Octahedral Frames for Feature-Aligned Cross Fields

Abstract

Fig. 1. A variety of feature-aligned cross fields computed using our novel cross field formulation. We present a method for designing smooth cross fields on surfaces that automatically align to sharp features of an underlying geometry. Our approach introduces a novel class of energies based on a representation of cross fields in the spherical harmonic basis. We provide theoretical analysis of these energies in the smooth setting, showing that they penalize deviations from surface creases while otherwise promoting intrinsically smooth fields. We demonstrate the applicability of our method to quad-meshing and include an extensive benchmark comparing our fields to other automatic approaches for generating feature-aligned cross fields on triangle meshes. CCS Concepts: • Computing methodologies → Mesh models; Shape analysis.

Reliable feature-line driven quad-remeshing

“…Also, it is difficult to ensure that this does not impact other requirements, such as the absence of fold-overs in the parametrization. Methods such as [Bommes et al 2013a;Campen et al 2015] introduce constraints to avoid fold-overs, while another class of solutions goes one step back in the pipeline [Diamanti et al 2015;Myles and Zorin 2013;Zhang et al 2020] and re-optimize the cross-field and provide better input for the following parameterization step. Instant Meshes [Jakob et al 2015] formulates an "extrinsic" parametrization approach that "naturally" exhibit a form of sharp-feature alignment.…”

“…We adopt the field propagation method described in [Diamanti et al 2014], but other similar solutions, e.g. [Jakob et al 2015;Zhang et al 2020] could be used interchangeably. The per-face cross-field determines irregular points at vertices.…”

Reliable feature-line driven quad-remeshing

“…Also, it is difficult to ensure that this does not impact other requirements, such as the absence of fold-overs in the parametrization. Methods such as [Bommes et al 2013a;Campen et al 2015] introduce constraints to avoid fold-overs, while another class of solutions goes one step back in the pipeline [Diamanti et al 2015;Myles and Zorin 2013;Zhang et al 2020] and re-optimize the cross-field and provide better input for the following parameterization step. Instant Meshes [Jakob et al 2015] formulates an "extrinsic" parametrization approach that "naturally" exhibit a form of sharp-feature alignment.…”

An optimally convergent smooth blended B-spline construction for semi-structured quadrilateral and hexahedral meshes

“…On quadrilateral meshes with extraordinary vertices the SB-splines are simply given by the weighted basis functions w B (x)B(x) and w Q (x)Q(x). In hexahedral meshes, the extraordinary edges and vertices usually form a connected network as illustrated in Figure 1c; see also relevant work on hexahedral meshing [15][16][17][18][19][20][21][22][23]. That is, the weight function w Q (x) = 1 − w B (x) has a support over the entire network and is decomposed as w Q (x) = i j w P i, j (x) + j w J j (x) into locally supported weight functions.…”

“…For instance, multivariate B-splines are defined only on structured quadrilateral and hexahedral meshes in 2D and 3D, respectively. Most industrial complex geometries cannot be parametrised with a structured mesh so that a limited number of singularities on the surface or inside the volume must be introduced [15][16][17][18][19][20][21][22][23]. These singularities manifest themselves as extraordinary vertices and edges in the mesh, see Figure 1.…”