Article

Fast exact and approximate geodesics on meshes

Authors:

Vitaly Surazhsky,

Tatiana Surazhsky,

Danil Kirsanov,

Steven J. Gortler,

Hugues HoppeAuthors Info & Claims

SIGGRAPH '05: ACM SIGGRAPH 2005 Papers

Pages 553 - 560

https://doi.org/10.1145/1186822.1073228

Published: 01 July 2005 Publication History

Abstract

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.

Supplementary Material

MP4 File (pps019.mp4)

Download
33.63 MB

References

[1]

Chen, J., and Han, Y. 1996. Shortest paths on a polyhedron; part I: computing shortest paths. Int. J. Comp. Geom. & Appl. 6 (2), 127--144.

[2]

Floater, M. S., Hormann, K., and Reimers, M. 2002. Parameterization of manifold triangulations. Approx. Theory X: Abstract and Classical Analysis, Vanderbilt University Press, Nashville, 197--209.

[3]

Floater, M. S. 2005. Chordal cubic spline interpolation is fourth order accurate. IMA J. Numer Analysis. to appear.

[4]

Funkhouser, T., Kazhdan, M., Shilane, P., Min, P., Kiefer, W., Tal, A., Rusinkiewicz, S., and Dobkin, D. 2004. Modeling by example. In Proc. of SIGGRAPH 2004, 652--663.

Digital Library

[5]

Goldberg, A. V., and Harrelson, C. 2005. Computing the shortest path: A* search meets graph theory. In Proc. of the 16th Annual ACMSIAM Symp. on Discrete Algorithms, 156--165.

Digital Library

[6]

Hershberger, J. E., and Suri, S. 1999. An optimal algorithm for Euclidean shortest paths in the plane. Siam J. Comp. 28 (6) 2215--2256.

Digital Library

[7]

Hilaga, M., Shinagawa, Y., Kohmura, T., and Kunii, T. L. 2001. Topology matching for fully automatic similarity estimation of 3d shapes. In Proc. of ACM SIGGRAPH 2001, 203--212.

Digital Library

[8]

Kanai, T., and Suzuki, H. 2001. Approximate shortest path on a polyhedral surface and its applications. Comp. -Aided Design 33, 11, 801--811.

[9]

Kaneva, B., and O'Rourke, J. 2000. An implementation of Chen & Han's shortest paths algorithm. In Proc. of the 12th Canadian Conf. on Comput. Geom., 139--146.

[10]

Kapoor, S. 1999. Efficient computation of geodesic shortest paths. In Proc. 32nd Annual ACM Symp. Theory Comput., 770--779.

Digital Library

[11]

Katz, S., and Tal, A. 2003. Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans. on Graphics 22, 3 (Jul), 954--961.

Digital Library

[12]

Kimmel, R., and Sethian, J. A. 1998. Computing geodesic paths on manifolds. Proc. of National Academy of Sci. 95(15) (July), 8431--8435.

[13]

Kirsanov, D. 2004. Minimal Discrete Curves and Surfaces. PhD thesis, Applied Math., Harvard University.

Digital Library

[14]

Kobbelt, L., Campagna, S., Vorsatz, J., and Seidel, H.-P. 1998. Interactive multiresolution modeling on arbitrary meshes. In Proc. of SIGGRAPH 98, 105--114.

Digital Library

[15]

Krishnamurthy, V., and Levoy, M. 1996. Fitting smooth surfaces to dense polygon meshes. In Proc. of SIGGRAPH 96, 313--324.

Digital Library

[16]

Lanthier, M., Maheshwari, A., and Sack, J.-R. 1997. Approximating weighted shortest paths on polyhedral surfaces. In Proc. 13th Annu. ACM Symp. Comput. Geom. 274--283.

Digital Library

[17]

Lee, H., Kim, L., Meyer, M., and Desbrun, M. 2001. Meshes on fire. In Eurographics Workshop on Comp. Animation and Simulation, 75--84.

Digital Library

[18]

Loop, C. T. 1987. Smooth subdivision surfaces based on triangles. Master's thesis, Mathematics, Univ. of Utah.

[19]

Martinez, D., Velho, L., and Carvalho, P. C. 2004. Geodesic paths on triangular meshes. In Proc. of SIBGRAPI/SIACG., 210--217.

Digital Library

[20]

Mitchell, J. S. B., Mount, D. M., and Papadimitriou, C. H. 1987. The discrete geodesic problem. SIAM J. of Computing 16(4), 647--668.

Digital Library

[21]

Mitchell, J. 2000. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, Eds. Elsevier Science, 633--702.

[22]

Novotni, M., and Klein, R. 2002. Computing geodesic distances on triangular meshes. In Proc. of WSCG'2002, 341--347.

[23]

Peyré, G., and Cohen, L. 2003. Geodesic re-meshing and parameterization using front propagation. In Proc. of VLSM'03, 33--40.

[24]

Peyré, G., and Cohen, L. 2005. Geodesic computations for fast and accurate surface remeshing and parameterization. In Progress in nonlinear differential equations and their applications, vol. 63, 157--171.

[25]

Pham-Trong, V., Szafran, N., and Biard, L. 2001. Pseudo-geodesics on three-dimensional surfaces and pseudo-geodesic meshes. Numerical Algorithms 26, 305--315.

[26]

Pohl, I. 1971. Bi-directional search. In Machine Intelligence, vol. 6, 124--140.

[27]

Polthier, K., and Schmies, M. 1998. Straightest geodesics on polyhedral surfaces. Mathematical Visualization. Ed: H. C. Hege, K. Polthier. Springer Verlag, 391--409.

[28]

Praun, E., Hoppe, H., and Finkelstein. A. 1999. Robust mesh watermarking. In Proc. of SIGGRAPH 99, 49--56.

Digital Library

[29]

Praun, E., Finkelstein, A., and Hoppe, H. 2000. Lapped textures. In Proc. of SIGGRAPH 2000, 465--470.

Digital Library

[30]

Reimers. M. 2004. Topics in Mesh based Modeling. PhD thesis, Mathematics, Univ. of Oslo.

[31]

Sander, P., Wood, Z., Gortler, S., Snyder, J., and Hoppe, H. 2003. Multi-chart geometry images. In Proc. of Symp. on Geometry Processing, 146--155.

Digital Library

[32]

Sifri, O., Sheffer, A., and Gotsman, C. 2003. Geodesic-based surface remeshing. In Proc. 12th Intnl. Meshing Roundtable, 189--199.

[33]

Sloan, P.-P. J., Rose, C. F., and Cohen, M. F. 2001. Shape by example. In ACM Symp. on Interactive 3D Graphics, 135--144.

Digital Library

[34]

Zhou, K., Snyder, J., Guo, B., and Shum, H.-Y. 2004. Iso-charts: Stretch-driven mesh parameterization using spectral analysis. In Proc. of the 2004 EG/ACM SIGGRAPH Symp. on Geometry Processing, 45--54.

Digital Library

[35]

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.

Digital Library

Cited By

Rawat SBiswas M(2022)Enhanced Heat Method for Computation of Geodesic Distance on Triangular Meshes2022 IEEE 9th Uttar Pradesh Section International Conference on Electrical, Electronics and Computer Engineering (UPCON)10.1109/UPCON56432.2022.9986375(1-5)Online publication date: 2-Dec-2022
https://doi.org/10.1109/UPCON56432.2022.9986375
Pillwein SMusialski P(2021)Generalized deployable elastic geodesic gridsACM Transactions on Graphics10.1145/3478513.348051640:6(1-15)Online publication date: 10-Dec-2021
https://dl.acm.org/doi/10.1145/3478513.3480516
Arora RSingh K(2021)Mid-Air Drawing of Curves on 3D Surfaces in Virtual RealityACM Transactions on Graphics10.1145/345909040:3(1-17)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3459090
Show More Cited By

Index Terms

Fast exact and approximate geodesics on meshes
1. Computing methodologies
  1. Computer graphics
    1. Image manipulation
    2. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Fast exact and approximate geodesics on meshes

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points ...
The k-max distance in graphs and images

A new distance in the graphs and images, called the k-max distance, is introduced.The length of path is defined as the sum of the k maximum edge costs along the path.The observations are presented that make the computation of the distance feasible.The ...
Computing Geodesic Distances in Tree Space

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann [Adv. Appl. Math., 27 (2001), pp. 733-767]. We show that the possible combinatorial types of shortest ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SIGGRAPH '05: ACM SIGGRAPH 2005 Papers

July 2005

826 pages

ISBN:9781450378253

DOI:10.1145/1186822

Editor:
Markus Gross
Eidgenössische Technische Hochschule Zürich

Copyright © 2005 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 July 2005

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Conference

SIGGRAPH05

Sponsor:

SIGGRAPH

SIGGRAPH05: Special Interest Group on Computer Graphics and Interactive Techniques Conference

July 31 - August 4, 2005

California, Los Angeles

Acceptance Rates

SIGGRAPH '05 Paper Acceptance Rate 98 of 461 submissions, 21%;

Overall Acceptance Rate 1,822 of 8,601 submissions, 21%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

50
Total Citations
View Citations
3,117
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)0

Reflects downloads up to 16 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Rawat SBiswas M(2022)Enhanced Heat Method for Computation of Geodesic Distance on Triangular Meshes2022 IEEE 9th Uttar Pradesh Section International Conference on Electrical, Electronics and Computer Engineering (UPCON)10.1109/UPCON56432.2022.9986375(1-5)Online publication date: 2-Dec-2022
https://doi.org/10.1109/UPCON56432.2022.9986375
Pillwein SMusialski P(2021)Generalized deployable elastic geodesic gridsACM Transactions on Graphics10.1145/3478513.348051640:6(1-15)Online publication date: 10-Dec-2021
https://dl.acm.org/doi/10.1145/3478513.3480516
Arora RSingh K(2021)Mid-Air Drawing of Curves on 3D Surfaces in Virtual RealityACM Transactions on Graphics10.1145/345909040:3(1-17)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3459090
Farias RKallmann M(2020)Optimal Path Maps on the GPUIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2019.290427126:9(2863-2874)Online publication date: 1-Sep-2020
https://doi.org/10.1109/TVCG.2019.2904271
Wróblewski AAndrzejczak J(2019)Wave propagation time optimization for geodesic distances calculation using the Heat MethodOpen Physics10.1515/phys-2019-002717:1(263-275)Online publication date: 8-Jun-2019
https://doi.org/10.1515/phys-2019-0027
Herholz PAlexa M(2019)Efficient Computation of Smoothed Exponential MapsComputer Graphics Forum10.1111/cgf.1360738:6(79-90)Online publication date: 14-Mar-2019
https://doi.org/10.1111/cgf.13607
Nascimento EPotje GMartins RChamone FCampos MBajcsy R(2019)GEOBIT: A Geodesic-Based Binary Descriptor Invariant to Non-Rigid Deformations for RGB-D Images2019 IEEE/CVF International Conference on Computer Vision (ICCV)10.1109/ICCV.2019.01010(10003-10011)Online publication date: Oct-2019
https://doi.org/10.1109/ICCV.2019.01010
He THuang HYi LZhou YWu CWang JSoatto S(2019)GeoNet: Deep Geodesic Networks for Point Cloud Analysis2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR.2019.00705(6881-6890)Online publication date: Jun-2019
https://doi.org/10.1109/CVPR.2019.00705
Li SLuo ZZhen MYao YShen TFang TQuan L(2019)Cross-Atlas Convolution for Parameterization Invariant Learning on Textured Mesh Surface2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR.2019.00630(6136-6145)Online publication date: Jun-2019
https://doi.org/10.1109/CVPR.2019.00630
Mirloo MEbrahimnezhad H(2018)Salient Point Detection in Protrusion Parts of 3D Object Robust to Isometric Variations3D Research10.1007/s13319-018-0155-19:1(1-18)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s13319-018-0155-1
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents