Efficient piecewise higher-order parametrization of discrete surfaces with local and global injectivity
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 be a triangle in with barycentric coordinates ; we will sometimes use as a shorthand for . A map of polynomial degree over this domain in Bernstein–Bézier form [63] is defined as with coefficients (control points) and triangular Bernstein basis functions
In Fig. 2 such triangular maps are illustrated for the linear case, , and the cubic case, . 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 per triangle – which in the classical linear case simply are the three vertices’ mapping coordinates (or texture coordinates) per triangle – and we consider 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. , of the same orientation, i.e. orientation preserving () or reversing (), and the sum of unsigned image sector angles is 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 , its images and under the two maps , associated with the two adjacent triangles are related by a translation plus rotation by some integer multiple of [12], [73]. Note that this is a weaker requirement than 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 “” 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)
- et al.
Curved, isoparametric, “quadrilateral” elements for finite element analysis
Int J Solids Struct
(1968) - et al.
Generation of curved high-order meshes with optimal quality and geometric accuracy
Proc. 25th Int. Meshing Roundtable
Procedia Eng
(2016) - et al.
Automatic curvilinear quality mesh generation driven by smooth boundary and guaranteed fidelity
Procedia Eng
(2014) - et al.
High-order curvilinear meshing using a thermo-elastic analogy
Comput Aided Des
(2016) - et al.
Curvilinear mesh generation using a variational framework
25th International Meshing Roundtable Special Issue: Advances in Mesh Generation
Comput Aided Des
(2018) - et al.
Adaptive mesh generation for curved domains
Appl Numer Math
(2005) - et al.
Towards curvilinear meshing in 3D: The case of quadratic simplices
Comput Aided Des
(2001) - et al.
Optimizing the geometrical accuracy of curvilinear meshes
J Comput Phys
(2016) - et al.
3D collision detection: a survey
Comput Graph
(2001) Triangular Bernstein–Bézier patches
Comput Aided Geom Design
(1986)
On the identification of symmetric quadrature rules for finite element methods
Comput Math Appl
Symmetric quadrature rules on a triangle
Comput Math Appl
Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves
Comput Aided Geom Design
Triwild: robust triangulation with curve constraints
ACM Trans Graph
Animation of deformable bodies with quadratic Bézier finite elements
ACM Trans Graph
Geometric optimization via composite majorization
ACM Trans Graph
Simplicial complex augmentation framework for bijective maps
ACM Trans Graph
Feature-based surface parameterization and texture mapping
ACM Trans Graph
Surface parameterization: a tutorial and survey
Mesh parameterization methods and their applications
Found Trends Comput Graph Vis
Discrete conformal mappings via circle patterns
ACM Trans Graph
Designing quadrangulations with discrete harmonic forms
Symp Geom Proc
QuadCover: Surface parameterization using branched coverings
Comput Graph Forum
Mixed-integer quadrangulation
ACM Trans Graph
Similarity maps and field-guided T-splines: a perfect couple
ACM Trans Graph
Seamless surface mappings
ACM Trans Graph
Orbifold tutte embeddings
ACM Trans Graph
Boundary first flattening
ACM Trans Graph
Bijective mappings of meshes with boundary and the degree in mesh processing
SIAM J Imaging Sci
Scalable locally injective mappings
ACM Trans Graph
Bijective parameterization with free boundaries
ACM Trans Graph
Progressive parameterizations
ACM Trans Graph
Blended cured quasi-newton for distortion optimization
ACM Trans Graph
Isometry-aware preconditioning for mesh parameterization
Comput Graph Forum
Accelerated quadratic proxy for geometric optimization
ACM Trans Graph
Analytic eigensystems for isotropic distortion energies
ACM Trans Graph
Stripe patterns on surfaces
ACM Trans Graph
Subdivision surfaces in character animation
Parameterizing subdivision surfaces
ACM Trans Graph
Subdivision exterior calculus for geometry processing
ACM Trans Graph
Curvilinear mesh generation in 3D
Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes
Int J Numer Methods Eng
Defining quality measures for high-order planar triangles and curved mesh generation
Cited by (13)
Error-bounded Edge-based Remeshing of High-order Tetrahedral Meshes
2021, CAD Computer Aided Design3D Bézier Guarding: Boundary-Conforming Curved Tetrahedral Meshing
2023, ACM Transactions on GraphicsError-bounded Image Triangulation
2023, Computer Graphics ForumA Course on Hex-Mesh Generation and Processing
2022, Proceedings - SIGGRAPH Asia 2022 CoursesHigh-Order Directional Fields
2022, ACM Transactions on GraphicsGlobally Injective Flattening via a Reduced Harmonic Subspace
2022, ACM Transactions on Graphics