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Improving Spatial Coverage while Preserving the Blue Noise of Point Sets
"... We explore the notion of a Wellspaced Bluenoise Distribution (WBD) of points, which combines two desirable properties. First, the point distribution is random, as measured by its spectrum having blue noise. Second, it is wellspaced in the sense that the minimum separation distance between samples ..."
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We explore the notion of a Wellspaced Bluenoise Distribution (WBD) of points, which combines two desirable properties. First, the point distribution is random, as measured by its spectrum having blue noise. Second, it is wellspaced in the sense that the minimum separation distance between samples is large compared to the maximum coverage distance between a domain point and a sample, i.e. its Voronoi cell aspect ratios 2βi are small. It is well known that maximizing one of these properties destroys the other: uniform random points have no aspect ratio bound, and the vertices of an equilateral triangular tiling have no randomness. However, we show that there is a lot of room in the middle to get good values for both. Maximal Poissondisk sampling provides β = 1 and blue noise. We show that a standard optimization technique can improve the wellspacedness while preserving randomness. Given a random point set, our Optβi algorithm iterates over the points, and for each point locally optimizes its Voronoi cell aspect ratio 2βi. It can improve βi to a large fraction of the theoretical bound given by a structured tiling: improving from 1.0 to around 0.8, about halfway to 0.58, while preserving most of the randomness of the original set. In terms of both β and randomness, the output of Optβi compares favorably to alternative point improvement techniques, such as centroidal Voronoi tessellation with a constant density function, which do not target β directly. We demonstrate the usefulness of our output through meshing and filtering applications. An open problem is constructing from scratch a WBD distribution with a guarantee of β < 1. Key words: optimization, Voronoi aspect ratio, disk, triangulation, 1.
Sifted Disks
, 2013
"... We introduce the Sifted Disk technique for locally resampling a point cloud in order to reduce the number of points. Two neighboring points are removed and we attempt to find a single random point that is sufficient to replace them both. The resampling respects the original sizing function; In tha ..."
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We introduce the Sifted Disk technique for locally resampling a point cloud in order to reduce the number of points. Two neighboring points are removed and we attempt to find a single random point that is sufficient to replace them both. The resampling respects the original sizing function; In that sense it is not a coarsening. The angle and edge length guarantees of a Delaunay triangulation of the points are preserved. The sifted point cloud is still suitable for texture synthesis because the Fourier spectrum is largely unchanged. We provide an efficient algorithm, and demonstrate that sifting uniform Maximal Poissondisk Sampling (MPS) and Delaunay Refinement (DR) points reduces the number of points by about 25%, and achieves a density about 1/3 more than the theoretical minimum. We show twodimensional stippling and meshing applications to demonstrate the significance of the concept.
Safe Steiner Points for Delaunay Refinement
, 2008
"... Summary. In mesh refinement for scientific computing or graphics, the input is a description of an input geometry, and the problem is to produce a set of additional “Steiner ” points whose Delaunay triangulation respects the input geometry, and whose points are wellspaced. Ideally, we would like th ..."
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Summary. In mesh refinement for scientific computing or graphics, the input is a description of an input geometry, and the problem is to produce a set of additional “Steiner ” points whose Delaunay triangulation respects the input geometry, and whose points are wellspaced. Ideally, we would like the minimum output size – as few Steiner points as possible. Ruppert showed how to come within a constant factor of optimal in two dimensions, and a number of techniques have been proposed to extend his work to various settings (higher dimension, curved features, fast runtime, parallelism, etc). Each extension needs a different choice of Steiner point, and thus its own tedious proof of the sizing guarantee. In this article, I show how to classify Steiner points as safe or potentially unsafe; any algorithm that chooses only safe points achieves the sizing guarantee. Furthermore, I show that it is easy to classify as safe the Steiner points chosen in most prior work. This work frees future meshing researchers to more easily consider varying the choice of Steiner points to achieve important new properties.
An Efficient Query Structure for Mesh Refinement
 CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY
, 2008
"... We are interested in the following mesh refinement problem: given an input set of points P in R^d, we would like to produce a goodquality triangulation by adding new points in P. Algorithms for mesh refinement are typically incremental: they compute the Delaunay triangulation of the input, and inse ..."
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We are interested in the following mesh refinement problem: given an input set of points P in R^d, we would like to produce a goodquality triangulation by adding new points in P. Algorithms for mesh refinement are typically incremental: they compute the Delaunay triangulation of the input, and insert points one by one. However, retriangulating after each insertion can take linear time. In this work we develop a query structure that maintains the mesh without paying the full cost of retriangulating. Assuming that the meshing algorithm processes badquality elements in increasing order of their size, our structure allows inserting new points and computing a restriction of the Voronoi cell of a point, both in constant time.
Research and Teaching Statement
"... My research areas are design and analysis of algorithms, computational geometry, mesh generation, and their applications. Geometric algorithms play a fundamental role in many areas such as scientic computing, graphics, visualization, computer aided design, computer vision, structural biology, and g ..."
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My research areas are design and analysis of algorithms, computational geometry, mesh generation, and their applications. Geometric algorithms play a fundamental role in many areas such as scientic computing, graphics, visualization, computer aided design, computer vision, structural biology, and geographic information systems. The overall goal of my research is to understand the mathematical structure of problems in these areas; and to design, analyze and implement ecient algorithmic solutions with both theoretical bounds and good performance in practice. This enables me to have an interdisciplinary impact on various scientic applications. Mesh Generation. A principal component of my research is meshes (triangulations), which are subdivisions of geometric domains into small and simple elements. Meshes are essential to numerical simulations. Two of their key requirements are to ensure that the shape of the mesh elements are of good quality and that the size of the mesh is small. Mesh element quality is critical in determining interpolation error in applications and hence is an important factor in the accuracy of the simulations as well as the convergence speed. Mesh size is also critical because between two meshes with the same quality bound, the one with fewer elements is generally preferred as it implies faster processing by the numerical algorithm in use. In collaboration with many researchers from