Curved optimal delaunay triangulation

Abstract

Meshes with curvilinear elements hold the appealing promise of enhanced geometric flexibility and higher-order numerical accuracy compared to their commonly-used straight-edge counterparts. However, the generation of curved meshes remains a computationally expensive endeavor with current meshing approaches: high-order parametric elements are notoriously difficult to conform to a given boundary geometry, and enforcing a smooth and non-degenerate Jacobian everywhere brings additional numerical difficulties to the meshing of complex domains. In this paper, we propose an extension of Optimal Delaunay Triangulations (ODT) to curved and graded isotropic meshes. By exploiting a continuum mechanics interpretation of ODT instead of the usual approximation theoretical foundations, we formulate a very robust geometry and topology optimization of Bézier meshes based on a new simple functional promoting isotropic and uniform Jacobians throughout the domain. We demonstrate that our resulting curved meshes can adapt to complex domains with high precision even for a small count of elements thanks to the added flexibility afforded by more control points and higher order basis functions.

Target Image Mask Correction Based on Skeleton Divergence

“…After a new set of directional corner parameters are obtained and utilized, the new vertex coordinate equation of the smoothed boundary curve is established to reconstruct the smoothed boundary curve. Secondly, based on the Delaunay triangulation [16], the boundary discrete curve is utilized to extract the skeleton of the global discrete region, and partition discrete pixel regions using a single connected region surrounded by skeletons and boundaries. Then, taking the skeleton as the divergent center of the internal pixel combination, the discrete pixel edge similarity of the local region is used to guide the internal discrete pixel merging.…”

“…Since Υ is a full-rank square matrix, the solution of the linear Equation 6is unique, and the vertex coordinate set of the curve C * after smoothing is obtained. Using the smoothed edge vector set, the image skeleton is then extracted simultaneously from the source and mask images, and skeleton processing proceeds using Delaunay triangulation [16]. The skeleton is then used as the center for local mask correction and localized area merging with edge guidance, for simultaneous optimization of source and mask images.…”

Tetrahedral meshing in the wild

“…Many algorithms have been proposed to improve the quality of an existing tetrahedral mesh by displacing vertices or changing the local connectivity [A. Freitag and Ollivier-Gooch 1998;Alexa 2019;Alliez et al 2005b;Canann et al 1996Canann et al , 1993Chen and Xu 2004;Faraj et al 2016;Feng et al 2018;Hu et al 2018;Lipman 2012]. Our method relies on the algorithm proposed in Hu et al [2018], which uses a set of local operations to optimize the conformal AMIPS energy [Fu et al 2015;Rabinovich et al 2017].…”

TriWild

“…The Optimal Delaunay Triangulation family choose a metric that harmonizes with Delaunay methods and their duality with Voronoi diagrams [Chen and Xu 2004]. While strategies exist to encourage these methods to huge input domain boundaries [Alliez et al 2005;Feng et al 2018], they require a good, boundary-preserving initial starting point and generally do not support internal features. Our method follows the strategy of Hu et al [2018] to create such an initial starting point for boundary and internal linear or curved features.…”