Abstract
We present a method for constructing seamless parametrization for surfaces of any genus that can handle any feasible cone configuration with any type of cones. The mapping is guaranteed to be locally injective, which is due to careful construction of a simple domain boundary polygon. The polygon’s complexity depends on the cones in the field, and it is independent of mesh geometry. The result is a small polygon that can be optimized prior to the interior mapping, which contributes to the robustness of the pipeline.
For a surface of genus >0, non-contractible loops play an important role, and their holonomies significantly affect mapping quality. We enable holonomy prescription, where local injectivity is guaranteed. Our prescription, however, is limited and cannot handle all feasible holonomies due to monotonicity constraints that keep our polygon simple. Yet this work is an important step toward fully solving the holonomy prescription problem.
Supplemental Material
Available for Download
Supplementary material
- 2015. Orbifold tutte embeddings. ACM Trans. Graph. 34, 6 (2015), 190–1.Google ScholarDigital Library .
- 2013a. Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32, 4 (2013), 98.Google ScholarDigital Library .
- 2013b. Quad-mesh generation and processing: A survey. Comput. Graph. Forum 32, 6 (2013), 51–76.Google ScholarDigital Library .
- 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (2009), 77:1–77:10.Google ScholarDigital Library .
- 2015. Quantized global parametrization. ACM Trans. Graph. 34, 6 (2015), 192:1–192:12.Google ScholarDigital Library .
- 2021. Efficient and robust discrete conformal equivalence with boundary. ACM Trans. Graph. 40, 6 (2021), 1–16.Google ScholarDigital Library .
- 2019. Seamless parametrization with arbitrary cones for arbitrary genus. ACM Trans. Graph. 39, 1 (2019), 1–19.Google ScholarDigital Library .
- 2017. Similarity maps and field-guided T-splines: A perfect couple. ACM Trans. Graph. 36, 4 (2017), 1–16.Google ScholarDigital Library .
- 2010. Applied Integer Programming: Modeling and Solution. John Wiley & Sons.Google Scholar .
- 2019. Quadrilateral mesh generation I: Metric based method. Comput. Methods Appl. Mech. Eng. 356 (2019), 652–668. https://scholar.googleusercontent.com/scholar.bib?q=info:N8rgiMNe2Q4J:scholar.google.com/&output=citation&scisdr=Cm2Nms06EJSSn4lAXrI:AGlGAw8AAAAAZH9FRrId00Dy9sGd5JyEp0mDShc&scisig=AGlGAw8AAAAAZH9FRsEUo3ManL1HbgV0YgmHg-g&scisf=4&ct=citation&cd=-1&hl=en.Google ScholarCross Ref .
- 2016. Bounded distortion parametrization in the space of metrics. ACM Trans. Graph. 35, 6 (2016), 215:1–215:16.Google ScholarDigital Library .
- 2013. An efficient computation of handle and tunnel loops via Reeb graphs. ACM Trans. Graphics 32, 4 (2013), 1–10.Google ScholarDigital Library .
- 2015. Integrable PolyVector fields. ACM Trans. Graph. 34, 4 (2015), 38:1–38:12.Google ScholarDigital Library .
- 2018. Quadrangulation through morse-parameterization hybridization. ACM Trans. Graph. 37, 4 (2018), 92:1–92:15.Google ScholarDigital Library .
- 2003. One-to-one piecewise linear mappings over triangulations. Math. Comp. 72, 242 (2003), 685–696.Google ScholarDigital Library .
- 2021. Discrete conformal equivalence of polyhedral surfaces. ACM Trans. Graph. 40, 4 (2021).Google ScholarDigital Library .
- 2006. Discrete one-forms on meshes and applications to 3D mesh parameterization. Comput. Aid. Geom. Des. 23, 2 (2006), 83–112.Google ScholarDigital Library .
- 1969. Planar maps with prescribed types of vertices and faces. Mathematika 16, 1 (1969), 28–36.Google ScholarCross Ref .
- 2018. Gurobi Optimizer Reference Manual. Retrieved from http://www.gurobi.com.Google Scholar .
- 2015. Instant field-aligned meshes. ACM Trans. Graph. 34, 6 (2015), 189–1.Google ScholarDigital Library .
- 1973. A theorem on the structure of cell–decompositions of orientable 2–manifolds. Mathematika 20, 1 (1973), 63–82.Google ScholarCross Ref .
- 2007. QuadCover: Surface parameterization using branched coverings. Comput. Graph. Forum 26, 3 (2007), 375–384.Google ScholarCross Ref .
- 2021. Direct seamless parametrization. ACM Trans. Graph. 40, 1 (2021), 1–14.Google ScholarDigital Library .
- 2022. Seamless parametrization of spheres with controlled singularities. Comput. Graph. Forum. 41, 1 (2022), 57–68.Google Scholar .
- 2014. Strict minimizers for geometric optimization. ACM Transactions on Graphics (TOG) 33, 6 (2014), 185.Google ScholarDigital Library .
- 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4 (2012), 108.Google ScholarDigital Library .
- 2004. Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6, 05 (2004), 765–780.Google ScholarCross Ref .
- 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4, Article
135 (2014), 14 pages.Google ScholarDigital Library . - 2012. Global parametrization by incremental flattening. ACM Trans. Graph. 31, 4 (2012), 109.Google ScholarDigital Library .
- 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4 (2013), 105.Google ScholarDigital Library .
- 2017. Scalable locally injective mappings. ACM Trans. Graph. 36, 4 (2017).Google ScholarDigital Library .
- 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4 (2006), 1460–1485.Google ScholarDigital Library .
- 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1 (2009), 1–11.Google ScholarDigital Library .
- 2008. N-symmetry direction field design. ACM Trans. Graph. 27, 2 (2008), 10.Google ScholarDigital Library .
- 2019. Progressive embedding. ACM Trans. Graph. 38, 4 (2019), 32.Google ScholarDigital Library .
- 2022. Which cross fields can be quadrangulated? Global parameterization from prescribed holonomy signatures. ACM Trans. Graph. 41, 4 (2022), 1–12.Google ScholarDigital Library .
- 2017. Geometric optimization via composite majorization. ACM Trans. Graph. 36, 4 (2017).Google ScholarDigital Library .
- 2015. Bijective parameterization with free boundaries. ACM Trans. Graph. 34, 4 (2015), 70:1–70:9.Google ScholarDigital Library .
- 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3 (2008), 77.Google ScholarDigital Library .
- 2015. Discrete conformal deformation: Algorithm and experiments. SIAM J. Imag. Sci. 8, 3 (2015), 1421– 1456.Google ScholarDigital Library .
- 2011. Simple quad domains for field aligned mesh parametrization. In Proceedings of SIGGRAPH Asia. 1–12.Google ScholarDigital Library .
- 1963. How to draw a graph. Proc. Lond. Math. Soc 13, 3 (1963), 743–768.Google ScholarCross Ref .
- 2016. Directional field synthesis, design, and processing. Comput. Graph. Forum 35, 2 (2016), 545–572.Google ScholarCross Ref .
- 2014. Locally injective parametrization with arbitrary fixed boundaries. TOG 33, 4 (2014), 75.Google ScholarDigital Library .
- 2020. Combinatorial construction of seamless parameter domains. Comput. Graph. Forum, 39 2020. 179–190.Google Scholar .
Index Terms
- Seamless Parametrization with Cone and Partial Loop Control
Recommendations
Seamless Parametrization with Arbitrary Cones for Arbitrary Genus
Seamless global parametrization of surfaces is a key operation in geometry processing, e.g., for high-quality quad mesh generation. A common approach is to prescribe the parametric domain structure, in particular, the locations of parametrization ...
The cut cone III: On the role of triangle facets
AbstractThe cut polytopePn is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph onn nodes. A well known class of facets ofPn arises from the triangle inequalities:xij +xik +xjk ≤ 2 andxij -xik -...
On the directed cut cone and polytope
In this paper we study the directed cut cone and polytope which are the positive hull and convex hull of all directed cut vectors of a complete directed graph, respectively. We present results on the polyhedral structure of these polyhedra. A relation ...
Comments