a surface is a real problem which can be solved by techniques well-known from computer Finstellb 
graphics. Given a smooth surface, a straightforward algorithm for computing the surface Rekristal 
area as well as the two curvature integrals could be based on conventional integration over 
this surface, [6]. Clemens 
The technique presented here is quite different from this approach. Due to Crofton’s inter- Es 
section formulae and Hadwiger’s recursive definition of the Euler number, we are able to 
estimate the surface area and the two curvature integrals without having to localize the 
surface, and hence, we do not need any model for the smoothness of the surface. Our tech- A 
nique is closely related to the method of estimating the surface density from the correlation 
function, as first discussed in [7]. However, it does not evaluate the full correlation function, An kaltgez 
and it involves a filtering that is not used when calculating correlation functions. The prob- oradienten 
lem reduces to the numerical integration of functions defined on the unit sphere. For both dienten ert 
approaches the accuracy of estimation depends on the numerical accuracy of integration (i.e. von außen 
it depends on the chosen quadrature rule). eine Temp 
The length of the vector of numbers of neighborhood configurations is equal to the total über 800° 
number of different configurations occurring in the binary image. Hence, the vector length gänge CIE 
does not depend on the image size itself, it depends only on the size of the applied filter erhöhl 
mask. This is a clear advantage over other techniques, in particular, for large spatial images a 
or when data have to be accumulated from a series of images of the same specimen but of Die ya 
different sizes. Hence, the ‘analysis step’ can be performed very easily and quickly, and the a 
algorithm for the statistical estimation of the geometric characteristics can be presented in "oo I 
a well-structured form. Seren 
By means of conventional 2d image analysis only a part of the four basic characteristics Vy, 
Sv, My, and Ky is available. If the the microstructure is macroscopically homogeneous and Expenme 
isotropic, the three characteristics Vyv, Sy, and My can be stereolocically estimated from 
a planar section of the microstructure, but if the microstrcture is not isotropic then only Der Wei 
the volume density Vi, can be unbiasedly estimated by means of 2d image analysis. In this sen mit ei 
respect, 3d image analysis has clear advantages over 2d image analysis. Finally, we remark were 
that 3d image analysis can be also very helpful in cases of macroscopically non-homogeneous ud Ce 
microstructures having a geometrical gradient. lographisc 
lenden K 
Whee 
References Ken 
[1] R. Schneider (1993) Conver Bodies: The Brunn-Minkowski Theory. Encyclopedia of 
Mathematics and Its Application, Vol. 44. Cambridge University Press, Cambridge. Ett. 
[2] C. Lang, J. Ohser, and R. Hilfer (2000) J. Microscopy 198, to be appeared. a 
[3] J. Ohser and W. Nagel (1996) J. Microscopy 184, S. 117-126. Bild 1 ze 
[4] W. Nagel ,J. Ohser, and K. Pischang (2000) J. Microscopy 198 , 54-62. Abstand ) 
[5] J. Ohser and F. Miicklich (2000) Statistical Analysis of Microstructures in Materials ans 
Science. J. Wiley & Sons. Chichester. iD } 
[6] D. Cohen-Or and A. Kaufman (1995) Graph. Mod. and Image Processing 57, 453-461. Ka 
[7] P. Debye, H.R. Anderson, and H. Brumberger (1957) J. Appl. Phys. 28, 679-683. nn 
Der Anl 
Bild 2 
Vom Ra 
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