Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Tsung-Ming Huang; Wei-Hung Liao; Wen-Wei Lin

doi:10.1515/jnma-2022-0072

40% Rabatt

auf Fachbücher bei De Gruyter Brill *

Artikel

Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Tsung-Ming Huang , Wei-Hung Liao und Wen-Wei Lin

Veröffentlicht/Copyright: 24. August 2023

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Journal of Numerical Mathematics Band 32 Heft 1

Abstract

Here, we extend the finite distortion problem from bounded domains in ℝ² to closed genus-zero surfaces in ℝ³ by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊² by minimizing the total area distortion energy on ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊². The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊². In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

Keywords: spherical equiareal parameterization; stretch energy minimization; R-linear convergence

JEL Classification: 68U05; 65D18; 65𝔼10

Funding statement: This work was partially supported by the Ministry of Science and Technology (MoST), the National Center for Theoretical Sciences, and the ST Yau Center in Taiwan. W.-W. Lin and T.-M. Huang were partially supported by MoST 110-2115-M-A49-004- and 110-2115-M-003-012-MY3, respectively.

References

[1] D. X. Gu’s home page, http://www3.cs.stonybrook.edu/~gu/, 2017.Suche in Google Scholar

[2] GitHub—alecjacobson/common-3d-test-models: Repository containing common 3D test models in original format with original source if known and obj mesh, https://github.com/alecjacobson/common-3d-test-models, 2021.Suche in Google Scholar

[3] L. V. Ahlfors, Lectures on Quasiconformal Mappings, Vol. 38, Amer. Math. Soc., 2006.10.1090/ulect/038Suche in Google Scholar

[4] K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48), Princeton University Press, 2008.10.1515/9781400830114Suche in Google Scholar

[5] K. Astala, T. Iwaniec, G. J. Martin, and J. Onninen, Extremal mappings of finite distortion, Proc. Lond. Math. Soc. 91 (2005), No. 3, 655–702.10.1112/S0024611505015376Suche in Google Scholar

[6] U. Baid, S. Ghodasara, and M. Bilelloetc, The RSNA-ASNR-MICCAI BraTS 2021 Benchmark on Brain Tumor Segmentation and Radiogenomic Classification, 2021.Suche in Google Scholar

[7] S. Bakas, H. Akbari, and A. Sotirasetc, Advancing The Cancer Genome Atlas glioma MRI collections with expert segmentation labels and radiomic features, Sci. Data 4 (2017), No. 1, 170117.10.1038/sdata.2017.117Suche in Google Scholar PubMed PubMed Central

[8] P. T. Choi, K. C. Lam, and L. M. Lui, FLASH: Fast Landmark Aligned Spherical Harmonic parameterization for Genus-0 closed brain surfaces, SIAM J. Imaging Sci. 8 (2015), No. 1, 67–94.10.1137/130950008Suche in Google Scholar

[9] I. Fonseca and W. Gangbo, Local invertibility of Sobolev functions, SIAM J. Math. Anal. 26 (1995), No. 2, 280–304.10.1137/S0036141093257416Suche in Google Scholar

[10] X. Gu, Y.Wang, T. F. Chan, P. M. Thompson, and S.-T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Trans. Med. Imaging 23 (2004), No. 8, 949–958.10.1109/TMI.2004.831226Suche in Google Scholar PubMed

[11] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven, Göttinger Nachr. 1907 (1907), 191–210.Suche in Google Scholar

[12] B. Lévy, S. Petitjean, N. Ray, and J. Maillot, Least squares conformal maps for automatic texture atlas generation, ACM Trans. Graph. 21 (2002), No. 3, 362–371.10.1145/566654.566590Suche in Google Scholar

[13] W. H. Liao, T. M. Huang, W. W. Lin, and M. H. Yueh, Convergence analysis of Dirichlet energy minimization for spherical conformal parameterizations, arXiv:2206.15167, 2022.Suche in Google Scholar

[14] L. M. Lui, X. D. Gu, and S. T. Yau, Convergence of an iterative algorithm for Teichmüller maps via harmonic energy optimization, Math. Comp. 84 (2015), No. 296, 2823–2842.10.1090/S0025-5718-2015-02962-7Suche in Google Scholar

[15] H. Poincaré, Sur l’uniformisation des fonctions analytiques, Acta Math. 31 (1908), 1–63.10.1007/BF02415442Suche in Google Scholar

[16] B. Springborn, P. Schröder, and U. Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph. 27 (2008), No. 3, 1–11.10.1145/1360612.1360676Suche in Google Scholar

[17] K. Su, L. Cui, K. Qian, N. Lei, J. Zhang, M. Zhang, and X. D. Gu, Area-preserving mesh parameterization for poly-annulus surfaces based on optimal mass transportation, Comput. Aided Geom. Design 46 (2016), 76–91.10.1016/j.cagd.2016.05.005Suche in Google Scholar

[18] S. Vodop’yanov and V. Gol’dshtein, Quasiconformal mappings and spaces of functions with generalized first derivatives, Sib. Math. J. 17 (1976), No. 3, 399–411.10.1007/BF00967859Suche in Google Scholar

[19] S. Yoshizawa, A. Belyaev, and H.-P. Seidel, A fast and simple stretch-minimizing mesh parameterization, In: Proceedings Shape Modeling Applications, 2004, IEEE, 2004, pp. 200–208.10.1109/SMI.2004.1314507Suche in Google Scholar

[20] M.-H. Yueh, T. Li, W.-W. Lin, and S.-T. Yau, A novel algorithm for volume-preserving parameterizations of 3-manifolds, SIAM J. Imag. Sci. 12 (2019), No. 2, 1071–1098.10.1137/18M1201184Suche in Google Scholar

[21] M.-H. Yueh, W.-W. Lin, C.-T.Wu, and S.-T. Yau, A novel stretch energy minimization algorithm for equiareal parameterizations, J. Sci. Comput. 78 (2019), No. 3, 1353–1386.10.1007/s10915-018-0822-7Suche in Google Scholar

[22] X. Zhao, Z. Su, X. D. Gu, A. Kaufman, J. Sun, J. Gao, and F. Luo, Area-preservation mapping using optimal mass transport, IEEE Trans. Vis. Comput. Graph. 19 (2013), No. 12, 2838–2847.10.1109/TVCG.2013.135Suche in Google Scholar PubMed

[23] G. Zou, J. Hu, X. D. Gu, and J. Hua, Authalic parameterization of general surfaces using Lie advection, IEEE Trans. Vis. Comput. Graph. 17 (2011), No. 12, 2005–2014.10.1109/TVCG.2011.171Suche in Google Scholar PubMed

Received: 2022-08-18

Revised: 2023-07-31

Accepted: 2023-08-10

Published Online: 2023-08-24

Published in Print: 2024-03-25

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/jnma-2022-0072

Schlagwörter für diesen Artikel

spherical equiareal parameterization; stretch energy minimization; R-linear convergence