local classical music station had a slogan reminding listeners that “all music was once new”. The same holds in computer science where the basic concepts that any undergraduate might be expected to learn were once the cutting edge of theory. It is easy to forget this. I approach theoretical computer science with the clear vision that I am searching for the algorithms and data structures that will be standard, ubiquitous, even “obvious ” in 10, 15, or 20 years. I do this by focusing on high impact areas of scientific importance where foundational questions remain wide open. My work started mainly in mesh generation, the essential preprocess for numerical solution of partial differential equations by the finite element method. Basic questions in mesh generation have lingered unanswered for decades, despite being the focus of a massive engineering effort distributed across major industries including big money players in automobiles, aviation, and petroleum. I have been working to answer the primary algorithmic questions about the construction of finite element meshes with an eye towards the major pain points for those using meshes in the field. Recently, I have leveraged my expertise in mesh generation, geometric algorithms, and
RuntimeEfficient Meshing for PiecewiseLinear Complexes ∗
, 2010
"... We present a new meshing algorithm to mesh an arbitrary piecewiselinear complex in three dimensions. The algorithm achieves an O(n log ∆ + m) runtime where n, m, and ∆ are the input size, the output size, and spread respectively. This runtime represents the first nontrivial runtime guarantee for t ..."
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We present a new meshing algorithm to mesh an arbitrary piecewiselinear complex in three dimensions. The algorithm achieves an O(n log ∆ + m) runtime where n, m, and ∆ are the input size, the output size, and spread respectively. This runtime represents the first nontrivial runtime guarantee for this class of input. The new algorithm extends prior work on runtimeefficient meshing by allowing the input to have acute input angles (called creases). Features meeting at creases are handled with protective collars. A new procedure is given for creating these collars in an unstructured fashion, without the need for expensive size precomputations as in prior work. Output tetrahedra have quality radiusedge ratios in a region away from the “creases”. Adjacent to creases, output tetrahedra have no large dihedral angles. The collar surface dividing these two regions is represented implicitly using surface reconstruction techniques. This new approach allows the algorithm to run in a single pass. 1
Towards Surface Reconstruction in Linear Time
, 2008
"... In a popular statement of the surface reconstruction problem in low dimensions, we have an input set of points drawn from a manifold (which is unknown), and are asked to compute an approximation to the manifold. Under some assumptions on the point cloud, the Restricted Delaunay Triangulation (RDT) a ..."
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In a popular statement of the surface reconstruction problem in low dimensions, we have an input set of points drawn from a manifold (which is unknown), and are asked to compute an approximation to the manifold. Under some assumptions on the point cloud, the Restricted Delaunay Triangulation (RDT) achieves this. To find the RDT, traditionally we first compute the Voronoi diagram, filter out parts of the diagram, and dualize what remains. The first step can take quadratic space and time in 3D or in higher dimension. Importing two recent results from mesh refinement, I show how to compute a smaller starting point, the Clipped Voronoi Diagram, that takes only linear space and can be found as quickly as sorting the points. If the inputs are in integer or floating point format, this takes only linear time. Filtering the clipped Voronoi diagram should only take linear additional time, but I leave the details for future work. 1 Sampling Assumptions The input is a set of samples S, all of them lying on an unknown manifold M. In order for it to be possible to find M, we need to impose conditions on S. Consider the medial axis of the manifold: all points in space equidistant to at least two points of M. At any point x on M, we can define a local feature size as the distance from x to the medial axis. Define the reach of M to be the minimum local feature size of any point;
Efficient Mesh Generation for Piecewise Linear Complexes
, 2009
"... The mesh generation problem is to output a set of tetrahedra that discretize an input geometry. The input is given as a piecewise linear complex (PLC), a set of points, lines, and polygons to which the output tetrahedra must conform. Additionally, a mesh generation algorithm must make guarantees on ..."
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The mesh generation problem is to output a set of tetrahedra that discretize an input geometry. The input is given as a piecewise linear complex (PLC), a set of points, lines, and polygons to which the output tetrahedra must conform. Additionally, a mesh generation algorithm must make guarantees on the quality and number of output tetrahedra. Downstream applications in scientific computing and visualization necessitate these guarantees on the mesh. Recent advances have led to provably correct algorithms for a number of input classes. Particular difficulties arise when the input contains creases, regions where input segments or polygons meet at acute angles. When the input is without creases, the mesh generation problem is better understood. Algorithms for such inputs exist with nearoptimal runtimes of O(n log ∆+m), where n and m are the size of the input and output, and ∆ is the ratio of largesttosmallest distances in the input geometry. The principle result of this thesis is to extend this result to the general case of piecewise linear complexes with creases. Correct algorithms to handle inputs with creases involve explicitly constructing