Multi-Scale Geometry Interpolation
Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mes...
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| Vydáno v: | Computer graphics forum Ročník 29; číslo 2; s. 309 - 318 |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford, UK
Blackwell Publishing Ltd
01.05.2010
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes. |
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| AbstractList | Interpolating vertex positions among triangle meshes with identical vertex-edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non-trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes. [PUBLICATION ABSTRACT] Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes. |
| Author | Hormann, K. Winkler, T. Drieseberg, J. Alexa, M. |
| Author_xml | – sequence: 1 givenname: T. surname: Winkler fullname: Winkler, T. organization: University of Lugano, Switzerland – sequence: 2 givenname: J. surname: Drieseberg fullname: Drieseberg, J. organization: TU Clausthal, Germany – sequence: 3 givenname: M. surname: Alexa fullname: Alexa, M. organization: TU Berlin, Germany – sequence: 4 givenname: K. surname: Hormann fullname: Hormann, K. organization: University of Lugano, Switzerland |
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| References_xml | – reference: Baran I., Vlasic D., Grinspun E., Popović J.: Semantic deformation transfer. ACM Trans. Graph. 28, 3 (Aug. 2009), 36:1-36:6. Proceedings of SIGGRAPH. – reference: Lee J.: Representing rotations and orientations in geometric computing. Computer Graphics and Applications 28, 2 (Mar.Apr. 2008), 75-83. – reference: Williams J. A., Bennamoun M.: Simultaneous registration of multiple point sets using orthonormal matrices. In Proceedings of the International Conference on Acoustics, Speech and Signal Processing 2000 ( Istanbul , Turkey , June 2000), vol. 4, pp. 2199-2202. – reference: Kircher S., Garland M.: Free-form motion processing. ACM Trans. Graph. 27, 2 (Apr. 2008), 12:1-12:13 – reference: Smith R. C., Pawlicki R., Kókai I., Finger J., Vetter T.: Navigating in a shape space of registered models. IEEE Trans. Vis. Comput. Graph. 13, 6 (Nov. 2007), 1552-1559. – reference: Lipman Y., Sorkine O., Levin D., Cohen-Or D.: Linear rotation-invariant coordinates for meshes. ACM Trans. Graph. 24, 3 (July 2005), 479-487. Proceedings of SIGGRAPH. – reference: Chu H.-K., Lee T.-Y.: Multiresolution mean shift clustering algorithm for shape interpolation. IEEE Trans. Vis. Comput. Graph. 15, 5 (Sept. 2009), 853-866. – reference: Pottmann H., Leopoldseder S., Hofer M.: Simultaneous registration of multiple views of a 3D object. In Photogrammetric Computer Vision, Kalliany R., Leberl F., (Eds.), vol. 34, Part 3A of Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. 2002, pp. 265-270. – reference: Sumner R. W., Zwicker M., Gotsman C., Popović J.: Mesh-based inverse kinematics. ACM Trans. Graph. 24, 3 (July 2005), 488-495. Proceedings of SIGGRAPH. – reference: Arun K. S., Huang T. S., Blostein S. D.: Least-squares fitting of two 3-D point sets. IEEE Trans. Pattern Anal. Mach. Intell. 9, 5 (Sept. 1987), 698-700. – reference: Alexa M.: Linear combination of transformations. ACM Trans. Graph. 21, 3 (July 2002), 380-387. Proceedings of SIGGRAPH. – reference: Sumner R. W., Popović J.: Deformation transfer for triangle meshes. ACM Trans. Graph. 23, 3 (Aug. 2004), 399-405. Proceedings of SIGGRAPH. – reference: Kilian M., Mitra N. J., Pottmann H.: Geometric modeling in shape space. ACM Trans. Graph. 26, 3 (July 2007), 64:1-64:8 Proceedings of SIGGRAPH. – reference: Floater M. S.: Mean value coordinates. Comput. Aided Geom. Des. 20, 1 (Mar. 2003), 19-27. – reference: Müller M., Heidelberger B., Teschner M., Gross M.: Meshless deformations based on shape matching. ACM Trans. Graph. 24, 3 (July 2005), 471-478. Proceedings of SIGGRAPH. – reference: Der K. G., Sumner R. W., Popović J.: Inverse kinematics for reduced deformable models. ACM Trans. Graph. 25, 3 (July 2006), 1174-1179. Proceedings of SIGGRAPH. – reference: Hasler N., Stoll C., Sunkel M., Rosenhahn B., Seidel H. P.: A statistical model of human pose and body shape. Comput. Graph. Forum 2, 28 (Mar. 2009). Proceedings of Eurographics 2009. – reference: Besl P. J., McKay N. D.: A method for registration of 3-D shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14, 2 (Feb. 1992), 239-256. – reference: Bergou M., Wardetzky M., Robinson S., Audoly B., Grinspun E.: Discrete elastic rods. ACM Trans. Graph. 27, 3 (Aug. 2008), 63:1-63:12. Proceedings of SIGGRAPH. – reference: Grassia S.: Practical parameterization of rotations using the exponential map. Journal of Graphics Tools 3, 3 (Mar. 1998), 29-48. – volume: 27 start-page: 12:1 issue: 2 year: Apr. 2008 end-page: 12:13 article-title: Free‐form motion processing publication-title: ACM Trans. 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| Snippet | Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex... Interpolating vertex positions among triangle meshes with identical vertex-edge graphs is a fundamental part of many geometric modelling systems. Linear vertex... |
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| SubjectTerms | Affine transformations Computer graphics Curvature Finite element method Geometry I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Hierarchy and geometric transformations Image processing systems Interpolation Matching Mathematical analysis Preserves Systems development Transformations |
| Title | Multi-Scale Geometry Interpolation |
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