A fast Voronoi-diagram algorithm with applications to geographical optimization problems

Iri, Masao; Murota, Kazuo; Ohya, Takao

doi:10.1007/bfb0008901

Cited by 83 publications

(46 citation statements)

References 9 publications

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“…Newton method solves the linear system of equations H δ X = −∇F to determine the search step δ X in each iteration, where H is the Hessian of the CVT function F. We note that the components of the gradient ∇F are given by (2)), and the explicit formulae for the second order derivatives, which are the elements of the Hessian, are available in [Iri et al 1984;Asami 1991]. …”

Section: Newton Methods and Quasi-newton Methodsmentioning

confidence: 99%

“…, x N ). Then the second derivatives of the CVT function are given by the following explicit formulae [Iri et al 1984;Asami 1991]:…”

Section: Preconditioned L-bfgs Methodsmentioning

confidence: 99%

“…It is well known that a faster convergence rate can be obtained by using higherorder methods (e.g., Newton method and its variants). However, due to the believed lack of smoothness of F and the apparent complexity in computing Hessian for a large-scale CVT problem, there have been few successful attempts in the literature in this direction [Iri et al 1984;.…”

Section: Variational Point Of Viewmentioning

confidence: 99%

“…Despite the recent result [Cortés et al 2005] that the CVT function F is C 1 , a major impediment to the development of fast Newton-like methods for CVT computation is the lack of understanding about the smoothness of F; for instance, it has long been believed [Iri et al 1984] that F is nonsmooth, that is, it is C 0 but not C 1 . In contrast, we will show that F is almost always C 2 .…”

Section: Variational Point Of Viewmentioning

confidence: 99%

“…This is probably due to the complicated piecewise nature of the CVT energy function and the (wrong) belief that this function is nonsmooth [Iri et al 1984]. One notable previous attempt at accelerating CVT computation is the Lloyd-Newton method .…”

Section: Introductionmentioning

confidence: 99%

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On centroidal voronoi tessellation—energy smoothness and fast computation

et al. 2009

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Centroidal Voronoi tessellation (CVT) is a fundamental geometric structure that finds many applications in computational science and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. Our goal is to develop efficient methods for CVT computation, justify the fast convergence of these methods theoretically and demonstrate their superiority with experimental examples in various cases. Specifically, it is shown that the CVT energy function has C 2 smoothness in convex domains and in most other commonly encountered domains with smooth density, correcting the view in the literature that this function is non-smooth (that is, merely C 0 but not C 1 ). Due to its C 2 smoothness, it is therefore possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We apply quasi-Newton methods to computing CVT and demonstrate their faster convergence than Lloyd's method and their better robustness than the Lloyd-Newton method, a previous attempt at CVT computation acceleration. The application of these results to surface remeshing in computer graphics is also studied.

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Section: Newton Methods and Quasi-newton Methodsmentioning

confidence: 99%

“…, x N ). Then the second derivatives of the CVT function are given by the following explicit formulae [Iri et al 1984;Asami 1991]:…”

Section: Preconditioned L-bfgs Methodsmentioning

confidence: 99%

Section: Variational Point Of Viewmentioning

confidence: 99%

Section: Variational Point Of Viewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On centroidal voronoi tessellation—energy smoothness and fast computation

et al. 2009

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Adaptive vector quantization using reinitialization method

Nishida

Kurogi

Saeki

2005

Electron. Comm. Jpn. Pt. II

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SUMMARYIn vector quantization (VQ), which is useful for the digital coding of large-volume signals such as for image or audio data, many conventional techniques had assumed that the input signal has a time-invariant probability distribution. However, in a real setting, since the stochastic nature of the environment such as input signal or sensor characteristics vary with time, several VQ techniques that can adapt to these kinds of input signal variations have been proposed in recent years. These techniques have problems such as descending to a local solution or slow adaptation. Therefore, in this paper, the authors use a technique called the reinitialization method to propose an adaptive VQ technique that adapts to input signal changes faster than conventional techniques. In this paper, they first show the reinitialization method for solving the local solution problem of the gradient method and then perform an experiment to compare the performance of conventional techniques and the proposed technique concerning VQ for two-dimensional and higher-dimensional vectors. The results verified that the proposed technique had higher adaptive capabilities than conventional techniques relative to temporal variations of the probability distribution of the input signal.

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Voronoi Diagrams in Facility Location

Sugihara¹

2001

Encyclopedia of Optimization

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A fast Voronoi-diagram algorithm with applications to geographical optimization problems

Cited by 83 publications

References 9 publications

On centroidal voronoi tessellation—energy smoothness and fast computation

On centroidal voronoi tessellation—energy smoothness and fast computation

Adaptive vector quantization using reinitialization method

Voronoi Diagrams in Facility Location

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