The Application of 3d Image Analysis to the
Characterization of Materials
Joachim Ohser
Institute of Industrial Mathematics*, Kaiserslautern
Abstract
The analysis of 3d images is a new technique which is of general interest in quantitative
metallography since there is a continuously growing number of 3d images of microstructures
usually taken by X-ray microtomograpy or confocal microscopy. A new method is presented
which is strictly based on integral-geometric formulae. By means of this approach the basic
characteristics can be expressed as inner products of two vectors where the first one carries
the ‘integrated local knowledge’ about the microstructure and the second one depends on
sine, Lepaig [973 the lateral resolution of the image as well as the quadrature rules used in the discretization
of the integral-geometric formulae. As an example of application we consider the analysis
of 3d microtomographic images obtained from the structures open nickel foam and sinter
Dresden 1989 copper material.
art, 1978
1 Basic Characteristics of Microstructures
mpufer fomogra-
urg, MD (US) In stochastic geometry, components (or phases) of microstructures are usually considered as
-100 A sets which may have ‘smooth’ surfaces almost everywhere. In the simplest case a component
raphe für die is described by a locally finite union of compact convex sets. We assume that the structure
56,5. 189-184 is microscopically heterogeneous but macroscopically homogeneous, i.e. a component « is
Jalen, Compe. 1 J deled as a macroscopically homogeneous random set. The homogeneity of a allows us
mar, Stuttgart to introduce the following geometric characteristics: the volume density Vy, the surface
sstörungsfieie ib density Sy, the specific integral of mean curvature My, and the specific integral of total
Er curvature Ky. These quantities play a central role in the quantitative characterization of
ire Prüfung oo microstructures components. Up to multiplicative constants, these geometric characteristics
a are the densities of the quermassintegrals defined for a homogeneous random set, and the
ame BI 0 list of these four basic geometric characteristics is complete in ‘some sense’ (see Hadwiger’s
SE characterization theorem, [1]). In other words, in this sense it is sufficient to characterize a
ae Ans macroscopically homogeneous microstructure (consisting of two components) using the basic
mon ori characteristics Vy, Sy, My, and Ky.
logie, Berlin 19% In particular, the basic characteristic of the second component of the microstructure can be
ha 4 computed from that of the first one. Let 3 denote the second component (in our example
«" Divlomarbett the pore space) then obviously Vy(a) = 1 — Vy(B) and Sv(a) = Sy (8) but, furthermore,
under some weak assumptions made for the smoothness of the interface between and 3 we
er-Doincere che get the relationships My (a) = —My (8) and Ky (a) = Kv(B), [2].
. Friedrich Schl
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