61 research outputs found
Computing Three-dimensional Constrained Delaunay Refinement Using the GPU
We propose the first GPU algorithm for the 3D triangulation refinement
problem. For an input of a piecewise linear complex and a
constant , it produces, by adding Steiner points, a constrained Delaunay
triangulation conforming to and containing tetrahedra mostly of
radius-edge ratios smaller than . Our implementation of the algorithm shows
that it can be an order of magnitude faster than the best CPU algorithm while
using a similar amount of Steiner points to produce triangulations of
comparable quality
VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains
Correspondences between geometric domains (mappings) are ubiquitous in computer graphics and engineering, both for a variety of downstream applications and as core building blocks for higher level algorithms. In particular, mapping a shape to a convex or star-shaped domain with simple geometry is a fundamental module in existing pipelines for mesh generation, solid texturing, generation of shape correspondences, advanced manufacturing etc. For the case of surfaces, computing such a mapping with guarantees of injectivity is a solved problem. Conversely, robust algorithms for the generation of injective volume mappings to simple polytopes are yet to be found, making this a fundamental open problem in volume mesh processing. VOLMAP is a large scale benchmark aimed to support ongoing research in volume mapping algorithms. The dataset contains 4.7K tetrahedral meshes, whose boundary vertices are mapped to a variety of simple domains, either convex or star-shaped. This data constitutes the input for candidate algorithms, which are then required to position interior vertices in the domain to obtain a volume map. Overall, this yields more than 22K alternative test cases. VOLMAP also comprises tools to process this data, analyze the resulting maps, and extend the dataset with new meshes, boundary maps and base domains. This article provides a brief overview of the field, discussing its importance and the lack of effective techniques. We then introduce both the dataset and its major features. An example of comparative analysis between two existing methods is also present
An Open Source Mesh Generation Platform for Biophysical Modeling Using Realistic Cellular Geometries
ABSTRACT Advances in imaging methods such as electron microscopy, tomography, and other modalities are enabling high-resolution reconstructions of cellular and organelle geometries. Such advances pave the way for using these geometries for biophysical and mathematical modeling once these data can be represented as a geometric mesh, which, when carefully conditioned, enables the discretization and solution of partial differential equations. In this study, we outline the steps for a naïve user to approach GAMer 2 , a mesh generation code written in C++ designed to convert structural datasets to realistic geometric meshes, while preserving the underlying shapes. We present two example cases, 1) mesh generation at the subcellular scale as informed by electron tomography, and 2) meshing a protein with structure from x-ray crystallography. We further demonstrate that the meshes generated by GAMer are suitable for use with numerical methods. Together, this collection of libraries and tools simplifies the process of constructing realistic geometric meshes from structural biology data. SIGNIFICANCE As biophysical structure determination methods improve, the rate of new structural data is increasing. New methods that allow the interpretation, analysis, and reuse of such structural information will thus take on commensurate importance. In particular, geometric meshes, such as those commonly used in graphics and mathematics, can enable a myriad of mathematical analysis. In this work, we describe GAMer 2 , a mesh generation library designed for biological datasets. Using GAMer 2 and associated tools PyGAMer and BlendGAMer , biologists can robustly generate computer and algorithm friendly geometric mesh representations informed by structural biology data. We expect that GAMer 2 will be a valuable tool to bring realistic geometries to biophysical models
An Open Source Mesh Generation Platform for Biophysical Modeling Using Realistic Cellular Geometries
Advances in imaging methods such as electron microscopy, tomography and other
modalities are enabling high-resolution reconstructions of cellular and
organelle geometries. Such advances pave the way for using these geometries for
biophysical and mathematical modeling once these data can be represented as a
geometric mesh, which, when carefully conditioned, enables the discretization
and solution of partial differential equations. In this study, we outline the
steps for a na\"ive user to approach GAMer 2, a mesh generation code written in
C++ designed to convert structural datasets to realistic geometric meshes,
while preserving the underlying shapes. We present two example cases, 1) mesh
generation at the subcellular scale as informed by electron tomography, and 2)
meshing a protein with structure from x-ray crystallography. We further
demonstrate that the meshes generated by GAMer are suitable for use with
numerical methods. Together, this collection of libraries and tools simplifies
the process of constructing realistic geometric meshes from structural biology
data.Comment: 6 pages and 4 figures. Supplemental Movie available upon reques
Surface Networks
We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs
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