Computing multidimensional Delaunay tessellations

Devijver, Pierre A.; Dekesel, Michel

doi:10.1016/0167-8655(83)90069-7

Cited by 12 publications

(5 citation statements)

References 8 publications

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“…In 2-D, the method generally accepted as the most efficient is the 'sweepline' algorithm of Fortune (1987). For three or more dimensions, the early methods are ones by Watson (1981), Bowyer (1981) and Devijver & Dekesel (1983). Many of the higher dimensional methods use the 'empty circle' property, which says that the circles (spheres in 3-D) passing through the vertices of Delaunay triangles contain no other nodes.…”

Section: Methods Of Calculating Delaunay Triangulationsmentioning

confidence: 99%

Geophysical parametrization and interpolation of irregular data using natural neighbours

Sambridge

Braun

McQueen

1995

Geophysical Journal International

330

213

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S U M M A R YAn approach is presented for interpolating a property of the Earth (for example temperature or seismic velocity) specified at a series of 'reference' points with arbitrary distribution in two or three dimensions. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into 'Delaunay' triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly interpolated anywhere in the medium using a method known as natural-neighbour interpolation. This method has the following useful properties: (1) the original function values are recovered exactly at the reference points; (2) the interpolation is entirely local (every point is only influenced by its natural-neighbour nodes); and (3) the derivatives of the interpolated function are continuous everywhere except at the reference points. In addition, the ability to handle highly irregular distributions of nodes means that large variations in the scale-lengths of the interpolated function can be represented easily. These properties make the procedure ideally suited for 'gridding' of irregularly spaced geophysical data, or as the basis of parametrization in inverse problems such as seismic tomography.We have extended the theory to produce expressions for the derivatives of the interpolated function. These may be calculated efficiently by modifying an existing algorithm which calculates the interpolated function using only local information. Full details of the theory and numerical algorithms are given. The new theory for function and derivative interpolation has applications to a range of geophysical interpolation and parametrization problems. In addition, it shows much promise when used as the basis of a finite-element procedure for numerical solution of partial differential equations.

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Section: Methods Of Calculating Delaunay Triangulationsmentioning

confidence: 99%

Geophysical parametrization and interpolation of irregular data using natural neighbours

Sambridge

Braun

McQueen

1995

Geophysical Journal International

330

213

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“…As a result, one can then construct the Voronoi cell by using the techniques, as described, for example, in Balinski (1961), Brostow et al (1978), Devijver and Dekesel (1983), Khang and Fujiwara (1989), Ziegler (1995), and Xu (2006). Now we will give two examples to show how the Voronoi cell associated with the ILS estimator (46) The corresponding two-and three-dimensional Voronoi cells are shown in Fig.…”

Section: Representation Of the Integer Ls Estimatormentioning

confidence: 97%

Mixed Integer Linear Models

2013

Handbook of Geomathematics

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Space geodesy has brought profound convenience to our daily life with positioning-based products on one hand and provided a challenging opportunity to contribute fundamentally to mathematics and statistics on the other hand. The use of carrier phase observables has given rise to new observation models we have never encountered in any course/lecture of statistics and/or adjustment theory. This chapter is to provide a tutorial on mixed integer linear models. We first classify real-valued and mixed integer linear models and, then, accordingly define the corresponding conventional and mixed integer least squares (ILS) problems. Since the weighted ILS problem or the closest point problem is NP-hard, we start with suboptimal integer solutions. Reduction and/or decorrelation methods are briefly reviewed for practical applications. Integer unknown parameters are then exactly solved under a general framework of integer programming (aided with decorrelation and/or reduction methods) and represented/estimated from the statistical point of view. As an indispensable and fundamental element of integer least squares estimator, the Voronoi cell is shown to be extremely complex, both computationally and in shape, and has to be bounded with figures of simple shape. As a direct application, we obtain lower and upper probabilistic bounds for the probability with which the integers are correctly estimated. Finally, we briefly discuss an integer hypothesis testing problem.

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“…We may note that a rough upper bound of is a rectangular box and has actually already been given by (12). In this section, we will find new bounds that would best approximate from inside and outside in a certain sense of optimality.…”

Section: Examplesmentioning

confidence: 99%

“…Given distinct points in -dimensional space, a number of algorithms have been proposed to find the corresponding Voronoi cells [7], [8], [12], [14], [63]. However, when the number of points tends to infinity so that they form a lattice, finding the Voronoi cell becomes difficult.…”

Section: A Defining Voronoi Cellsmentioning

confidence: 99%

Voronoi cells, probabilistic bounds, and hypothesis testing in mixed integer linear models

2006

IEEE Trans. Inform. Theory

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Computing multidimensional Delaunay tessellations

Cited by 12 publications

References 8 publications

Geophysical parametrization and interpolation of irregular data using natural neighbours

Geophysical parametrization and interpolation of irregular data using natural neighbours

Mixed Integer Linear Models

Voronoi cells, probabilistic bounds, and hypothesis testing in mixed integer linear models

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