On shortest paths in polyhedral spaces
SIAM Journal on Computing
Approximation algorithms for shortest path motion planning
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Shortest paths on a polyhedron
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Approximate Euclidean shortest path in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Star Unfolding of a Polytope with Applications
SIAM Journal on Computing
A Note on the Complexity of Dijkstra's Algorithm for Graphs with Weighted Vertices
IEEE Transactions on Computers
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
ACM Transactions on Graphics (TOG)
Maintaining all-pairs approximate shortest paths under deletion of edges
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
All Pairs Shortest Paths in weighted directed graphs ? exact and almost exact algorithms
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
New results on shortest paths in three dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Iso-charts: stretch-driven mesh parameterization using spectral analysis
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Similarity-based surface modelling using geodesic fans
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Fast exact and approximate geodesics on meshes
ACM SIGGRAPH 2005 Papers
Geodesic Remeshing Using Front Propagation
International Journal of Computer Vision
An optimal-time algorithm for shortest paths on a convex polytope in three dimensions
Proceedings of the twenty-second annual symposium on Computational geometry
Interactive decal compositing with discrete exponential maps
ACM SIGGRAPH 2006 Papers
Parallel algorithms for approximation of distance maps on parametric surfaces
ACM Transactions on Graphics (TOG)
Improving Chen and Han's algorithm on the discrete geodesic problem
ACM Transactions on Graphics (TOG)
Shortest Path Problems on a Polyhedral Surface
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Non-rigid registration under isometric deformations
SGP '08 Proceedings of the Symposium on Geometry Processing
Editable polycube map for GPU-based subdivision surfaces
I3D '11 Symposium on Interactive 3D Graphics and Games
Efficiently Computing Exact Geodesic Loops within Finite Steps
IEEE Transactions on Visualization and Computer Graphics
Texture brush: an interactive surface texturing interface
Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
Saddle vertex graph (SVG): a novel solution to the discrete geodesic problem
ACM Transactions on Graphics (TOG)
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Computing discrete geodesics on polyhedral surfaces plays an important role in computer graphics. In contrast to the well-studied "single-source, all-destination" discrete geodesic problem, little progress has been reported to the all-pairs geodesic, i.e., computing the geodesic distance between arbitrary two points on the surface. To our knowledge, the existing all-pairs geodesic algorithms have very high computational cost, thus, can not be applied to real-world models, which usually contain thousands of vertices. In this paper, we propose an efficient algorithm to approximate the all-pairs geodesic on triangular meshes. The pre-processing step takes O(mn2 log n) time for the input mesh with n vertices and m samples, where m (≪ n) is specified by the user, usually between a few hundred and several thousand. In the query step, our algorithm can compute the approximate geodesic distance between arbitrary pair of points (not necessarily mesh vertices) in O(1) time. Furthermore, the geodesic path and the geodesic distance field can be approximated in linear time. Both theoretical analysis and experimental results on real-world models demonstrate that our algorithm is efficient and accurate. We demonstrate the efficacy of our algorithm on the interactive texture mapping by using discrete exponential map.