The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.06 - June (2009 vol.31)
pp: 1006-1016
Jonathan R. Polimeni , Harvard Medical School, Charlestown
Mukund Balasubramanian , Boston University, Boston
ABSTRACT
We present two algorithms for computing distances along convex and non-convex polyhedral surfaces. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimal-geodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.
INDEX TERMS
Curve, surface, solid, and object representations, Differential geometry, Flat maps, Triangular meshes, Surface-based analysis, Computational geometry
CITATION
Jonathan R. Polimeni, Mukund Balasubramanian, "Exact Geodesics and Shortest Paths on Polyhedral Surfaces", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 6, pp. 1006-1016, June 2009, doi:10.1109/TPAMI.2008.213
REFERENCES
[1] E. Schwartz and B. Merker, “Flattening Cortex: An Optimal Computer Algorithm and Comparisons with Physical Flattening of the Opercular Surface of Striate Cortex,” Soc. for Neuroscience Abstracts, 1985.
[2] E.L. Schwartz, A. Shaw, and E. Wolfson, “A Numerical Solution to the Generalized Mapmaker's Problem,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 1005-1008, 1989.
[3] G. Zigelman, R. Kimmel, and N. Kiryati, “Texture Mapping Using Surface Flattening via Multidimensional Scaling,” IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 2, pp. 198-207, Apr.-June 2002.
[4] S. Katz and A. Tal, “Hierarchical Mesh Decomposition Using Fuzzy Clustering and Cuts,” Proc. ACM SIGGRAPH '03, pp. 954-961, 2003.
[5] V. Krishnamurthy and M. Levoy, “Fitting Smooth Surfaces to Dense Polygon Meshes,” Proc. ACM SIGGRAPH '96, pp. 313-324, 1996.
[6] M. Lanthier, A. Maheshwari, and J.R. Sack, “Approximating Shortest Paths on Weighted Polyhedral Surfaces,” Algorithmica, vol. 30, no. 4, pp. 527-562, 2001.
[7] F.A. Jolesz, W.E. Lorensen, H. Shinmoto, H. Atsumi, S. Nakajima, P. Kavanaugh, P. Saiviroonporn, S. Seltzer, S. Silverman, M. Phillips, and R. Kikinis, “Interactive Virtual Endoscopy,” Am. J. Radiology, vol. 169, pp. 1229-1237, 1997.
[8] M. Sharir and A. Schorr, “On Shortest Paths in Polyhedral Surfaces,” SIAM J. Computing, vol. 15, no. 1, pp. 193-215, 1986.
[9] D.M. Mount, “On Finding Shortest Paths on Convex Polyhedra,” Technical Report 1495, Dept. of Computer Science, Univ. of Maryland, Baltimore, 1984.
[10] J. O'Rourke, S. Suri, and H. Booth, “Shortest Paths on Polyhedral Surfaces,” Lecture Notes in Computer Science, vol. 182, pp. 243-254, Springer, 1985.
[11] S. Kapoor, “Efficient Computation of Geodesic Shortest Paths,” Proc. 31st Ann. ACM Symp. Theory of Computing, pp. 770-779, 1999.
[12] J. O'Rourke, “Computational Geometry Column 35,” Int'l J. Computational Geometry and Applications, vol. 9, no. 4-5, pp. 513-515, 1999.
[13] E. Wolfson and E.L. Schwartz, “Computing Minimal Distances on Arbitrary Polyhedral Surfaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 1001-1005, 1989.
[14] J. Hershberger and S. Suri, “Practical Methods for Approximating Shortest Paths on a Convex Polytope in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^{3}$ ,” Computational Geometry, vol. 10, no. 1, pp. 31-46, 1998.
[15] P.K. Agarwal, S. Har-Peled, M. Sharir, and K.R. Varadarajan, “Approximating Shortest Paths on a Convex Polytope in Three Dimensions,” J. ACM, vol. 44, no. 4, pp. 567-584, 1997.
[16] K.R. Varadarajan and P.K. Agarwal, “Approximating Shortest Paths on a Nonconvex Polyhedron,” SIAM J. Computing, vol. 30, no. 4, pp. 1321-1340, 2000.
[17] S. Har-Peled, “Constructing Approximate Shortest Path Maps in Three Dimensions,” SIAM J. Computing, vol. 28, no. 4, pp. 1182-1197, 1999.
[18] T. Kanai and H. Suzuki, “Approximate Shortest Path on a Polyhedral Surface and Its Applications,” Computer-Aided Design, vol. 33, no. 11, pp. 801-811, 2001.
[19] M. Kageura and K. Shimada, “Finding the Shortest Path for Quality Assurance of Electric Components,” J. Mechanical Design, vol. 126, pp. 1017-1026, 2004.
[20] R. Kimmel and J.A. Sethian, “Computing Geodesic Paths on Manifolds,” Proc. Nat'l Academy of Sciences USA, vol. 95, no. 15, pp.8431-8435, 1998.
[21] J.A. Sethian and A. Vladimirsky, “Fast Methods for the Eikonal and Related Hamilton-Jacobi Equations on Unstructured Meshes,” Proc. Nat'l Academy of Sciences USA, vol. 97, no. 11, pp.5699-5703, 2000.
[22] M. Novotni and R. Klein, “Computing Geodesic Paths on Triangular Meshes,” Proc. 10th Int'l Conf. in Central Europe on Computer Graphics, Visualization, and Computer Vision, pp. 341-347, 2002.
[23] A. Bartesaghi and G. Sapiro, “A System for the Generation of Curves on 3D Brain Images,” Human Brain Mapping, vol. 14, no. 1, pp. 1-15, 2001.
[24] N. Khaneja, M. Miller, and U. Grenander, “Dynamic Programming Generation of Curves on Brain Surfaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 11, pp. 1260-1265, Nov. 1998.
[25] O.P. Hinds, J.R. Polimeni, M.L. Blackwell, C.J. Wiggins, G.C. Wiggins, A. van der Kouwe, L.L. Wald, E.L. Schwartz, and B. Fischl, “Reconstruction and Analysis of Human V1 by Imaging the Stria of Gennari Using MRI at 7T,” Soc. for Neuroscience Abstracts, 2005.
[26] J.R. Polimeni, O.P. Hinds, M. Balasubramanian, A. van der Kouwe, L.L. Wald, A.M. Dale, B. Fischl, and E.L. Schwartz, “The Human V1-V2-V3 Visuotopic Map Complex Measured via fMRI at 3 and 7 Tesla,” Soc. for Neuroscience Abstracts, 2005.
[27] D.J. Struik, Lectures on Classical Differential Geometry, second ed. Addison-Wesley, 1961.
[28] M.P. do Carmo, Differential Geometry of Curves and Surfaces. Prentice Hall, 1976.
[29] J. McCleary, Geometry from a Differentiable Viewpoint. Cambridge Univ. Press, 1994.
[30] D.R. Smith, Variational Methods in Optimization. Prentice Hall, 1974.
[31] I.M. Gelfand and S.V. Fomin, Calculus of Variations. Prentice Hall, 1963.
[32] M.R. Garey, D.S. Johnson, F.P. Preparata, and R.E. Tarjan, “Triangulating a Simple Polygon,” Information Processing Letters, vol. 7, no. 4, pp. 175-179, 1978.
[33] R.E. Tarjan and C.J.V. Wyk, “An $O(n \log\log n)$ -Time Algorithm for Triangulating a Simple Polygon,” SIAM J. Computing, vol. 17, no. 1, pp. 143-178, 1988.
[34] L.C. Kinsey, Topology of Surfaces. Springer, 1993.
[35] H. Edelsbrunner, Geometry and Topology of Mesh Generation. Cambridge Univ. Press, 2001.
[36] J.S.B. Mitchell, D.M. Mount, and C.H. Papadimitriou, “The Discrete Geodesic Problem,” SIAM J. Computing, vol. 16, no. 4, pp. 647-668, 1987.
[37] K. Polthier and M. Schmies, “Straightest Geodesics on Polyhedral Surfaces,” Math. Visualization: Algorithms, Applications, and Numerics, H.-C. Hege and K. Polthier, eds., pp. 135-150, Springer, 1998.
[38] E.W. Dijkstra, “A Note on Two Problems in Connection with Graphs,” Numerische Mathematik, vol. 1, pp. 269-271, 1959.
[39] R.W. Floyd, “Algorithm 97: Shortest Path,” Comm. ACM, vol. 5, p.345, 1962.
[40] J. Chen and Y. Han, “Shortest Paths on a Polyhedron,” Int'l J. Computational Geometry and Applications, vol. 6, pp. 127-144, 1996.
[41] V. Surazhsky, T. Surazhsky, D. Kirsanov, S.J. Gortler, and H. Hoppe, “Fast Exact and Approximate Geodesics on Meshes,” ACM Trans. Graphics, Proc. ACM SIGGRAPH '05, vol. 24, no. 3, pp.553-560, 2005.
[42] B. Kaneva and J. O'Rourke, “An Implementation of Chen and Han's Shortest Paths Algorithm,” Proc. 12th Canadian Conf. Computational Geometry, pp. 139-146, 2000.
[43] H. Fuchs, Z.M. Kedem, and S.P. Uselton, “Optimal Surface Reconstruction from Planar Contours,” Comm. ACM, vol. 20, no. 10, pp. 693-702, 1977.
[44] O.P. Hinds, J.R. Polimeni, and E.L. Schwartz, “Brain Surface Reconstruction from Slice Contours,” NeuroImage, vol. 31, no. 1, p.S445, 2006.
[45] B. Fischl, A. Liu, and A.M. Dale, “Automated Manifold Surgery: Constructing Geometrically Accurate and Topologically Correct Models of the Human Cerebral Cortex,” IEEE Trans. Medical Imaging, vol. 20, no. 1, pp. 70-80, 2001.
[46] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C. Cambridge Univ. Press, 1988.
[47] M. Balasubramanian, J.R. Polimeni, and E.L. Schwartz, “Quasi-Isometric Flattening of Large-Scale Cortical Surfaces,” Soc. for Neuroscience Abstracts, 2005.
[48] M. Balasubramanian, J.R. Polimeni, and E.L. Schwartz, “Quantitative Evaluation and Comparison of Cortical Flattening Algorithms,” Soc. for Neuroscience Abstracts, 2006.
[49] B. Fischl, M.I. Sereno, and A.M. Dale, “Cortical Surface-Based Analysis II: Inflation, Flattening and a Surface-Based Coordinate System,” NeuroImage, vol. 9, no. 2, pp. 195-207, 1999.
[50] E.L. Schwartz, “Computational Studies of the Spatial Architecture of Primate Visual Cortex: Columns, Maps, and Protomaps,” Primary Visual Cortex in Primates, vol. 10, series Cerebral Cortex, A.Peters and K. Rockland, eds., pp. 359-411, Plenum Press, 1994.
22 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool