Discretized Riemannian Delaunay Triangulations

Rouxel-Labbé, Mael; Wintraecken, Mathijs; Boissonnat, Jean‐Daniel

doi:10.1016/j.proeng.2016.11.026

Cited by 13 publications

(14 citation statements)

References 26 publications

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“…We used the same arguments as in (64), except for the second inequality which is based on assumption (ii) of Theorem 2.4. Likewise, ifq ∈ E \ Ω then M λ,δ ≤ −U h (p) ≤ C bd (h + 4C Lip δ).…”

Section: F Doubling Of Variablesmentioning

confidence: 99%

“…• A Riemannian metric on a domain of R d is described by a field M of positive definite tensors, and gives rise to the generalized eikonal equation du M −1 = 1. Numerical methods for Riemannian distance computation have applications in geometry processing [64], optics [37], statistics with the Fisher-Rao distance, ... In image processing and segmentation, anisotropic Riemannian metrics are often used to favor paths aligned with tubular structures of interest [38,7,18].…”

Section: Introductionmentioning

confidence: 99%

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Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Mirebeau¹

2019

SIAM J. Numer. Anal.

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Section: F Doubling Of Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Mirebeau¹

2019

SIAM J. Numer. Anal.

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“…Notions related to the canvas will explicitly carry canvas in the name (for example, an edge of C is a canvas edge). In our analysis, we shall assume that the canvas is a dense triangulation, although weaker and more efficient structures can be used (see Section 9 and [23]).…”

Section: Discrete Riemannian Structuresmentioning

confidence: 99%

“…and its dual complex (in black) realized with straight simplices of a two-dimensional domain endowed with a hyperbolic shock-based metric field. Right, the discrete Riemannian Voronoi diagram and the dual complex realized with curved simplices of the "chair" surface endowed with a curvature-based metric field [23].…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams

Boissonnat¹,

Rouxel-Labbé²,

Wintraecken³

2019

SIAM J. Comput.

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The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R 2 and on surfaces embedded in R 3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Ω under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.

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“…An ordinary Delaunay-based triangulation, as used for simple point sets, is obviously unsuitable for our purpose, as the anisotropic extent of the individual Gaussian kernels calls for different metrics for the assessment of local distance relations. One way is to approach this problem as general Delaunay triangulation for a Riemannian manifold, whose local metric is stretched according to the Gaussians' covariance tensors [Budninskiy et al 2016;Rouxel-Labbé et al 2016]. However, such strategies lead to volume meshes, from which the extraction of a surface structure relies on input sampling guarantees we cannot provide in practice, and ends up being even harder than the triangulation problem itself [Amenta et al 2001].…”

Section: Topological Inferencementioning

confidence: 99%

Gaussian-product subdivision surfaces

2019

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Probabilistic distribution models like Gaussian mixtures have shown great potential for improving both the quality and speed of several geometric operators. This is largely due to their ability to model large fuzzy data using only a reduced set of atomic distributions, allowing for large compression rates at minimal information loss. We introduce a new surface model that utilizes these qualities of Gaussian mixtures for the definition and control of a parametric smooth surface. Our approach is based on an enriched mesh data structure, which describes the probability distribution of spatial surface locations around each vertex via a Gaussian covariance matrix. By incorporating this additional covariance information, we show how to define a smooth surface via a nonlinear probabilistic subdivision operator based on products of Gaussians, which is able to capture rich details at fixed control mesh resolution. This entails new applications in surface reconstruction, modeling, and geometric compression.

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Discretized Riemannian Delaunay Triangulations

Cited by 13 publications

References 26 publications

Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms

Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams

Gaussian-product subdivision surfaces

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