Skip to main content
Springer Nature Link
Log in

Voronoi-Based Medial Axis Approximation from Samples: Issues and Solutions

  • Chapter
Transactions on Computational Science XX

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 8110))

Abstract

Continuous curves are approximated by sample points, which carry the shape information of the curve. If sampling is sufficiently dense, the sample points can be used to extract the structural properties of the curve (e.g., crust, medial axis, etc.). This article focuses on approximation of medial axis from sample points. Especially, we review the methods that approximate the medial axis using Voronoi diagram. Such methods are extremely sensitive to noise and boundary perturbations. Thus, a pre- or post-processing step is needed to filter irrelevant branches of the medial axis, which are introduced in this article. We, then, propose a new medial axis approximation algorithm that automatically avoids irrelevant branches through labeling sample points. The results indicate that our method is stable, easy to implement, robust and able to handle sharp corners, even in the presence of significant noise and perturbations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
USD 18.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Blum, H., et al.: A Transformation for Extracting New Descriptors of Shape. Models for the Perception of Speech and Visual Form 19, 362–380 (1967)

    Google Scholar 

  2. Blum, H., Nagel, R.N.: Shape Description Using Weighted Symmetric Axis Features. Pattern Recognition 10, 167–180 (1978)

    Article  MATH  Google Scholar 

  3. Bookstein, F.L.: The Line-Skeleton. Computer Graphics and Image Processing 11, 123–137 (1979)

    Article  Google Scholar 

  4. Brandt, J.W., Jain, A.K., Ralph Algazi, V.: Medial Axis Representation and Encoding of Scanned Documents. Journal of Visual Communication and Image Representation 2, 151–165 (1991)

    Article  Google Scholar 

  5. Gross, L.M.: Transfinite Surface Interpolation over Voronoi Diagrams. PhD Thesis, Arizona State University (1995)

    Google Scholar 

  6. Chou, J.J.: Numerical Control Milling Machine Tool path Generation for Regions Bounded by Free Form Curves and Surfaces. PhD Thesis, University of Utah Salt Lake City, UT, USA (1989)

    Google Scholar 

  7. O’rourke, J.: Computational Geometry in C. Cambridge University (1998)

    Google Scholar 

  8. Gursoy, H.N., Patrikalakis, N.M.: Automatic Coarse and Fine Surface Mesh Generation Scheme Based on Medial Axis Transform: Part I Algorithms. Engineering with Computers (New York) 8, 121–137 (1992)

    Article  Google Scholar 

  9. Sherbrooke, E.C., Patrikalakis, N.M., Brisson, E.: An Algorithm for the Medial Axis Transform of 3d Polyhedral Solids. IEEE Transactions on Visualization and Computer Graphics 2, 44–61 (1996)

    Article  Google Scholar 

  10. Hisada, M., Belyaev, A.G., Kunii, T.L.: A Skeleton-Based Approach for Detection of Perceptually Salient Features on Polygonal Surfaces. Computer Graphics Forum 21, 689–700 (2002)

    Article  Google Scholar 

  11. Hoffmann, C.M.: Geometric and Solid Modeling: An Introduction. Morgan Kaufmann (1989)

    Google Scholar 

  12. Gold, C.: Crust and Anti-Crust: A One-Step Boundary and Skeleton Extraction Algorithm. In: Proceedings of the Fifteenth Annual Symposium on Computational Geometry, pp. 189–196 (1999)

    Google Scholar 

  13. Gold, C., Snoeyink, J.: A One-Step Crust and Skeleton Extraction Algorithm. Algorithmica 30, 144–163 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gold, C., Dakowicz, M.: The Crust and Skeleton–Applications in GIS. In: Proceedings of 2nd International Symposium on Voronoi Diagrams in Science and Engineering, pp. 33–42 (2005)

    Google Scholar 

  15. Lam, L., Lee, S.W., Suen, C.Y.: Thinning Methodologies-a Comprehensive Survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 869–885 (1992)

    Google Scholar 

  16. Ogniewicz, R.L., Kübler, O.: Hierarchic Voronoi Skeletons. Pattern Recognition 28, 343–359 (1995)

    Article  Google Scholar 

  17. Borgefors, G.: Centres of Maximal Discs in the 5-7-11 Distance Transform. In: Proceedings of the Scandinavian Conference on Image Analysis, vol. 1, pp. 105–105 (1993)

    Google Scholar 

  18. Arcelli, C., Frucci, M.: Reversible Skeletonization by (5, 7, 11)-Erosion. In: Proceedings of the International Workshop on Visual Form: Analysis and Recognition, pp. 21–28 (1992)

    Google Scholar 

  19. Chou, J.J.: Voronoi Diagrams for Planar Shapes. IEEE Computer Graphics and Applications 15, 52–59 (1995)

    Article  Google Scholar 

  20. Ramanathan, M., Gurumoorthy, B.: Constructing Medial Axis Transform of Planar Domains with Curved Boundaries. Computer-Aided Design 35, 619–632 (2003)

    Article  Google Scholar 

  21. Amenta, N., Bern, M.W., Eppstein, D.: The Crust and the Beta-Skeleton: Combinatorial Curve Reconstruction. Graphical Models and Image Processing 60, 125–135 (1998)

    Article  Google Scholar 

  22. Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Prentice Hall, Upper Saddle River (2002)

    Google Scholar 

  23. Russ, J.C.: The Image Processing Handbook. CRC Press (2002)

    Google Scholar 

  24. Wenger, R.: Shape and Medial Axis Approximation from Samples. PhD Thesis. The Ohio State University (2003)

    Google Scholar 

  25. Cheng, S.W., Funke, S., Golin, M., Kumar, P., Poon, S.H., Ramos, E.: Curve Reconstruction from Noisy Samples. Computational Geometry 31, 63–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dey, T.K., Wenger, R.: Fast Reconstruction of Curves with Sharp Corners. International Journal of Computational Geometry and Applications 12, 353–400 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ledoux, H.: Modelling Three-Dimensional Fields in Geo-Science with the Voronoi Diagram and Its Dual. Ph.D Thesis. School of Computing, University of Glamorgan, Pontypridd, Wales, UK (2006)

    Google Scholar 

  28. Gavrilova, M., Ratschek, H., Rokne, J.G.: Exact Computation of Delaunay and Power Triangulations. Reliable Computing 6, 39–60 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Karimipour, F., Delavar, M.R., Frank, A.U.: A Simplex-Based Approach to Implement Dimension Independent Spatial Analyses. Journal of Computer and Geosciences 36, 1123–1134 (2010)

    Article  Google Scholar 

  30. Giesen, J., Miklos, B., Pauly, M.: Medial Axis Approximation of Planar Shapes from Union of Balls: A Simpler and More Robust Algorithm. In: Canad. Conf. Comput. Geom., pp. 105–108 (2007)

    Google Scholar 

  31. Edelsbrunner, H.: The Union of Balls and Its Dual Shape. Discrete & Computational Geometry 13, 415–440 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Amenta, N., Choi, S., Kolluri, R.K.: The Power Crust. In: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 249–266 (2001)

    Google Scholar 

  33. Amenta, N., Kolluri, R.K.: The Medial Axis of a Union of Balls. Computational Geometry 20, 25–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Attali, D., Montanvert, A.: Computing and Simplifying 2d and 3d Continuous Skeletons. Computer Vision and Image Understanding 67, 261–273 (1997)

    Article  Google Scholar 

  35. Miklos, B., Giesen, J., Pauly, M.: Medial Axis Approximation from Inner Voronoi Balls: A Demo of the Mesecina Tool. In: Proceedings of the Twenty-third Annual Symposium on Computational Geometry, pp. 123–124 (2007)

    Google Scholar 

  36. Tam, R.C.: Voronoi Ball Models for Computational Shape Applications. PhD thesis, The University of British Columbia (2004)

    Google Scholar 

  37. Karimipour, F., Delavar, M.R., Frank, A.U.: A Mathematical Tool to Extend 2D Spatial Operations to Higher Dimensions. In: Gervasi, O., Murgante, B., Laganà, A., Taniar, D., Mun, Y., Gavrilova, M.L. (eds.) ICCSA 2008, Part I. LNCS, vol. 5072, pp. 153–164. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  38. Alliez, P., Devillers, O., Snoeyink, J.: Removing Degeneracies by Perturbing the Problem or Perturbing the World. Reliable Computing 6, 61–79 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mokhtarian, F., Mackworth, A.: A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves (Pdf). IEEE Transactions on Pattern Analysis and Machine Intelligence 14 (1992)

    Google Scholar 

  40. Siddiqi, K., Bouix, S., Tannenbaum, A., Zucker, S.W.: Hamilton-Jacobi Skeletons. International Journal of Computer Vision 48, 215–231 (2002)

    Article  MATH  Google Scholar 

  41. Siddiqi, K., Kimia, B.B., Shu, C.W.: Geometric Shock-Capturing Eno Schemes for Subpixel Interpolation, Computation and Curve Evolution. Graphical Models and Image Processing 59, 278–301 (1997)

    Article  Google Scholar 

  42. Attali, D., Montanvert, A.: Modeling Noise for a Better Simplification of Skeletons. In: Proceedings of the International Conference on Image Processing, vol. 3, pp. 13–16 (1996)

    Google Scholar 

  43. Attali, D., di Baja, G., Thiel, E.: Pruning Discrete and Semicontinuous Skeletons. In: Braccini, C., Vernazza, G., DeFloriani, L. (eds.) ICIAP 1995. LNCS, vol. 974, pp. 488–493. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  44. Malandain, G., Fernández-Vidal, S.: Euclidean Skeletons. Image and Vision Computing 16, 317–327 (1998)

    Article  Google Scholar 

  45. Chazal, F., Lieutier, A.: The “Lambda Medial Axis”. Graphical Models 67, 304–331 (2005)

    Article  MATH  Google Scholar 

  46. Attali, D., Montanvert, A.: Semicontinuous Skeletons of 2d and 3d Shapes. In: Aspects of Visual Form Processing, pp. 32–41 (1994)

    Google Scholar 

  47. Bai, X., Latecki, L.J.: Discrete skeleton evolution. In: Yuille, A.L., Zhu, S.-C., Cremers, D., Wang, Y. (eds.) EMMCVPR 2007. LNCS, vol. 4679, pp. 362–374. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  48. Giesen, J., Miklos, B., Pauly, M., Wormser, C.: The Scale Axis Transform. In: Proceedings of the 25th Annual Symposium on Computational Geometry, pp. 106–115 (2009)

    Google Scholar 

  49. Mesecina: computational geometry you can see, http://www.balintmiklos.com/mesecina

  50. Karimipour, F., Ghandehari, M.: A Stable Voronoi-Based Algorithm for Medial Axis Extraction through Labeling Sample Points. In: Proceedings of the 9th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2012), New Jersey, USA, pp. 109–114 (2012)

    Google Scholar 

  51. Ghandehari, M., Karimipour, F.: Voronoi-Based Curve Reconstruction: Issues and Solutions. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part II. LNCS, vol. 7334, pp. 194–207. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  52. Karimipour, F., Ghandehari, M., Ledoux, H.: Medial Axis Approximation of River Network for Catchment Area Delineation. In: Proceedings of the International Workshop on Geoinformation Advances. Lecture Notes in Geoinformation and Cartography (LNG&C), p. 223. Springer, Johor (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran, Iran

    Farid Karimipour & Mehran Ghandehari

Authors
  1. Farid Karimipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehran Ghandehari
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Calgary, Calgary, AB, Canada

    Marina L. Gavrilova

  2. CloudFabriQ Ltd., Warnford Court, 29 Throgmorton Street, EC2N 2AT, London, UK

    C. J. Kenneth Tan

  3. Rutgers University, New Brunswick, NJ, USA

    Bahman Kalantari

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Karimipour, F., Ghandehari, M. (2013). Voronoi-Based Medial Axis Approximation from Samples: Issues and Solutions. In: Gavrilova, M.L., Tan, C.J.K., Kalantari, B. (eds) Transactions on Computational Science XX. Lecture Notes in Computer Science, vol 8110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41905-8_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-41905-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41904-1

  • Online ISBN: 978-3-642-41905-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics