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PDE-Based Medial Axis Extraction and Shape Manipulation of Arbitrary Meshes

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Abstract

Shape skeletonization (i.e., medial axis extraction) is powerful in many visual computing applications, such as pattern recognition, object segmentation, registration, and animation. In this paper, the authors expand the use of diffusion equations combined with distance field information to approximate medial axes of arbitrary 3D solids represented by polygonal meshes based on their differential properties. It offers an alternative but natural way for medial axis extraction for commonly used 3D polygonal models. By solving the PDE along time axis, this system can not only quickly extract diffusion-based medial axes of input meshes, but also allow users to visualize the extraction process at each time step. In addition, the proposed model provides users a set of manipulation toolkits to sculpt extracted medial axes, then use diffusion-based techniques to recover corresponding deformed shapes according to the original input datasets. This skeleton-based shape manipulation offers a fast and easy way for animation and deformation of complicated mesh objects.

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References

  1. H. Blum, A Transformation for Extracting New Descriptions of Shape, Models for the Perception of Speech and Visual Form, MIT Press, Boston, USA, 1967, 362–380.

    Google Scholar 

  2. H. Blum, Biological shape and visual science, Journal of Theoretical Biology, 1973, 38(2): 205–287.

    Article  MathSciNet  Google Scholar 

  3. S. Bouix and K. Siddiqi, Divergence-Based Medial Surfaces, European Conference on Computer Vision (ECCV), Dublin, Ireland, 2000.

  4. Q. Zhou, T. Ju, and S. Hu, Topology repair of solid models using skeletons, IEEE Transaction on Visualization and Computer Graphics, 2007, 13(4): 675–685.

    Article  Google Scholar 

  5. T. Ju, Q. Zhou, and S. Hu, Editing the Topology of 3D Models by Sketching, SIGGRAPH’07: ACM SIGGRAPH 2007 papers, ACM, New York, NY, USA, San Diego, California, 2007, 42.

  6. G. Turk, Generating textures on arbitrary surfaces using reaction-diffusion, Computer Graphics, 1991, 25(4): 289–298.

    Article  MathSciNet  Google Scholar 

  7. A. Witkin and M. Kass, Reaction-Diffusion Textures, SIGGRAPH 91, Las Vegas, NV, USA, 1991, 299–308.

  8. M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, Image Inpainting, SIGGRAPH 2000, New Orleans, LA USA, 2000, 417–424.

  9. P. Pérez, M. Gangnet, and A. Blake, Poisson image editing, ACM Transactions on Graphics (TOG), 2003, 22(3): 313–318.

    Article  Google Scholar 

  10. M. I. G. Bloor and M. J. Wilson, Using partial differential equations to generate free-form surfaces, Computer Aided Design, 1990, 22(4): 202–212.

    Article  MATH  Google Scholar 

  11. H. Du and H. Qin, Dynamic PDE surface design using geometric and physical constraints, Graphical Models, 2005, 67(1): 43–71.

    Article  MATH  Google Scholar 

  12. H. Du and H. Qin, Free-form geometric modeling by integrating parametric and implicit PDEs, IEEE Transaction on Visualization and Computer Graphics, 2007, 13(3): 549–561.

    Article  Google Scholar 

  13. H. Du and H. Qin, Medial axis extraction and shape manipulation of solid objects using parabolic PDEs, in Proceedings of the ninth ACM Symposium on Solid Modeling and Applications, Genova, Italy, 2004.

  14. C. Arcelli and G. Sanniti di Baja, A width-independent fast thinning algorithm, IEEE Transaction on Pattern Analysis and Machine Intelligence, 1985, 7(4): 463–474.

    Article  Google Scholar 

  15. T. Lee and R. Kashyap, Building skeleton models via 3D medial surface/axis thinning algorithm, CVGIP: Graphical Models and Image Processing, 1994, 56(6): 462–478.

    Article  Google Scholar 

  16. A. Manzanera, T. M. Bernard, F. Preteux, and B. Longuet, Medial Faces from a Concise 3D Thinning Algorithm, ICCV’99, Kerkyra, Greece, 1999, 337–343.

  17. C. Arcelli and G. Sanniti di Baja, Ridge points in Euclidean distance maps, Pattern Recognition Letters, 1992, 13(4): 237–243.

    Article  Google Scholar 

  18. I. Bitter, A. E. Kaufman, and M. Sato, Penalized-distance volumetric skeleton algorithm, IEEE Transaction on Visualization and Computer Graphics, 2001, 7(3): 195–206.

    Article  Google Scholar 

  19. M. Foskey, M. C. Lin, and Dinesh Manocha, Effient Computation of a Simplified Medial Axis, in Proceedings of the 8th ACM Symposium on Solid Modeling and Applications, ACM Press, Seattle, Washington, USA, 2003, 96–107.

    Chapter  Google Scholar 

  20. J. Gomez and O. Faugeras, Reconciling Distance Functions and Level Sets, Technical Report TR 3666, INRIA, 1999.

  21. F. Leymarie and M.D. Levine, Simulating the grassfire transform using an active contour model, IEEE Transaction on Pattern Analysis and Machine Intelligence, 1992, 14(1): 56–57.

    Article  Google Scholar 

  22. M. Schmitt, Some examples of algorithms analysis in computational geometry by means of mathematical morphology techniques, Lecture Notes in Computer Science, Geometry and Robotics, Springer-Verlag, 1989, 391: 225–246.

    Google Scholar 

  23. N. Amenta, S. Choi, and R. K. Kolluri, The Power Crust, in Proceedings of the 6th ACM Symposium on Solid Modeling and Applications, ACM Press, Ann Arbor, Michigan, United States, 2001, 249–266

    Chapter  Google Scholar 

  24. T. K. Dey and W. Zhao, Approximate medial axis as a voronoi subcomplex, in Proceedings of the 7th ACM symposium on Solid Modeling and Applications, ACM Press, Saarbrcken, Germany, 2002, 356–366.

    Google Scholar 

  25. J. A. Goldak, X. Yu, A. Knight, and L. Dong, Constructing discrete medial axis of 3-D objects, Int. J. Computational Geometry and Its Applications, 1991, 1(3): 327–339.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Näf, O. Kübler, R. Kikinis, M. E. Shenton, and G. Szekely, Characterization and recognition of 3D organ shape in medical image analysis using skeletonization, IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, 1996, 139.

  27. R. Ogniewicz, Discrete Voronoi Skeletons, Hartung-Gorre Verlag Konstanz, Germany, 1993.

  28. D. Sheehy, C. Armstrong, and D. Robinson, Shape description by medial surface construction, IEEE Transaction on Visualization and Computer Graphics, 1996, 2(1): 62–72.

    Article  Google Scholar 

  29. E. C. Sherbrooke, N. M. Patrikalakis, and E. Brisson, Computation of the medial axis transform of 3-D polyhedra, in Proceedings of the 3rd ACM Symposium on Solid Modeling and Applications, ACM Press, Salt Lake City, Utah, United States, 1995, 187–200.

    Chapter  Google Scholar 

  30. E. C. Sherbrooke, N. Patrikalakis and E. Brisson, An algorithm for the medial axis transform of 3D polyhedral solids, IEEE Transaction on Visualization and Computer Graphics, 1996, 2(1): 44–61.

    Article  Google Scholar 

  31. R. Kimmel, D. Shaked, N. Kiryati, and A. M. Bruckstein, Skeletonization via distance maps and level sets, Computer Vision and Image Understanding: CVIU, 1995, 62(3): 382–391.

    Article  Google Scholar 

  32. W. C. Ma, F. C. Wu, and M. Ouhyoung, Skeleton extraction of 3D objects with radial basis functions, Shape Modeling International, Seoul, Korea, 2003, 207–215.

  33. J. Bloomenthal and C. Lim, Skeletal Methods of Shape Manipulation, Shape Modeling International, Aizu-Wakamatsu, Japan, 1999, 44–47.

  34. E. Zauderer, Partial Differential Equations of Applied Mathematics, Second Edition, John Wiley & Sons, New York, 1988.

  35. J. Stam and E. Fiume, Depicting Fire and Other Gaseous Phenomena Using Diffusion Professes, SIGGRAPH 95, Los Angeles, CA, USA, 1995, 129–136.

  36. M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr, Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 99, Los Angeles, CA, USA, 1999, 317–323.

  37. T. Preußer and M. Rumpf, Anisotropic Nonlinear Diffusion in Flow Visualization, IEEE Computer Society Press, Los Alamitos, USA, 1999, 325–332.

  38. U. Diewald, T. Preußer, and M. Rumpf, Anisotropic diffusion in vector field visualization on euclidean domains and surfaces, IEEE Transaction on Visualization and Computer Graphics, 2000, 6(2): 139–149.

    Article  Google Scholar 

  39. G. Taubin, A Signal Processing Approach to Fair Surface Design, SIGGRAPH 1995, Los Angeles, CA, USA, 1995, 351–358.

  40. L. Kobbelt, S. Campagna, J. Vorsatz, and H. P. Seidel, Interactive Multi-Resolution Modeling on Arbitrary Meshes, ACM Press, New York, 1998, 105–114.

    Google Scholar 

  41. D. Zorin and P. Shröder, Subdivision for Modeling and Animation, ACM Press, New York, 2000.

    Google Scholar 

  42. P. Veron and J. C. Leon, Static polyhedron simplification using error measurements, Computer-Aided Design, 1997, 29(4): 287–298.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Sterne Kessler Goldstein & Fox, P.L.L.C., Washington, 20005, USA

    Haixia DU

  2. National Library of Medicine, National Institutes of Health, Bethesda, 20894, USA

    Terry YOO

  3. Computer Science Department, Stony Brook University, Stony Brook, NY, 11794, USA

    Hong QIN

Authors
  1. Haixia DU
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  2. Terry YOO
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  3. Hong QIN
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Corresponding author

Correspondence to Haixia DU.

Additional information

This research was supported in part by the National Science Foundation (NSF) Information Technology Research under Grant No. IIS-0082035, the NSF under Grant No. IIS-0097646, Alfred P. Sloan Fellowship, Honda Initiation Award, and an appointment of Haixia Du to the NLM Research Participation Program sponsored by the National Library of Medicine and administered by the Oak Ridge Institute for Science and Education.

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DU, H., YOO, T. & QIN, H. PDE-Based Medial Axis Extraction and Shape Manipulation of Arbitrary Meshes. J Syst Sci Complex 21, 609–625 (2008). https://doi.org/10.1007/s11424-008-9138-2

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  • DOI: https://doi.org/10.1007/s11424-008-9138-2

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