Faster Approximations of Shortest Geodesic Paths on Polyhedra Through Adaptive Priority Queue

William Robson Schwartz, Pedro Jussieu Rezende, Helio Pedrini

Abstract

Computing shortest geodesic paths is a crucial problem in several application areas, including robotics, medical imaging, terrain navigation and computational geometry. This type of computation on triangular meshes helps to solve different tasks, such as mesh watermarking, shape classification and mesh parametrization. In this work, a priority queue based on a bucketing structure is applied to speed up graph-based methods that approximates shortest geodesic paths on polyhedra. Initially, the problem is stated, some of its properties are discussed and a review of relevant methods is presented. Finally, we describe the proposed method and show several results and comparisons that confirm its benefits.

References

  1. Agarwal, P., Har-Peled, S., and Karia, M. (2002). Computing Approximate Shortest Paths on Convex Polytopes. Algorithmica, 33:227-242.
  2. Ahuja, R. K., Mehlhorn, K., Orlin, J. B., and Tarjan, R. E. (1990). Faster Algorithms for the Shortest Path Problem. Journal of the ACM, 37:213-223.
  3. Aleksandrov, L., Lanthier, M., Maheshwari, A., and Sack, J.-R. (1998). An e -approximation for Weighted Shortest Paths on Polyhedral Surfaces. In Proc. 6th Scandinavian Workshop on Algorithm Theory - Lecture Notes in Computer Science, volume 1432, pages 11- 22.
  4. Aleksandrov, L., Maheshwari, A., and Sack, J.-R. (2005). Determining Approximate Shortest Paths on Weighted Polyhedral Surfaces. Journal of the ACM, 52(1):25-53.
  5. Barbehenn, M. (1998). A Note on the Complexity of Dijkstra's Algorithm for Graphs with Weighted Vertices. IEEE Transactions on Computers, 47:263.
  6. Bose, P., Maheshwari, A., Shu, C., and Wuhrer, S. (2011). A Survey of Geodesic Paths on 3D Surfaces. Computational Geometry, 44(9):486-498.
  7. Bronstein, A., Bronstein, M., Kimmel, R., Mahmoudi, M., and Sapiro, G. (2010). A Gromov-Hausdorff Framework with Diffusion Geometry for TopologicallyRobust Non-rigid Shape Matching. International Journal of Computer Vision, 89:266-286.
  8. Chen, J. and Han, Y. (1990). Shortest Paths on a Polyhedron. In Proc. 6th Annual Symposium on Computational Geometry, pages 360-369.
  9. Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. (2001). Introduction to Algorithms. MIT Press and McGraw-Hill, second edition.
  10. Dijkstra, E. W. (1959). A note on two problems in connection with graphs. Numerische Mathematik, 1:269- 271.
  11. Hamza, A. B. and Krim, H. (2003). Geodesic Object Representation and Recognition. In Proc. International Conference on Discrete Geometry for Computer Imagery, volume 2, pages 378-387.
  12. Hershberger, J. and Suri, S. (1993). Efficient Computation of Euclidean Shortest Paths in the Plane. In Proc. 34th Annual Symposium on Foundations of Computer Science, pages 508-517, Palo Alto, CA, USA.
  13. Hilaga, M., Shinagawa, Y., Kohmura, T., and Kunit, T. L. (2001). Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes. In Proc. Conference on Computer Graphics (SIGGRAPH), pages 203-212.
  14. Hwang, Y. K. and Ahuja, N. (1992). Gross Motion Planning: A Survey. ACM Computer Survey, 24(3):219- 291.
  15. Kamousi, P., Lazard, S., Maheshwari, A., and Wuhrer, S. (2013). Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph. arXiv.org.
  16. Kanai, T. and Suzuki, H. (2001). Approximate Shortest Path on a Polyhedral Surface and its Applications. Computer-Aided Design, 33(11):801-811.
  17. Kapoor, S. (1999). Efficient Computation of Geodesic Shortest Paths. In Proc. 31st Annual ACM Symposium on Theory of Computing, pages 770-779, Atlanta-GA, USA.
  18. Kimmel, R. and Sethian, J. A. (1998). Computing Geodesic Paths on Manifolds. Proc. National Academy of Sciences of the United States of America, 95(15):8431- 8435.
  19. Li, Z., Jin, Y., Jin, X., and Ma, L. (2012). Approximate Straightest Path Computation and its Application in Parameterization. The Visual Computer, 28(1):63-74.
  20. Lozano-Prez, T. and Wesley, M. A. (1979). An Algorithm for Planning Collision-Free Paths among Polyhedral Obstacles. Communications of the ACM, 22(10):560- 570.
  21. Mitchell, J. S. B., Mount, D. M., and Papadimitriou, C. H. (1987). The Discrete Geodesic Problem. SIAM Journal on Computing, 16(4):647-668.
  22. Mount, D. (1985). On Finding Shortest Paths in Convex Polyhedra. Technical Report 1495, University of Maryland, Baltimore, USA.
  23. Mount, D. (1986). Storing the Subdivision of a Polyhedral Surface. In Second Annual Symposium on Computational Geometry, pages 150-158.
  24. Novotni, M. and Klein, R. (2002). Computing Geodesic Distances on Triangular Meshes. In Proc. 10th International Conference in Central Europe on Computer Graphics, pages 341-347.
  25. Onclinx, V., Lee, J., Wertz, V., and Verleysen, M. (2010). Dimensionality Reduction by Rank Preservation. In International Joint Conference on Neural Networks, pages 1 -8.
  26. O'Rourke, J., Suri, S., and Booth, H. (1985). Shortest Paths on Polyhedral Surfaces. In Proc. 2nd Symposium of Theoretical Aspects of Computer Science, pages 243- 254.
  27. Papadimitriou, C. H. (1985). An Algorithm for ShortestPath Motion in Three Dimensions. Information Processing Letters, 20:259-263.
  28. Peyré, G., Péchaud, M., Keriven, R., and Cohen, L. D. (2010). Geodesic Methods in Computer Vision and Graphics. Foundations and Trends in Computer Graphics and Vision, 5:197-397.
  29. Rabin, J., Peyré, G., and Cohen, L. D. (2010). Geodesic Shape Retrieval via Optimal Mass Transport. In Proceedings of the 11th European conference on Computer Vision: Part V, pages 771-784, Heraklion, Crete, Greece. Springer-Verlag.
  30. Sharir, M. and Schorr, A. (1986). On Shortest Paths in Polyhedral Spaces. SIAM Journal on Computing, 15(1):193-215.
  31. Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S. J., and Hoppe, H. (2005). Fast Exact and Approximate Geodesics on Meshes. ACM Transactions on Graphics, 24(3):553-560.
  32. Tenenbaum, J. B., de Silva, V., and Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319-2323.
  33. Varadarajan, K. R. and Agarwal, P. K. (2000). Approximating Shortest Paths on a Nonconvex Polyhedron. SIAM Journal on Computing, 30(4):1321-1340.
  34. Wang, K., Lavoue, G., Denis, F., and Baskurt, A. (2008). A Comprehensive Survey on Three-Dimensional Mesh Watermarking. IEEE Transactions on Multimedia, 10(8):1513-1527.
  35. Ying, X., Wang, X., and He, Y. (2013). Saddle Vertex Graph (SVG): A Novel Solution to the Discrete Geodesic Problem. ACM Transactions on Graphics, 32(6):170:1-170:12.
  36. Zigelman, G., Kimmel, R., and Kiryati, N. (2002). Texture Mapping Using Surface Flattening via Multidimensional Scaling. IEEE Transactions on Visualization and Computer Graphics, 8(2):198-207.
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Paper Citation


in Harvard Style

Schwartz W., Rezende P. and Pedrini H. (2015). Faster Approximations of Shortest Geodesic Paths on Polyhedra Through Adaptive Priority Queue . In Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2015) ISBN 978-989-758-089-5, pages 371-378. DOI: 10.5220/0005260903710378


in Bibtex Style

@conference{visapp15,
author={William Robson Schwartz and Pedro Jussieu Rezende and Helio Pedrini},
title={Faster Approximations of Shortest Geodesic Paths on Polyhedra Through Adaptive Priority Queue},
booktitle={Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2015)},
year={2015},
pages={371-378},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005260903710378},
isbn={978-989-758-089-5},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 10th International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2015)
TI - Faster Approximations of Shortest Geodesic Paths on Polyhedra Through Adaptive Priority Queue
SN - 978-989-758-089-5
AU - Schwartz W.
AU - Rezende P.
AU - Pedrini H.
PY - 2015
SP - 371
EP - 378
DO - 10.5220/0005260903710378