THE EULER NUMBER FROM THE DISTANCE FUNCTION

Authors

  • Ximo Gual-Arnau University Jaume I of Castellón

DOI:

https://doi.org/10.5566/ias.v32.p175-181

Keywords:

critical points, distance function, Euler number, stereology, tangent counts

Abstract

We present a new method to obtain the Euler number of a domain based on the tangent counts of concentric spheres in ℝ³ (or circles in ℝ², with respect to the center O, that sweeps the domain. This method is derived from the Poincaré-Hopf Theorem, when the index of critical points of the square of the distance function with respect to the origin O are considered.

References

J.W. Bruce, P.J. Giblin (1984). Curves and singularities, Cambridge University Press.

R. T. De Hoff (1987). Use of the disector to estimate the Euler characteristics of three-dimensional micro structures. Acta Stereol. 6, 133-140.

X. Gual-Arnau and J. J. Nun ̃o-Ballesteros (2001). A Stereological Version of the Gauss-Bonnet Formula. Geometriae Dedicata. 84, 253-260.

H. J. G. Gundersen, R. W. Boyce, J.R. Nyengaard, A. Odgaard (1993). The conneulor: unbiased estimation of connectivity using physical disectors under projection, Bone 14, 217–222.

H. Hadwiger (1957). Vorlesungen, U ̈ber Inhalt, Oberfla ̈che und Isoperimetrie, Springer-Verlag, Berlin.

M. Morse (1929). Singular points of vector fields under general boundary conditions, American Journal of Mathematics 51, 165–178.

J. Osher, W. Nagel (1996). The estimation of the

Euer-Poincare ́ characteristic from observations on parallel sections, J. Microsopy 184 , 117–126.

J. Rataj (2004). On estimation of the Euler number by projections on thin slabs, Adv. Appl. Prob. (SGSA) 36 , 715–724.

L. A. Santaló (1976). Integral Geometry and Geometric Probability, Addison-Wesley Publishing Company Inc., London.

E.Teufel (1982). Differential Topology and the Computation of Total Absolute Curvature, Math. Ann 258, 471–480.

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Published

2013-11-12

Issue

Section

Original Research Paper

How to Cite

Gual-Arnau, X. (2013). THE EULER NUMBER FROM THE DISTANCE FUNCTION. Image Analysis and Stereology, 32(3), 175-181. https://doi.org/10.5566/ias.v32.p175-181