SIGGRAPH 2022

Developments in the field of parametrization-based quad mesh generation on surfaces have been impactful over the past decade. In this context, an important advance has been the replacement of error-prone rounding in the generation of integer-grid maps, by robust quantization methods. In parallel, parametrization-based hex mesh generation for volumes has been advanced. In this volumetric context, however, the state-of-the-art still relies on fragile rounding, not rarely producing defective meshes, especially when targeting a coarse mesh resolution. We present a method to robustly quantize volume parametrizations, i.e., to determine guaranteed valid choices of integers for 3D integer-grid maps. Inspired by the 2D case, we base our construction on a non-conforming cell decomposition of the volume, a 3D analogue of a T-mesh. In particular, we leverage the motorcycle complex, a recent generalization of the motorcycle graph, for this purpose. Integer values are expressed in a differential manner on the edges of this complex, enabling the efficient formulation of the conditions required to strictly prevent forcing the map into degeneration. Applying our method in the context of hexahedral meshing, we demonstrate that hexahedral meshes can be generated with significantly improved flexibility.

GMP 2022

Polycube mapping is an attractive approach for the generation of all-hexahedral meshes with a fully regular interior, i.e. free of internal singular edges or vertices. It is based on determining a low distortion map between the input model and a polycube domain, which then pulls back the regular voxel grid to form a hexahedral mesh for the model. Automatically finding an appropriate polycube domain for a given model, however, is a challenging problem. Existing algorithms are either very sensitive to the embedding and orientation of the input model, restricted to only subclasses of possible domains, or depend crucially on some initialization because they rely on a non-convex optimization formulation. This can easily lead to unsatisfactory and unnecessary corners and edges in the polycube structure. We present a novel approach to the problem of finding high-quality polycube domains. It is based on an entirely intrinsic formulation as a mixed integer optimization problem, which can be tackled by solving a series of simple convex problems, each of which can be solved to the global optimum. Experiments demonstrate that our method avoids many of the undesired corners and surface irregularities common to many previous methods.

Eurographics Symposium on Geometry Processing 2021

We present a highly practical, efficient, and versatile approach for computing approximate geodesic distances. The method is designed to operate on triangle meshes and a set of point sources on the surface. We also show extensions for all kinds of geometric input including inconsistent triangle soups and point clouds, as well as other source types, such as lines. The algorithm is based on the propagation of virtual sources and hence easy to implement. We extensively evaluate our method on about 10000 meshes taken from the Thingi10k and the Tet Meshing in the Wild data sets. Our approach clearly outperforms previous approximate methods in terms of runtime efficiency and accuracy. Through careful implementation and cache optimization, we achieve runtimes comparable to other elementary mesh operations (e.g. smoothing, curvature estimation) such that geodesic distances become a "first-class citizen" in the toolbox of geometric operations. Our method can be parallelized and we observe up to 6× speed-up on the CPU and 20× on the GPU. We present a number of mesh processing tasks easily implemented on the basis of fast geodesic distances. The source code of our method will be provided as a C++ library under the MIT license. Note: we are currently in the process of cleaning up and documenting the source code. A basic implementation can already be found in the supplemental material.