Skip to main content

Extended morphometric analysis of neuronal cells with Minkowski valuations


Minkowski valuations provide a systematic framework for quantifying different aspects of morphology. In this paper we apply vector- and tensor-valued Minkowski valuations to neuronal cells from the cat's retina in order to describe their morphological structure in a comprehensive way. We introduce the framework of Minkowski valuations, discuss their implementation for neuronal cells and show how they can be used to characterize cells of different morphological categories. We also provide a comparison to a Sholl analysis.

This is a preview of subscription content, access via your institution.


  • S. Douady, Y. Coulder, Phys. Rev. Lett. 68, 2098 (1992)

    Article  ADS  Google Scholar 

  • L. da F. Costa, E.T.M. Manoel, Neuroinformatics 1, 65 (2003)

    Article  Google Scholar 

  • H. Wässle, B.B. Boycott, R.B. Illing, Phil. Trans. R. Soc. 212, 177 (1981)

    Google Scholar 

  • Y. Fukuda, C.F. Hsiao, M. Watanabe, H. Ito, J. Neurophysiol. 52, 999 (1984)

    Google Scholar 

  • A. van Ooyen, J. Duijnhouwer, M.W.H. Remme, J. van Pelt, Network: Comput. Neural Syst. 13, 311 (2002)

    Article  Google Scholar 

  • H.B.M. Uylings, J. van Pelt, Network: Comput. Neural Syst. 13, 397 (2002)

    Article  Google Scholar 

  • L. da F. Costa, T.J. Velte, Journal of Comparative Neurology 404, 33 (1999)

    Article  Google Scholar 

  • G.A. Ascoli, J.L. Krichmar, Neurocomputing 48, 1003 (2000)

    Article  Google Scholar 

  • R.C. Coelho, L. da F. Costa, Neurocomputing 48, 555 (2001)

    Article  MathSciNet  Google Scholar 

  • S. Peng, B. Urbanc, L. Cruz, B.T. Hyman, H.E. Stanley, Proc. Nat. Ac. Sci. 100, 3847 (2003)

    Article  ADS  Google Scholar 

  • F. Rieke, D. Warland, R. de Ruyyter van Steveninck, W. Bialek, Spikes: Exploring the Neural Code (Bradford Books, 1999)

  • K. Morigiwa, M. Tauchi, Y. Fukuda, Neurosci. Res. 10, S131 (1989)

  • R.C. Coelho, L. da F. Costa, Applied Signal Processing 3, 163 (1996)

    Google Scholar 

  • L. da F Costa, E.T.M. Manoel, F. Faucereau, J. Chelly, J. van Pelt, G. Ramakers, Network: Comput. Neural Syst. 13, 283 (2002)

    Article  Google Scholar 

  • T.G. Smith, G.D. Lange, W.B. Marks, J. Neuroci. Methods 69, 133 (1996)

    MATH  Google Scholar 

  • E.P. Rodrigues, M.S. Barbosa, L. da F. Costa, Phys. Rev. E (2004)

  • R.M. Cesar, L. da F. Costa, Biological Cybernetics 79, 347 (1998)

    MATH  Article  Google Scholar 

  • M.S. Barbosa, E.S. Bernardes, L. da F. Costa, Phys. Rev. E 67 (2003)

  • M.S. Barbosa, L. da F. Costa et al., Eur. Phys. J. B 37, 109 (2003)

    Article  ADS  Google Scholar 

  • P. Soille, Morphological Image Analysis. Principles and Applications (Springer, Berlin, 1999)

  • K. Michelsen, H. de Raedt, Phys. Rep. 347, 461 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  • S. Alesker, Geom. Dedicata 74, 241 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  • R. Schneider, Rend. Circ. Mat. Ser II, Suppl. 65, 355 (2000)

    Google Scholar 

  • R. Schneider, R. Schuster, Rend. Circ. Mat. Ser II, Suppl. 70, 295 (2002)

    Google Scholar 

  • C. Beisbart, R. Dahlke, K. Mecke, H. Wagner, in Morphology of Condensed Matter. Physics and Geometry of Spatial Complex Systems, edited by K. Mecke, D. Stoyan (Springer, 2002), Vol. 600 of Lecture Notes in Physics, pp. 238–260, arXiv:physics/0203072

  • C. Beisbart, T. Buchert, H. Wagner, Physica A 293, 592 (2001)

    MATH  Article  ADS  Google Scholar 

  • R.H. Masland, Nature Neuroscience 4, 877 (2001)

    Article  Google Scholar 

  • B.J. O'Brien, T. Isayama, R. Richardson, D.M. Berson, J. Phys. 538.3, 787 (2002)

    Google Scholar 

  • H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer Verlag, Berlin, 1957)

  • W. Weil, in Convexity and its Applications, edited by P.M. Gruber, J.M. Wills (Birkhäuser, Basel, 1983), pp. 360–412

  • S. Alesker, Ann. of Math. 149, 977 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  • H. Hadwiger, C. Meier, Mathematische Nachrichten 56, 361 (1974)

    MathSciNet  Google Scholar 

  • C. Beisbart, K. Mecke, unpublished Manuscript 2005

  • C. Beisbart, Ph.D. thesis, Ludwig–Maximilians–Universität München (2001), archive/00000483/01/Beisbart_Claus.pdf

  • K. Mecke, T. Buchert, H. Wagner, Astronomy and Astrophysics 288, 697 (1994)

    ADS  Google Scholar 

  • M. Pu, D.M. Berson, T. Pan, The journal of Neuroscience 14 (1994)

  • D.M. Berson, M. Pu, E.V. Famiglietti, Journal of Comparative Neurology 399, 269 (1998)

    Article  Google Scholar 

  • R.C.C. et al., Real-Time Imaging 8, 213 (2002)

    Article  Google Scholar 

  • D. Fenchel, C.R. Acad. Sci. Paris 203, 647 (1936), in French

    MATH  Google Scholar 

  • A.D. Alexandrov, Matem. Sb. SSSR 2, 1205 (1937), in Russian, summary in German

    Google Scholar 

  • J. Schmalzing, T. Buchert, A.L. Melott, V. Sahni, B.S. Sathyaprakash, S.F. Shandarin, ApJ 526, 568 (1999)

    Article  ADS  Google Scholar 

  • L. da F. Costa, R.M.C. Jr., Shape Analysis and Classification: Theory and Practice (CRC, 2000)

  • N. Schmitz, Iwarp: Gimp Plugin,

  • J.H. Ward, J. Am. Stat. Ass. 58, 236 (1963)

    Article  Google Scholar 

  • K.L. Whitford, P. Dijkhuizen, F. Polleux, A. Ghosh, Annu. Rev. Neurosci. 25, 127 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to C. Beisbart.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beisbart, C., Barbosa, M., Wagner, H. et al. Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur. Phys. J. B 52, 531–546 (2006).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • 07.05.Kf Data analysis: algorithms and implementation; data management
  • 87.19.La Neuroscience
  • 02.40.Ft Convex sets and geometric inequalities