Practical Anisotropic Geodesy
The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative...
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| Vydané v: | Computer graphics forum Ročník 32; číslo 5; s. 63 - 71 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Oxford, UK
Blackwell Publishing Ltd
01.08.2013
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| Predmet: | |
| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics – which more recently gained interest in several application areas – is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well‐known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short‐Term Vector Dijkstra. This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods. |
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| AbstractList | The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low-level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms - some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics - which more recently gained interest in several application areas - is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well-known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short-Term Vector Dijkstra. This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods. The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics – which more recently gained interest in several application areas – is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well‐known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short‐Term Vector Dijkstra . This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods. The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low-level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms - some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics - which more recently gained interest in several application areas - is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well-known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short-Term Vector Dijkstra. This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods. [PUBLICATION ABSTRACT] |
| Author | Heistermann, Martin Kobbelt, Leif Campen, Marcel |
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| Cites_doi | 10.1007/s11633-007-0008-5 10.1145/1559755.1559761 10.1145/262839.263101 10.1016/j.cagd.2011.06.003 10.1109/TMI.2002.1009386 10.1145/2185520.2185606 10.1007/978-3-540-73273-0_57 10.1007/s00607-007-0249-8 10.1137/0216045 10.1145/98524.98601 10.1007/s00371-009-0362-0 10.1109/TVCG.2012.29 10.1109/CVPR.2009.5206703 10.1007/s11263-010-0331-0 10.1145/2516971.2516977 10.1109/70.62043 10.1137/S0036142901392742 10.1073/pnas.95.15.8431 10.1073/pnas.090060097 10.1145/1141911.1141930 10.1145/1409625.1409626 10.1007/11566465_23 10.1109/9.412624 |
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| Copyright | 2013 The Author(s) Computer Graphics Forum © 2013 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Schmidt R.: Stroke parameterization. Comp. Graph. Forum 32, 2 (2013). – reference: Seong J.-K., Jeong W.-K., Cohen E.: Curvature-based anisotropic geodesic distance computation for parametric and implicit surfaces. The Visual Computer 25, 8 (Apr. 2009), 743-755. – reference: Fisher M., Springborn B., Schröder P., Bobenko A. I.: An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing. Computing 81, 2-3 (2007), 199-213. – reference: Kovacs D., Myles A., Zorin D.: Anisotropic quadrangulation. Comp. Aided Geom. Design 28, 8 (2011), 449-462. – reference: Tang J., Wu G.-S., Zhang F.-Y., Zhang M.-M.: Fast approximate geodesic paths on triangle mesh. International Journal of Automation and Computing 4, 1 (Jan. 2007), 8-13. – reference: Sethian J. A., Vladimirsky A.: Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms. SIAM J. Num. Anal. 41, 1 (2004), 325-363. – reference: Tsitsiklis J. N.: Globally Optimal Trajectories. IEEE Transactions on Automatic Control 40, 9 (1995), 1528-1538. – reference: Parker G. J. M., Wheeler-Kingshott C. A. M., Barker G. J.: Estimating distributed anatomical brain connectivity using fast marching methods and diffusion tensor imaging. IEEE Trans. Med. Imaging 21, 5 (2002), 505-512. – reference: Kimmel R., Sethian J. A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95, 15 (1998), 8431-8435. – reference: Benmansour F., Cohen L. D.: Tubular structure segmentation based on minimal path method and anisotropic enhancement. Int. J. of Computer Vision 92, 2 (2011), 192-210. – reference: Schmidt R., Grimm C., Wyvill B.: Interactive Decal Compositing with Discrete Exponential Maps. ACM Transactions on Graphics 25, 3 (2006), 605-613. – reference: Mitchell J. S., Mount D. M., Papadimitriou C. H.: The Discrete Geodesic Problem. SIAM Journal on Computing 16, 4 (1987), 647-668. – reference: Pichon E., Westin C.-F., Tannenbaum A. R.: A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography. Medical image computing and computer-assisted intervention 8, Pt 1 (Jan. 2005), 180-7. – reference: Xin S.-Q., Wang G.-J.: Improving Chen and Han's algorithm on the discrete geodesic problem. ACM Trans. Graph. 28, 4 (2009), 104:1-104:8. – reference: Surazhsky V., Surazhsky T., Kirsanov D., Gortler S. J., Hoppe H.: Fast exact and approximate geodesics on meshes. In SIGGRAPH (2005), vol. 24. – reference: Sethian J. A., Vladimirsky A.: Fast methods for the Eikonal and related Hamilton-Jacobi equations on unstructured meshes. Proc. Nat. Acad. Sci. 97, 11 (2000), 5699-703. – reference: Yoo S. W., Seong J.-K., Sung M.-H., Shin S. Y., Cohen E.: A triangulation-invariant method for anisotropic geodesic map computation on surface meshes. IEEE Trans. Vis. Comput. Graph. 18, 10 (2012), 1664-1677. – reference: Bougleux S., Peyré G., Cohen L. D.: Anisotropic Geodesics for Perceptual Grouping and Domain Meshing. In ECCV (2008), vol. 5303, pp. 129-142. – reference: Campen M., Bommes D., Kobbelt L.: Dual Loops Meshing: Quality Quad Layouts on Manifolds. ACM Transactions on Graphics 31, 4 (2012), 110:1-110:11. – reference: Lanthier M., Maheshwari A., Sack J.-R.: Shortest Anisotropic Paths on Terrains. ICAL 26 (1999), 523-533. – reference: Weber O., Devir Y. S., Bronstein A. M., Bronstein M. M., Kimmel R.: Parallel algorithms for approximation of distance maps on parametric surfaces. ACM Transactions on Graphics 27, 4 (2008), 104:1-104:16. – volume: 92 start-page: 192 issue: 2 year: 2011 end-page: 210 article-title: Tubular structure segmentation based on minimal path method and anisotropic enhancement publication-title: Int. J. of Computer Vision – volume: 4 start-page: 8 issue: 1 year: Jan. 2007 end-page: 13 article-title: Fast approximate geodesic paths on triangle mesh publication-title: International Journal of Automation and Computing – volume: 25 start-page: 605 issue: 3 year: 2006 end-page: 613 article-title: Interactive Decal Compositing with Discrete Exponential Maps publication-title: ACM Transactions on Graphics – volume: 41 start-page: 325 issue: 1 year: 2004 end-page: 363 article-title: Ordered Upwind Methods for Static Hamilton‐Jacobi Equations: Theory and Algorithms publication-title: SIAM J. Num. Anal. – volume: 16 start-page: 647 issue: 4 year: 1987 end-page: 668 article-title: The Discrete Geodesic Problem publication-title: SIAM Journal on Computing – volume: 26 start-page: 523 year: 1999 end-page: 533 article-title: Shortest Anisotropic Paths on Terrains publication-title: ICAL – start-page: 274 year: 1997 end-page: 283 – volume: 81 start-page: 199 issue: 2–3 year: 2007 end-page: 213 article-title: An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing publication-title: Computing – volume: 31 start-page: 110:1 issue: 4 year: 2012 end-page: 110:11 article-title: Dual Loops Meshing: Quality Quad Layouts on Manifolds publication-title: ACM Transactions on Graphics – volume: 28 start-page: 104:1 issue: 4 year: 2009 end-page: 104:8 article-title: Improving Chen and Han's algorithm on the discrete geodesic problem publication-title: ACM Trans. Graph. – volume: 18 start-page: 1664 issue: 10 year: 2012 end-page: 1677 article-title: A triangulation‐invariant method for anisotropic geodesic map computation on surface meshes publication-title: IEEE Trans. Vis. Comput. Graph. – volume: 21 start-page: 505 issue: 5 year: 2002 end-page: 512 article-title: Estimating distributed anatomical brain connectivity using fast marching methods and diffusion tensor imaging publication-title: IEEE Trans. Med. Imaging – start-page: 687 year: 2007 end-page: 699 – year: 1990 – volume: 32 issue: 2 year: 2013 article-title: Stroke parameterization publication-title: Comp. Graph. Forum – volume: 27 start-page: 104:1 issue: 4 year: 2008 end-page: 104:16 article-title: Parallel algorithms for approximation of distance maps on parametric surfaces publication-title: ACM Transactions on Graphics – volume: 97 start-page: 5699 issue: 11 year: 2000 end-page: 703 article-title: Fast methods for the Eikonal and related Hamilton‐Jacobi equations on unstructured meshes publication-title: Proc. Nat. Acad. Sci. – volume: 95 start-page: 8431 issue: 15 year: 1998 end-page: 8435 article-title: Computing geodesic paths on manifolds publication-title: Proc. Natl. Acad. 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| Snippet | The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and... The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low-level building block in countless Computer Graphics and... |
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| SubjectTerms | Accuracy Algorithms Anisotropy Approximation Computation Computer graphics Computer science Euclidean space I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling Mathematical analysis Studies Vectors (mathematics) |
| Title | Practical Anisotropic Geodesy |
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