Abstract
Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative curve length error under map to assess its quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion.
QIS perfectly equidistributes distortion over the entire domain for the ground-truth test (unit hemisphere flattening) and, when it is not possible, tends to create zones where all cells have the same distortion. Such zones correspond to fragments of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to the de Boor equidistribution principle, which asks for an integral of a certain error indicator function to be the same over each mesh cell.
Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed extensive testing: on more than 10,000 domains QIS was reliably better than the competing methods. In summary, we reliably build 2D and 3D mesh deformations with the smallest known distortion estimates for very stiff problems.
Supplemental Material
Available for Download
Supplementary material
- 2013. Injective and bounded distortion mappings in 3D. ACM Trans. Graph. 32, 4, Article
106 (July 2013), 14 pages.DOI: Google ScholarDigital Library . - 2000. The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm. Springer US, Boston, MA, 197–232.
DOI: Google ScholarCross Ref . - 1969. The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys. 9 (1969), 139–166.Google ScholarCross Ref .
- 1976. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 4 (1976), 337–403.Google ScholarCross Ref .
- 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, Article
77 (July 2009), 10 pages.DOI: Google ScholarDigital Library . - 2003. Bi-Lipschitz parameterization of surfaces. Math. Ann. 327, 1 (2003), 135–169.Google ScholarCross Ref .
- 1856. Sur la construction des cartes géographiques. Bulletin de la classe physico-mathématique de l’Académie Impériale des sciences de Saint-Pétersbourg VIV (1856), 257–261.
Reprinted in P. L. Tchebychef, Œuvres I, Chelsea, New York, 1962, pp. 233–236 and 239–247. Google Scholar . - 2016. Bounded distortion parametrization in the space of metrics. ACM Trans. Graph. 35, 6, Article
215 (Nov. 2016), 16 pages.DOI: Google ScholarDigital Library . - 1982. Sur les lois de comportement en elasticite non-lineaire compressible. C.R. Acad. Sci. Paris Ser.II 295 (1982), 423–426.Google Scholar .
- 1985. Unilateral problems in nonlinear, three-dimensional elasticity. Arch. Rational Mech. Anal. 87 (1985), 319–338.
DOI: Google ScholarCross Ref . - 1988. Mathematical Elasticity: Three-Dimensional Elasticity. Number v. 1 in
Studies in mathematics and its applications . North-Holland.Google Scholar . - 1973. Good Approximation by Splines with Variable Knots. Birkhäuser, Basel, 57–72.
DOI: Google ScholarCross Ref . - 1938. Chebyshev projection for the Soviet Union (in Russian). Geodesist 10 (1938), 4–14. Retrieved from https://elib.rgo.ru/safe-view/123456789/222079/1/MDAwMDAzMDZfR2VvZGV6aXN0IOKEljEwLnBkZg==Google Scholar .
- 2020. Lifting simplices to find injectivity. ACM Trans. Graph. 39, 4, Article
120 (July 2020), 17 pages.DOI: Google ScholarDigital Library . - 1861. LIII. Explanation of a projection by balance of errors for maps applying to a very large extent of the earth’s surface; and comparison of this projection with other projections. London, Edinburgh, Dublin Philos. Mag. J. Sci. 22, 149 (1861), 409–421.
DOI: Google ScholarCross Ref - 2021. Guaranteed globally injective 3D deformation processing. ACM Trans. Graph. (SIGGRAPH) 40, 4, Article
75 (2021).Google ScholarDigital Library . - 2016. Computing inversion-free mappings by simplex assembly. ACM Trans. Graph. 35, 6, Article
216 (Nov. 2016), 12 pages.DOI: Google ScholarDigital Library . - 2015. Computing locally injective mappings by advanced MIPS. ACM Trans. Graph. 34, 4, Article
71 (July 2015), 12 pages.DOI: Google ScholarDigital Library . - 2000. The barrier method for constructing quasi-isometric grids. Comput. Math. Math. Phys. 40 (2000), 1617–1637.Google Scholar .
- 1999. Regularization of the barrier variational method of grid generation. Comput. Math. Math. Phys. 39, 9 (1999), 1426–1440.Google Scholar .
- 2021. Foldover-free maps in 50 lines of code. ACM Trans. Graph. 40, 4 (2021).
DOI: Google ScholarDigital Library . - 2019. Hypoelastic stabilization of variational algorithm for construction of moving deforming meshes. In Optimization and Applications, , , , , , and (Eds.). Springer International Publishing, Cham, 497–511.Google Scholar .
- 2014. Variational method for untangling and optimization of spatial meshes. J. Comput. Appl. Math. 269 (2014), 24–41.
DOI: Google ScholarCross Ref . - 1994. Quasi-isometric parametrization of a curvilinear quadrangle and a metric of constant curvature. Matematicheskie Trudy 26 (1994), 3–19.Google Scholar .
- 1911. Démonstration d’un théorème de Tchébychef généralisé (in French). Journal für die reine und angewandte Mathematik 140 (1911), 247–251.Google ScholarCross Ref .
- 1952. Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bureau Stand. 49 (1952), 409–435.Google ScholarCross Ref .
- 2000. MIPS: An efficient global parametrization method. In Curve and Surface Design. Vanderbilt University Press.Google Scholar .
- 2001. Variational mesh adaptation: Isotropy and equidistribution. J. Comput. Phys. 174, 2 (
Dec. 2001), 903–924.DOI: Google ScholarDigital Library . - 2000. Control of cell shapes in the course of grid generation. Zh. Vychisl. Mat. Mat. Fiz. 40 (Jan. 2000), 1662–1684.Google Scholar .
- 1988. A mechanical model for a new grid generation method in computational fluid dynamics. Comput. Methods Appl. Mech. Eng. 66, 3 (1988), 323–338.Google ScholarDigital Library .
- 2014. Controlling singular values with semidefinite programming. ACM Trans. Graph. 33, 4 (2014).
DOI: Google ScholarDigital Library . - 2015. Large-scale bounded distortion mappings. ACM Trans. Graph. 34, 6 (2015).Google ScholarDigital Library .
- 2014. Strict minimizers for geometric optimization. ACM Trans. Graph. 33, 6, Article
185 (Nov. 2014), 14 pages.DOI: Google ScholarDigital Library . - 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4, Article
108 (July 2012), 13 pages.DOI: Google ScholarDigital Library . - 1969. A problem in cartography. Amer. Math. Month. 76, 10 (1969), 1101–1112. http://www.jstor.org/stable/2317182Google ScholarCross Ref .
- 1957. On ideal locking materials. Trans. Soc. Rheol. 1, 1 (1957), 169–175.Google ScholarCross Ref .
- 2017. Scalable locally injective mappings. ACM Trans. Graph. 36, 2, Article
16 (Apr. 2017), 16 pages.DOI: Google ScholarDigital Library . - 1966. Bounds on moduli of continuity for certain mappings. Siberian Math. J. 7, 5 (1966), 879–886.Google ScholarCross Ref .
- 1996. A variational approach to optimal meshes. Numer. Math. 72, 4 (1996), 523–540.Google ScholarCross Ref .
- 2013. Locally injective mappings. Comput. Graph. Forum 32, 5 (2013).Google ScholarDigital Library .
- 2007. As-Rigid-as-Possible surface modeling. In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP’07). Eurographics Association, Goslar, DEU, 109–116.Google Scholar .
- 2002. Bounded-distortion piecewise mesh parameterization. In Proceedings of the IEEE Conference on Visualization (VIS’02).355–362.
DOI: Google ScholarCross Ref . - 2019. Practical foldover-free volumetric mapping construction. Comput. Graph. Forum 38, 7 (2019), 287–297.
DOI: arXiv:https://onlinelibrary.wiley.com/doi/pdf/ 10.1111/cgf.13837 Google ScholarCross Ref . - 1853. Théorie des mécanismes connus sous le nom de parallélogrammes. Imprimerie de l’Académie impériale des sciences.Google Scholar .
- 2011. Convergence of de Boor’s algorithm for the generation of equidistributing meshes. IMA J. Numer. Anal. 31 (Apr. 2011), 580–596.
DOI: Google ScholarCross Ref . - 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Trans. Math. Software 23, 4 (1997), 550–560.Google ScholarDigital Library .
Index Terms
- In the Quest for Scale-optimal Mappings
Recommendations
Foldover-free maps in 50 lines of code
Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the ...
Injective and bounded distortion mappings in 3D
We introduce an efficient algorithm for producing provably injective mappings of tetrahedral meshes with strict bounds on their tetrahedra aspect-ratio distortion.
The algorithm takes as input a simplicial map (e.g., produced by some common deformation ...
Bounded distortion harmonic mappings in the plane
We present a framework for the computation of harmonic and conformal mappings in the plane with mathematical guarantees that the computed mappings are C∞, locally injective and satisfy strict bounds on the conformal and isometric distortion. Such ...
Comments