research-article

Volumetric parameterization and trivariate b-spline fitting using harmonic functions

Authors:

Mike KirbyAuthors Info & Claims

SPM '08: Proceedings of the 2008 ACM symposium on Solid and physical modeling

Pages 269 - 280

https://doi.org/10.1145/1364901.1364938

Published: 02 June 2008 Publication History

Abstract

We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes T. J. 2005], Imput data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to [Martin and Cohen 2001] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis.

References

[1]

Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. vol. 22, 485--493.

Digital Library

[2]

Arbarello, E., Cornalba, M., Griffiths, P., and Harris, J. 1938. Topics in the theory of algebraic curves.

[3]

Axelsson, O. 1994. Iterative Solution Methods. Cambridge University Press, Cambridge.

Digital Library

[4]

Binford, T. O. 1971. Visual perception by computer. In Proceedings of the IEEE Conference on Systems and Controls.

[5]

Chuang, J.-H., Ahuja, N., Lin, C.-C., Tsai, C.-H., and Chen, C.-H. 2004. A potential-based generalized cylinder representation. Computers&Graphics 28, 6 (December), 907--918.

Digital Library

[6]

Cohen, E., Riesenfeld, R. F., and Elber, G. 2001. Geometric modeling with splines: an introduction. A. K. Peters, Ltd., Natick, MA, USA.

Digital Library

[7]

Davis, P. J. 1979. Circulant Matrices. John Wiley & Sons, Inc.

[8]

Dong, S., Kircher, S., and Garland, M. 2005. Harmonic functions for quadrilateral remeshing of arbitrary manifolds. In Computer Aided Geometric Design.

Digital Library

[9]

Edelsbrunner, H. 1980. Dynamic data structures for orthogonal intersection queries. Technical Report F59, Inst. Informationsverarb., Tech. Univ. Graz.

[10]

Floater, M. S., and Hormann, K. 2005. Surface parameterization: a tutorial and survey. In Advances in multiresolution for geometric modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds. Springer Verlag, 157--186.

[11]

Floater, M. 2003. Mean value coordinates. Computer Aided Design 20, 1, 19--27.

Digital Library

[12]

Freitag, L., and Plassmann, P. 1997. Local optimization-based simplicial mesh untangling and improvement. Technical report. Mathematics and Computer Science Division, Argonne National Laboratory.

[13]

Grimm, C. M., and Hughes, J. F. 1995. Modeling surfaces of arbitrary topology using manifolds. In SIGGRAPH '95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 359--368.

Digital Library

[14]

Gu, X., and Yau, S.-T. 2003. Global conformal surface parameterization. In SGP '03: Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 127--137.

Digital Library

[15]

Hormann, K., and Greiner, G. 2000. Quadrilateral remeshing. In Proceedings of Vision, Modeling, and Visualization 2000, infix, Saarbrücken, Germany, B. Girod, G. Greiner, H. Niemann, and H.-P. Seidel, Eds., 153--162.

[16]

Hua, J., He, Y., and Qin, H. 2004. Multiresolution heterogeneous solid modeling and visualization using trivariate simplex splines. In SM '04: Proceedings of the ninth ACM symposium on Solid modeling and applications, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 47--58.

Digital Library

[17]

Hughes T. J., Cottrell J. A., B. Y. 2005. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 4135--4195.

[18]

Hughes, T. J. R. 2000. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover.

[19]

Jaillet, F., Shariat, B., and Vandorpe, D. 1997. Periodic b-spline surface skinning of anatomic shapes. In 9th Canadian Conference in Computational Geometry.

[20]

Lazarus, F., and Verroust, A. 1999. Level set diagrams of polyhedral objects. In SMA '99: Proceedings of the fifth ACM symposium on Solid modeling and applications, ACM Press, New York, NY, USA, 130--140.

Digital Library

[21]

Li, X., Guo, X., Wang, H., He, Y., Gu, X., and Qin, H. 2007. Harmonic volumetric mapping for solid modeling applications. In Symposium on Solid and Physical Modeling, 109--120.

Digital Library

[22]

Loop, C. 1994. Smooth spline surfaces over irregular meshes. In SIGGRAPH '94: Proceedings of the 21st annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 303--310.

Digital Library

[23]

Lyche, T., and Morken, K. 1988. A data reduction strategy for splines with applications to the approximation of functions and data. IMA J. of Numerical Analysis 8, 2, 185--208.

[24]

Martin, W., and Cohen, E. 2001. Representation and extraction of volumetric attributes using trivariate splines. In Symposium on Solid and Physical Modeling, 234--240.

Digital Library

[25]

Martin, T., Cohen, E., and Kirby, M. 2008. A comparison between isogeometric analysis versus fem applied to a femur. to be submitted.

[26]

Milnor, J. 1963. Morse theory. In Annual of Mathematics Studies, Princeton University Press, Princeton, NJ, vol. 51.

[27]

Ni, X., Garland, M., and Hart, J. C. 2004. Fair morse functions for extracting the topological structure of a surface mesh. In Proc. SIGGRAPH.

Digital Library

[28]

Schreiner, J., Scheidegger, C., Fleishman, S., and Silva, C. 2006. Direct (re)meshing for efficient surface processing. Computer Graphics Forum (Proceedings of Eurographics 2006) 25, 3, 527--536.

[29]

Sederberg T. W., Zheng J., N. A. 2003. T-splines and t-nurccs. In Proc. SIGGRAPH.

Digital Library

[30]

Sheffer, A., Praun, E., and Rose, K. 2006. Mesh parameterization methods and their applications. Foundations and Trends in Computer Graphics and Vision 2, 2.

Digital Library

[31]

Shinagawa, Y., Kunii, T. L., and Kergosien, Y. L. 1991. Surface coding based on morse theory. IEEE Comput. Graph. Appl. 11, 5, 66--78.

Digital Library

[32]

Si, H. 2005. Tetgen: A quality tetrahedral mesh generator and three-dimensional delaunay triangulator.

[33]

Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In ACM/EG Symposium on Geometry Processing.

Digital Library

[34]

Verroust, A., and Lazarus, F. 2000. Extracting skeletal curves from 3d scattered data. The Visual Computer 16, 1, 15--25.

Digital Library

[35]

Wang, Y., Gu, X., Thompson, P. M., and Yau, S.-T. 2004. 3d harmonic mapping and tetrahedral meshing of brain imaging data. In Proc. Medical Imaging Computing and Computer Assisted Intervention (MICCAI), St. Malo, France, Sept. 26--30 2004.

[36]

Zhou, X., and Lu, J. 2005. Nurbs-based galerkin method and application to skeletal muscle modeling. In SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modeling, ACM Press, New York, NY, USA, 71--78.

Digital Library

Cited By

Fang LStals L(2024)Data-based adaptive refinement of finite element thin plate splineJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115975451:COnline publication date: 8-Aug-2024
https://dl.acm.org/doi/10.1016/j.cam.2024.115975
Hinderink SCampen M(2023)Galaxy Maps: Localized Foliations for Bijective Volumetric MappingACM Transactions on Graphics10.1145/359241042:4(1-16)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592410
Cherchi GLivesu M(2023)VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base DomainsComputer Graphics Forum10.1111/cgf.1491542:5Online publication date: 10-Aug-2023
https://doi.org/10.1111/cgf.14915
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Index Terms

Volumetric parameterization and trivariate b-spline fitting using harmonic functions
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models
2. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
    2. Numerical analysis
      1. Computations on polynomials

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cover image ACM Conferences

SPM '08: Proceedings of the 2008 ACM symposium on Solid and physical modeling

June 2008

423 pages

ISBN:9781605581064

DOI:10.1145/1364901

Conference Chairs:
Eric Haines
Autodesk
,
Morgan McGuire
Williams College

Copyright © 2008 ACM.

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Published: 02 June 2008

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SPM08: SPM '08 - ACM Solid and Physical Modeling Symposium

June 2 - 4, 2008

New York, Stony Brook

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Cited By

Fang LStals L(2024)Data-based adaptive refinement of finite element thin plate splineJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115975451:COnline publication date: 8-Aug-2024
https://dl.acm.org/doi/10.1016/j.cam.2024.115975
Hinderink SCampen M(2023)Galaxy Maps: Localized Foliations for Bijective Volumetric MappingACM Transactions on Graphics10.1145/359241042:4(1-16)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592410
Cherchi GLivesu M(2023)VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base DomainsComputer Graphics Forum10.1111/cgf.1491542:5Online publication date: 10-Aug-2023
https://doi.org/10.1111/cgf.14915
Pan MZou RTong WGuo YChen F(2023) -smooth planar parameterization of complex domains for isogeometric analysis Computer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.116330417(116330)Online publication date: Dec-2023
https://doi.org/10.1016/j.cma.2023.116330
Gunpinar ELivesu MAttene M(2023)Exploration of 3D motorcycle complexes from hexahedral meshesComputers & Graphics10.1016/j.cag.2023.06.005114(105-115)Online publication date: Aug-2023
https://doi.org/10.1016/j.cag.2023.06.005
Thomas DEngvall LSchmidt STew KScott M(2022)U-splines: Splines over unstructured meshesComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2022.115515401(115515)Online publication date: Nov-2022
https://doi.org/10.1016/j.cma.2022.115515
Peterka TNashed YGrindeanu IMahadevan VYeh RLenz D(2022)Multivariate Functional Approximation of Scientific DataIn Situ Visualization for Computational Science10.1007/978-3-030-81627-8_17(375-397)Online publication date: 5-May-2022
https://doi.org/10.1007/978-3-030-81627-8_17
Limkilde AEvgrafov AGravesen JMantzaflaris A(2021)Practical isogeometric shape optimization: parametrization by means of regularizationJournal of Computational Design and Engineering10.1093/jcde/qwaa093Online publication date: 14-Jan-2021
https://doi.org/10.1093/jcde/qwaa093
Zheng YChen F(2021)Volumetric Boundary Correspondence for Isogeometric Analysis Based on Unbalanced Optimal TransportComputer-Aided Design10.1016/j.cad.2021.103078140:COnline publication date: 1-Nov-2021
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Updegrove AShadden SWilson N(2019)Construction of Analysis-Suitable Vascular Models Using Axis-Aligned PolycubesJournal of Biomechanical Engineering10.1115/1.4040773141:6(060906)Online publication date: 22-Apr-2019
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