Software for C1 Surface Interpolation

Lawson, Charles L.

doi:10.1016/b978-0-12-587260-7.50011-x

Cited by 557 publications

(331 citation statements)

References 15 publications

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“…Among all triangulations of a fixed two-dimensional vertex set, the Delaunay triangulation is optimal by a variety of criteria-maximizing the smallest angle in the triangulation [28], minimizing the largest circumcircle among the triangles [4], and minimizing a property called the roughness of the triangulation [35,37]. A twodimensional CDT shares these same optimality properties, if it is compared with every other constrained triangulation of the same PSLG [4,29].…”

Section: Interpolation Criteria Optimized By Cdtsmentioning

confidence: 94%

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General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Shewchuk

2007

Discrete Comput Geom

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Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely shaped" triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with "internal" boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

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Section: Interpolation Criteria Optimized By Cdtsmentioning

confidence: 94%

“…Delaunay triangulations have virtues when they are used to interpolate multivariate functions [13,28,37,50], including a tendency to favor "round" simplices over "skinny" ones. However, some applications rely on the presence of faces that represent specified discontinuities, as illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Shewchuk

2007

Discrete Comput Geom

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“…At NIST, an adaptive version of the "insertion method" of Lawson [55] has been implemented. For details see Witzgall et al 2004 [86].…”

Section: Triangulation (Tin) Methodsmentioning

confidence: 99%

Construction object identification from LADAR scans:

Gilsinn¹,

Witzgall²,

Cheok³

et al. 2005

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Laser Scanning devices (a.k.a. LADAR for Laser Detection and Ranging) are primarily used in construction projects to capture as-built data for documentation and re-engineering. These systems can rapidly generate unstructured point clouds of scene data within the field of view and line-of-sight of the LADAR. The Construction Metrology and Automation Group (CMAG) at NIST is interested in using these high-resolution scanned data to identify and locate construction objects of interest. The test operation is large-scale pick-and-place of structural steel, where the data from the scans are used to identify the general pose, position and orientation of a target object for initial positioning of a robotic crane. Close-in control of the crane using high framerate flash LADAR is a separate but related area of CMAG research.This initial study describes an experiment in which an I-beam on a concrete floor surface is scanned, and the resulting point cloud data are used to calculate its pose. Multiple scans of two sizes of I-beams were taken with the I-beam placed at various angles relative to the scanner. Two approaches for segmenting potential target objects are described: binning and triangular irregular networks (TINs). Using the method of principal components analysis, the pose of potential target objects is determined. Bounding boxes are then formed around these objects and compared to an ideal bounding box generated from the known geometric specifications of the Ibeam of interest. A measure of best fit is used to determine which scanned object most closely fits the bounding box of the ideal I-beam. A separate laser-based site measurement system (SMS) was used to measure points on the I-beams to form reference data for estimating the closeness of fit of computed pose of the I-beam to measured pose of the I-beam. Three spheres were located in the scan field as a means of registering the scan and SMS axes to one another.

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“…For the generation of the regular grid v(x, z), we first have to know in which triangle a particular gridpoint is located, and then to interpolate within this triangle. The efficient search of the encircling triangle is performed by the walking triangle or trifind algorithm (Lawson 1977;Lee & Schachter 1980;Sambridge et al 1995) without the need to check all triangles. The interpolation within a particular triangle is done by barycentric interpolation well known from computer graphics (e.g.…”

Section: O D E L Pa R a M E T R I Z At I O Nmentioning

confidence: 99%

Bayesian inversion of refraction seismic traveltime data

Ryberg

Haberland

2017

Geophysical Journal International

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S U M M A R YWe apply a Bayesian Markov chain Monte Carlo (McMC) formalism to the inversion of refraction seismic, traveltime data sets to derive 2-D velocity models below linear arrays (i.e. profiles) of sources and seismic receivers. Typical refraction data sets, especially when using the far-offset observations, are known as having experimental geometries which are very poor, highly ill-posed and far from being ideal. As a consequence, the structural resolution quickly degrades with depth. Conventional inversion techniques, based on regularization, potentially suffer from the choice of appropriate inversion parameters (i.e. number and distribution of cells, starting velocity models, damping and smoothing constraints, data noise level, etc.) and only local model space exploration. McMC techniques are used for exhaustive sampling of the model space without the need of prior knowledge (or assumptions) of inversion parameters, resulting in a large number of models fitting the observations. Statistical analysis of these models allows to derive an average (reference) solution and its standard deviation, thus providing uncertainty estimates of the inversion result. The highly non-linear character of the inversion problem, mainly caused by the experiment geometry, does not allow to derive a reference solution and error map by a simply averaging procedure. We present a modified averaging technique, which excludes parts of the prior distribution in the posterior values due to poor ray coverage, thus providing reliable estimates of inversion model properties even in those parts of the models. The model is discretized by a set of Voronoi polygons (with constant slowness cells) or a triangulated mesh (with interpolation within the triangles). Forward traveltime calculations are performed by a fast, finite-difference-based eikonal solver. The method is applied to a data set from a refraction seismic survey from Northern Namibia and compared to conventional tomography. An inversion test for a synthetic data set from a known model is also presented.

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Software for C1 Surface Interpolation

Cited by 557 publications

References 15 publications

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Construction object identification from LADAR scans:

Bayesian inversion of refraction seismic traveltime data

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