Exact geodesics and shortest paths on polyhedral surfaces

IEEE Transactions on Pattern Analysis and Machine Intelligence

Volumn 31, Issue 6, 2009, Pages 1006-1016

  Balasubramanian, Mukund ^a   Polimeni, Jonathan R ^b   Schwartz, Eric L ^a,c  

^a BOSTON UNIVERSITY   (United States)

^b HARVARD MEDICAL SCHOOL   (United States)

^c BOSTON UNIVERSITY SCHOOL OF MEDICINE   (United States)

Author keywords

Computational geometry; Differential geometry; Flat maps; Surface based analysis; Triangular meshes

Indexed keywords

COMPUTING DISTANCES; CONVEX POLYHEDRAL; CORRECT SOLUTIONS; DIFFERENTIAL GEOMETRY; DISTANCE COMPUTATIONS; FLAT MAPS; GEODESIC DISTANCES; GEODESIC PATHS; ISOMETRIC SURFACES; POLYHEDRAL SURFACES; RUN-TIME; SHORTEST PATHS; SURFACE-BASED ANALYSIS; TRIANGULAR MESHES;

ALGORITHMS; COMPUTATIONAL GEOMETRY; GEODESY;

SURFACE ANALYSIS;

ALGORITHM; ARTICLE; ARTIFICIAL INTELLIGENCE; AUTOMATED PATTERN RECOGNITION; COMPUTER ASSISTED DIAGNOSIS; IMAGE ENHANCEMENT; METHODOLOGY; REPRODUCIBILITY; SENSITIVITY AND SPECIFICITY; THREE DIMENSIONAL IMAGING;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; IMAGING, THREE-DIMENSIONAL; PATTERN RECOGNITION, AUTOMATED; REPRODUCIBILITY OF RESULTS; SENSITIVITY AND SPECIFICITY;

EID: 65549159747     PISSN: 01628828     EISSN: None     Source Type: Journal    
DOI: 10.1109/TPAMI.2008.213     Document Type: Article

References (50)

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. 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.