Skip to main content
SpringerLink
Log in

Untangling the Blum Medial Axis Transform

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

For over 30 years, Blum's Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with form, but it is one that is notoriously difficult to apply in a robust and stable way. It is well documented how a tiny change to an object's boundary can cause a large change in its MAT. There has also been great difficulty in using the MAT to decompose an object into a hierarchy of parts reflecting the natural parts-hierarchy that we perceive. This paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts. A small change in an object's boundary corresponds to a small change in its substance but may involve a large change in its connection information. The problems with Blum's MAT are generated because it does not explicitly represent this dichotomy of information. To use the Blum MAT to it's full potential, this paper presents a method for separating the substance and connection information of an object. This provides a natural parts-hierarchy while eliminating instabilities due to small boundary changes. The method also allows for graded, fuzzy classifications of object parts to match the ambiguity in human perception of many objects.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • August, J., Siddiqi, K., and Zucker, S. 1999. Ligature instabilities in the perceptual organization of form. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition. Fort Collins, Colorado, pp. 42–48.

  • Blum, H. 1967. A transformation for extracting new descriptors of form. In Models for the Perception of Speech and Visual Form, W. Whaten-Dunn (Ed.). MIT Press: Cambridge, MA, pp. 362–380.

    Google Scholar 

  • Blum, H. and Nagel, R. 1978. Form description using weighted symmetric axis features. Pattern Recognition, 10: 167–180.

    Google Scholar 

  • Brady, M. and Asada, H. 1984. Smoothed local symmetries and their implementation. International Journal of Robotics Research, 3(3):36–61.

    Google Scholar 

  • Bruce, J., Giblin, P., and Gibson, C. 1985. Symmetry sets. Proc. Royal Society Edinburgh, 101A:163–186.

    Google Scholar 

  • Burbeck, C., Pizer, S., Morse, B., Ariely, D., Zauberman, G., and Rolland, J. 1996. Linking object boundaries at scale—A common mechanism for size and form judgements. Vision Research, 36(3):361–372.

    Google Scholar 

  • Katz, R. 2002. Form Metrics for Interactive Rendering via Figural Models of Perception. Dissertation, University of North Carolina at Chapel Hill, Chapel Hill, NC.

  • Kovacs, I. and Julesz, B. 1994. Perceptual sensitivity maps within globally defined visual forms. Nature, 370:644–646.

    Google Scholar 

  • Marr, D. and Nishihara, K. 1978. Representation and recognition of the spatial organization of three dimensional structure. Proc. of the Royal Society of London, 200:269–294.

    Google Scholar 

  • Ogniewicz, R. and Ilg, M. 1992.Voronoi skeletons: Theory and applications. Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 63–69.

  • Orban, G., Kato, H., and Bishop, P., 1979a. End-zone region in receptive fields of hypercomplex and other striate neurons in the cat. Journal of Neurophysiology, 42:818–832.

    Google Scholar 

  • Orban, G.A., Kato, H., and Bishop, P.O. 1979b. Dimensions and properties of end-zone inhibitory areas in receptive fields of hypercomplex cells in cat striate cortex. Journal of Neurophysiology, 42:833–849.

    Google Scholar 

  • Pizer, S., Oliver, W., and Bloomberg, S. 1987. Hierarchical form description via the multiresolution symmetric axis transform. IEEE Trans. on Pattern Analysis and Machine Intelligence, 9(4):505–511.

    Google Scholar 

  • Pizer, S., Eberly, D., Fritsch, D., and Morse, B. 1998. Zoom-invariant vison of figural form: The mathematics of cores. Computer Vision and Image Understanding, 69(1):53–71.

    Google Scholar 

  • Shaked, D. and Bruckstein, A. 1998. Pruning medial axes. Computer Vision and Image Understanding, 69(2):156–169.

    Google Scholar 

  • Szekely, G. 1996. Form Characterization by Local Symmetries. Habilitationsschrift: Institut fur Kommunikationstechnik, Fachgruppe Bildwissenschaft, ETH, Zurich.

Download references

Author information

Authors and Affiliations

  1. University of North Carolina at Chapel Hill, USA

    Robert A. Katz & Stephen M. Pizer

Authors
  1. Robert A. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen M. Pizer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katz, R.A., Pizer, S.M. Untangling the Blum Medial Axis Transform. International Journal of Computer Vision 55, 139–153 (2003). https://doi.org/10.1023/A:1026183017197

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026183017197

Search

Navigation