Elsevier

Computer-Aided Design

Volume 127, October 2020, 102862
Computer-Aided Design

Efficient piecewise higher-order parametrization of discrete surfaces with local and global injectivity

https://doi.org/10.1016/j.cad.2020.102862Get rights and content

Abstract

The parametrization of triangle meshes, in particular by means of computing a map onto the plane, is a key operation in computer graphics. Typically, a piecewise linear setting is assumed, i.e., the map is linear per triangle. We present a method for the efficient computation and optimization of piecewise nonlinear parametrizations, with higher-order polynomial maps per triangle. We describe how recent advances in piecewise linear parametrization, in particular efficient second-order optimization based on majorization, as well as practically important constraints, such as local injectivity, global injectivity, and seamlessness, can be generalized to this higher-order regime. Not surprisingly, parametrizations of higher quality, i.e., lower distortion, can be obtained that way, as we demonstrate on a variety of examples.

Introduction

An essential operation in computer graphics and geometry processing applications is the parametrization of discrete surfaces, in particular by means of mapping a triangle mesh onto a planar domain. Prominent use cases are remeshing and texturing. Such parametrizations are most commonly computed, represented, and processed in a piecewise linear manner, with a linear (or affine) map per triangle. By contrast, we describe a method to efficiently generate and optimize low-distortion parametrizations using piecewise nonlinear, higher-order mapping functions.

The increased flexibility provided by this higher-order polynomial representation enables parametrizations of lower distortion and higher visual quality, as demonstrated in Fig. 1. The use of higher-order basis functions on unstructured simplicial meshes has been investigated for a long time [1], and advantages have been demonstrated in various contexts such as simulation [2] or deformation [3] before. We focus here for the first time on the question how and to what extent the problem of injective surface parametrization can benefit in terms of reduced parametric distortion — a key aspect in this domain. This will help estimating whether the benefit is worth the added cost in concrete use cases.

Our method has relations to and shares some technical aspects with algorithms for the generation or optimization of curved planar meshes, which are of interest in the field of finite element computations. At the same time, it has novel features and details, which are of particular relevance in the parametrization context — but may benefit the field of curved mesh generation as well.

Concretely, in this paper we concisely describe how to:

  • practically evaluate popular distortion objectives in the higher-order setting — which, due to the Jacobian not being constant per triangle, requires additional effort;

  • apply the concept of composite majorization [4] in this setting, enabling efficient second-order optimization using the distortion objective’s gradient and Hessian;

  • ensure local injectivity of the parametrization – which in the nonlinear setting is a per-point, not a per-triangle property – using an improved injectivity condition evaluation;

  • ensure, optionally, global injectivity (or bijectivity) by means of a nonlinear version of an ambient space triangulation[5], [6];

  • compute global higher-order parametrizations with cone singularities and seamless transitions, as these are important for purposes such as quadrilateral remeshing;

  • modify meshes by means of edge flips for purposes of mesh optimization while preserving the validity of an underlying higher-order parametrization.

Section snippets

Related work

Many of the underlying techniques we make use of have been proposed and employed before in other contexts, e.g., in piecewise linear parametrization or in higher-order mesh generation, as reviewed below. We adapt, improve, specialize, or generalize these to our problem setting, suitably combine them to form an efficient method, and spell out the necessary details specific to the parametrization problem in a self-contained manner.

Higher-order map

Let t be a triangle in R3 with barycentric coordinates ξ=(ξ0,ξ1,1ξ0ξ1); we will sometimes use ξ2 as a shorthand for 1ξ0ξ1. A map f:tR2 of polynomial degree n over this domain in Bernstein–Bézier form [63] is defined as f(ξ)=u(ξ)v(ξ)=i+j+k=nuijkvijkBijkn(ξ),with n(n+1)2 coefficients (control points) cijk=(uijk,vijk)R2 and triangular Bernstein basis functions Bijkn(ξ)=n!i!j!k!ξ0iξ1jξ2k.

In Fig. 2 such triangular maps are illustrated for the linear case, n=1, and the cubic case, n=3. The

Efficient optimization

Main objective of this paper is to show how nonlinear parametrization maps for triangle meshes can be generated and optimized for low distortion, possibly subject to various constraints. The degrees of freedom are the coefficients cijk per triangle – which in the classical linear case simply are the three vertices’ mapping coordinates (or texture coordinates) per triangle – and we consider ĒSDM as the distortion to be minimized.

This optimization objective is nonlinear, and we have a

Local injectivity

An important property of parametrizations, required by most applications, is local injectivity. A continuous piecewise map is locally injective iff its Jacobian per piece is non-singular, i.e. detJf0, of the same orientation, i.e. orientation preserving (detJf>0) or reversing (detJf<0), and the sum of unsigned image sector angles is 2π around every vertex [65] — though the latter condition is not relevant for all applications. We always assume orientation preservation in the following,

Global injectivity

While local injectivity of maps is the property required, e.g., by certain mesh generation or spline construction techniques, for other applications, e.g., texturing, global injectivity is important.

A locally injective map is globally injective (bijective) if the image of the boundary is not self-intersecting [65]. When starting optimization from an initially globally injective map, global injectivity can thus be guaranteed by ensuring that no boundary self-intersections arise, cf. Section 2.4.

Seamlessness

A parametrization is called seamless if for each edge e, its images f1(e) and f2(e) under the two maps f1, f2 associated with the two adjacent triangles are related by a translation plus rotation by some integer multiple of π2 [12], [73]. Note that this is a weaker requirement than C0 continuity (which requires an identity relation), cf. Section 3.2. This enables the global parametrization of surfaces with arbitrary topology.

To enable the representation of seamless parametrizations with

Results

In order to obtain some understanding of the benefits as well as the cost of using higher-order parametrization, we apply the techniques described to a range of models, in various resolutions, in various constraint scenarios, and with various parameter settings (degree, quadrature). Our optimization is generally initialized using an injective piecewise linear parametrization obtained using Tutte’s embedding onto a disk of equal area. For experiments with seamlessness constraints (Section 8.2),

Efficiency.

The main factor responsible for longer computation times compared to linear parametrization is not the increased number of variables (e.g. 6 or 10 control points for degree 2 or 3, compared to 3 for degree 1) but the increased number of summands entering energy, Jacobian, and Hessian due to quadrature (notice the “M×Q” in Eq. (9)). The setup of the Hessian matrix, rather than the subsequent solve (13), thus easily becomes the bottleneck. Preliminary experiments suggest that significant

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (89)

  • WitherdenF.D. et al.

    On the identification of symmetric quadrature rules for finite element methods

    Comput Math Appl

    (2015)
  • WandzuratS. et al.

    Symmetric quadrature rules on a triangle

    Comput Math Appl

    (2003)
  • GoldmanR.N. et al.

    Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves

    Comput Aided Geom Design

    (1984)
  • HuY. et al.

    Triwild: robust triangulation with curve constraints

    ACM Trans Graph

    (2019)
  • BargteilA.W. et al.

    Animation of deformable bodies with quadratic Bézier finite elements

    ACM Trans Graph

    (2014)
  • ShtengelA. et al.

    Geometric optimization via composite majorization

    ACM Trans Graph

    (2017)
  • JiangZ. et al.

    Simplicial complex augmentation framework for bijective maps

    ACM Trans Graph

    (2017)
  • ZhangE. et al.

    Feature-based surface parameterization and texture mapping

    ACM Trans Graph

    (2005)
  • FloaterM.S. et al.

    Surface parameterization: a tutorial and survey

  • ShefferA. et al.

    Mesh parameterization methods and their applications

    Found Trends Comput Graph Vis

    (2006)
  • KharevychL. et al.

    Discrete conformal mappings via circle patterns

    ACM Trans Graph

    (2006)
  • TongY. et al.

    Designing quadrangulations with discrete harmonic forms

    Symp Geom Proc

    (2006)
  • KälbererF. et al.

    QuadCover: Surface parameterization using branched coverings

    Comput Graph Forum

    (2007)
  • BommesD. et al.

    Mixed-integer quadrangulation

    ACM Trans Graph

    (2009)
  • CampenM. et al.

    Similarity maps and field-guided T-splines: a perfect couple

    ACM Trans Graph

    (2017)
  • AigermanN. et al.

    Seamless surface mappings

    ACM Trans Graph

    (2015)
  • AigermanN. et al.

    Orbifold tutte embeddings

    ACM Trans Graph

    (2015)
  • SawhneyR. et al.

    Boundary first flattening

    ACM Trans Graph

    (2018)
  • LipmanY.

    Bijective mappings of meshes with boundary and the degree in mesh processing

    SIAM J Imaging Sci

    (2014)
  • RabinovichM. et al.

    Scalable locally injective mappings

    ACM Trans Graph

    (2017)
  • SmithJ. et al.

    Bijective parameterization with free boundaries

    ACM Trans Graph

    (2015)
  • LiuL. et al.

    Progressive parameterizations

    ACM Trans Graph

    (2018)
  • ZhuY. et al.

    Blended cured quasi-newton for distortion optimization

    ACM Trans Graph

    (2018)
  • ClaiciS. et al.

    Isometry-aware preconditioning for mesh parameterization

    Comput Graph Forum

    (2017)
  • KovalskyS.Z. et al.

    Accelerated quadratic proxy for geometric optimization

    ACM Trans Graph

    (2016)
  • SmithB. et al.

    Analytic eigensystems for isotropic distortion energies

    ACM Trans Graph

    (2019)
  • KnöppelF. et al.

    Stripe patterns on surfaces

    ACM Trans Graph

    (2015)
  • DeRoseT. et al.

    Subdivision surfaces in character animation

  • HeL. et al.

    Parameterizing subdivision surfaces

    ACM Trans Graph

    (2010)
  • de GoesF. et al.

    Subdivision exterior calculus for geometry processing

    ACM Trans Graph

    (2016)
  • DeyS. et al.

    Curvilinear mesh generation in 3D

  • Remacle J-F, Toulorge T, Lambrechts J, Weill J-C. Robust untangling of curvilinear meshes. In: Proc. 21st Int. Meshing...
  • Gargallo-PeiróA. et al.

    Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes

    Int J Numer Methods Eng

    (2015)
  • RocaX. et al.

    Defining quality measures for high-order planar triangles and curved mesh generation

  • Cited by (13)

    • Error-bounded Image Triangulation

      2023, Computer Graphics Forum
    • A Course on Hex-Mesh Generation and Processing

      2022, Proceedings - SIGGRAPH Asia 2022 Courses
    • High-Order Directional Fields

      2022, ACM Transactions on Graphics
    View all citing articles on Scopus
    View full text