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Computing Center-Lines: An Application of Vector Field Topology

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Topology-Based Methods in Visualization II

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Summary

Curve-skeletons of 3-D objects are medial axes shrunk to a single line. There are several applications for curve-skeletons. For example, animation of 3-D objects, such as an animal or a human, as well as planning of flight paths for virtual colonoscopy. Other applications are the extraction of center lines within blood vessels where center lines are used to quantitatively measure vessel length, vessel diameter, and angles between vessels. The described method computes curve-skeletons based on a vector field that is orthogonal to the object's boundary surface. A topological analysis of this field then yields the center lines of the curve-skeletons. In contrast to previous methods, the vector field does not need to be computed for every sampled point of the entire volume. Instead, the vector field is determined only on the sample points on the boundary surface of the objects. Since most of the computational time was spent on calculating the force field in previous methods, the proposed approach requires significantly less time compared to previous vector-based techniques while still achieving a better accuracy and robustness compared to methods based on Voronoi tessellations.

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References

  1. N. Ahuja, J.-H. Chuang, Shape Representation Using a Generalized Potential Field Model. IEEE Trans. Pattern Analysis and Machine Intelligence, 19(2):169–176, 1997.

    Article  Google Scholar 

  2. N. Amenta, S. Choi, R.-K. Kolluri, The Power Crust, Proc. of 6th ACM Symp. on Solid Modeling, pp. 249–260, 2001.

    Google Scholar 

  3. G. Bertrand and Z. Aktouf, A three-dimensional thinning algorithm using subfields, Vision Geometry III, 2356:113–124. SPIE, 1994.

    Google Scholar 

  4. I. Bitter, A. E. Kaufman, M. Sato, Penalized-Distance Volumetric Skeleton Algorithm, IEEE Trans. Visualization and Comp. Graphics, 7(3), 2001.

    Google Scholar 

  5. G. Borgefors, On Digital Distance Transforms in Three Dimensions, Computer Vision and Image Understanding 64(3):368–376, 1996.

    Article  Google Scholar 

  6. G. Borgefors, I. Nyström, G. S. Di Baja, Computing skeletons in three dimensions, Pattern Recognition, 32(7), 1999.

    Google Scholar 

  7. S. Bouix, K. Siddiqi, Divergence-Based Medial Surfaces, ECCV 1842:603–618, Springer-Verlag, 2000.

    Article  Google Scholar 

  8. D. Brunner, G. Brunnett, Mesh Segmentation Using the Object Skeleton Graph, Proc. IASTED International Conf. on Computer Graphics and Imaging, 48–55, ACTA Press 2004.

    Google Scholar 

  9. J. F. Canny, A Computational Approach to Edge Detection. IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-8, No. 6, pp. 679–698, 1986.

    Google Scholar 

  10. M. Couprie and R. Zrour, Discrete Bisector Function and Euclidean Skeleton, Lecture Notes in Computer Science, vol. 3429, Springer-Verlag, 2005.

    Google Scholar 

  11. N. D. Cornea, D. Silver, P. Min, Curve-Skeleton Applications. In Proceedings IEEE Visualization, pp. 95–102, 2005.

    Google Scholar 

  12. N. D. Cornea, D. Silver, X. Yuan, R. Balasubramanian, Computing Hierarchical Curve-Skeletons of 3D Objects. The Visual Computer 21(11):945–955, Springer-Verlag, 2005.

    Article  Google Scholar 

  13. T. K. Dey and S. Goswami. Tight Cocone: A water-tight surface reconstructor. Proc. 8th ACM Sympos. Solid Modeling Applications, 127–134. Journal version in J. of Computing and Infor. Sci. Engin. Vol. 30, 2003, 302–307.

    Article  Google Scholar 

  14. U. Eckhardt, G. Maderlechner, Invariant Thinning, Pattern Recognition and Artificial Intellicgence (7):1115–1144, 1993.

    Article  Google Scholar 

  15. N. Gagvani and D. Silver, Parameter Controlled Volume Thinning, Graphical Models and Image Processing, 61(3):149–164, 1999.

    Article  Google Scholar 

  16. N. Gagvani and D. Silver, Animating volumetric models, Academic Press Professional 63(6):443–458, 2001.

    MATH  Google Scholar 

  17. P. Golland, W. E. L. Grimson, Fixed Topology Skeletons, IEEE CVPR, 2000.

    Google Scholar 

  18. W. Gong and G. Bertrand, A simple parallel 3d thinning algorithm. Proc. IEEE Pattern Recognition, 188–190, 1990.

    Google Scholar 

  19. T. He, L. Hong, D. Chen, Z. Liang, Reliable Path for Virtual Endoscopy: Ensuring Complete Examination of Human Organs, IEEE Trans. Visualization and Comp. Graphics, 7(4):333–342, 2001.

    Article  Google Scholar 

  20. M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974.

    Google Scholar 

  21. R. Jain, R. Kasturi, B. G. Schunck. Machine Vision. McGraw-Hill,Inc., New York, 1995.

    Google Scholar 

  22. A. Kanitsar, D. Fleischmann, R. Wegenkittl, P. Felkel, E. Gröller, CPR: Curved Planar Reformation. Proc. IEEE Visualization, pp. 37–44, 2002.

    Google Scholar 

  23. T. Lee and R. L. Kashyap, Building skeleton models via 3-d medial surface/axis thinning algorithms. CVGIP: Graphical Models and Image Processing, 56(6):462–478, November 1994.

    Article  Google Scholar 

  24. S. Lobregt and P. W. Verbeek and F. C. A. Groen, Three-Dimensional Skele-tonization: Principle and Algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2(1): 75–77, 1980.

    Article  Google Scholar 

  25. C. Lohou and G. Bertrand,A 3D 12-subiteration thinning algorithm based on P-simple points, Discrete Applied Mathematics 139:171–195, Elsevier, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  26. V. Luboz, X. Wu, K. Krissian, C. F. Westin, R. Kikinis, S. Cotin, S. Dawson,A segmentation and reconstruction technique for 3D vascular structures, MICCAI 2005, Lecture Notes in Computer Science 3749:43–50, 2005.

    Article  Google Scholar 

  27. C. M. Ma and M. Sonka,A fully parallel 3d thinning algorithm and its applications. Computer Vision and Image Understanding, 64(3):420–433, 1996.

    Article  Google Scholar 

  28. G. Malandain, S. Fernandez-Vidal,Euclidean Skeletons, Image and Vision Computing, vol. 16:317–327, 1998.

    Article  Google Scholar 

  29. A. Manzanera, T. Bernard, F. Preteux, B. Longuet,A unified mathematical framework for a compact and fully parallel n-D skeletonization procedure, Vision Geometry VIII, Vol. 3811: 57–68, SPIE, 1999.

    Google Scholar 

  30. K. Palagyi, A. Kuba,Directional 3D Thinning using 8 Subiterations, Proc. Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science 1568:325–336, 1999.

    Article  Google Scholar 

  31. K. Palagyi and A. Kuba.A parallel 3d 12-subiteration thinning algorithm.Graphical Models and Image Proc., 61(4):199–221, 1999.

    Article  Google Scholar 

  32. D. Perchet, C. I. Fetita, F. Preteux,Advanced navigation tools for virtual bron-choscopy, Proc. SPIE Conf. on Image Processing: Algorithms and Systems III, vol. 5298, 2004.

    Google Scholar 

  33. C. Pudney,Distance-Ordered Homotopic Thinning: A Skeletonization Algorithm for 3D Digital Images, Computer Vision and Image Understanding, 72(3):404–413, 1998.

    Article  Google Scholar 

  34. P. K. Saha, B. B. Chaudhuri, D. Dutta Majumder,A new shape preserving parallel thinning algorithm for 3d digital images. Pattern Recognition, 30(12):1939–1955, 1997.

    Article  Google Scholar 

  35. H. Schirmacher, M. Zöckler, D. Stalling, H. Hege,Boundary Surface Shrinking -a Continuous Approach to 3D Center Line Extraction, Proc. of IMDSP, 25–28, 1998.

    Google Scholar 

  36. J. A. Sethian,Fast Marching Methods, SIAM Review, 41(2):199–235, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  37. H. Si,TetGen, A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator, WIAS Technical Report No. 9, 2004.

    Google Scholar 

  38. H. Sundar, D. Silver, N. Gagvani, S. Dickinson,Skeleton Based Shape Matching and Retrieval, Proc. Shape Modeling Int'l, 2003.

    Google Scholar 

  39. K. Suresh,Automating the CAD/CAE Dimensional Reduction Process, ACM Symp. On Solid Modeling and Applications, 2003.

    Google Scholar 

  40. S. Svensson, I. Nystrom, G. Sanniti di Baja,Curve Skeletonization of Surfacelike Objects in 3D Images Guided by Voxel Classification, Pattern Recognition Letters, 23 (12):1419–1426, 2002.

    Article  MATH  Google Scholar 

  41. A. Telea, A. Vilanova,A robust level-set algorithm for centerline extraction, Eurographics/IEEE Symp. On Data Visualization, pp. 185–194, 2003.

    Google Scholar 

  42. Y. F. Tsao and K. S. Fu,A parallel thinning algorithm for 3d pictures. Computer Vision, Graphics and Image Proc., 17:315–331, 1981.

    Article  Google Scholar 

  43. L. Wade, R. E. Parent,Automated generation of control skeletons for use in animation, The Visual Computer 18(2):97–110, 2002.

    Article  MATH  Google Scholar 

  44. M. Wan, F. Dachille, A. Kaufman,Distance-Field Based Skeletons for Virtual Navigation, IEEE Visualization 2001, pp. 239–246, 2001.

    Google Scholar 

  45. T. Wischgoll, Gerik Scheuermann,Detection and Visualization of Planar Closed Streamlines, IEEE Transactions on Visualization and Computer Graphics, 7(2): 165–172, 2001.

    Article  Google Scholar 

  46. T. Wischgoll,Closed Streamlines in Flow Visualization, Ph.D. Thesis, Univer-sität Kaiserslautern, Germany, 2002.

    Google Scholar 

  47. Z. Yu, C. Bajaj,A Segmentation-Free Approach for Skeletonization of GrayScale Images via Anisotropic Vector Diffusion, CVPR 2004, pp. 415–420, 2004.

    Google Scholar 

  48. Y. Zhou, A. Kaufman, A. W. Toga,Three-dimensional Skeleton and Centerline Generation Based on an Approximate Minimum Distance Field, The Visual Computer, 14, pp. 303–314, 1998.

    Article  Google Scholar 

  49. Y. Zhou, A. W. Toga,Efficient skeletonization of volumetric objects, IEEE Trans. Visualization and Comp. Graphics, 5(3):196–209, 1999.

    Article  Google Scholar 

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Authors and Affiliations

  1. Wright State University, USA

    Thomas Wischgoll

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  1. Thomas Wischgoll
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Editors and Affiliations

  1. Konrad-Zuse-Zentrum für Informationstechnik Berlin Devision Scientific Computing, Department Visualization and Data Analysis, Takustraβe 7, Berlin, 14195, Germany

    Hans-Christian Hege

  2. Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, Berlin, 14195, Germany

    Konrad Polthier

  3. Fakultät für Mathematik und Informatik Institut für Informatik, Universität Leipzig, Postfach 100920, Leipzig, 04009, Germany

    Gerik Scheuermann

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Wischgoll, T. (2009). Computing Center-Lines: An Application of Vector Field Topology. In: Hege, HC., Polthier, K., Scheuermann, G. (eds) Topology-Based Methods in Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88606-8_13

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