Abstract of ``Stereology as Inverse Problem''
Stereology is the part of imaging science in which the
three-dimensional structure of a body is determined from
two-dimensional views.
Although it is relatively
easy to determine volume information from 2-D slices, it is nontrivial
in general to determine other physical properties such as internal
surface areas unless the medium
is known to have some simple symmetry
such as isotropy.
For this reason, stereology can be viewed as a type of inverse problem.
In earlier work I showed that an anisotropic
spatial correlation function of a random porous medium could be used to
compute the specific surface area when it is stationary as well as anisotropic
by first performing a three-dimensional radial average and
then taking the first derivative with respect to lag at the origin.
This result generalized the earlier result for isotropic porous media
of Debye et al. (1957).
Here I provide more detailed information about the use of
spatial correlation functions for anisotropic porous media and in
particular I show that, for stationary anisotropic media, the specific
surface area can be related to the derivative of the two-dimensional
radial average of the correlation function measured from
cross sections taken through the anisotropic medium. The main
concept is first illustrated
using a simple pedagogical example for an anisotropic distribution of
spherical voids. Then, a general derivation of formulas relating the
derivative of the planar correlation functions to surface integrals is
presented. When the surface normal is uniformly distributed (as is
the case for any distribution of spherical voids), my formulas can be
used to relate specific surface area to easily measureable
quantities from any single cross section. When the surface normal is
not distributed uniformly (as would be the case for an oriented
distribution of ellipsoidal voids), my results show how to obtain
valid estimates of specific surface area by averaging measurements on three
orthogonal cross sections.
One important general observation for porous media is that the
surface area from nearly flat cracks may be underestimated from
measurements on orthogonal cross sections if any of the cross sections
happen to lie in the plane of the cracks. This result is illustrated
by taking the very small aspect ratio (penny-shaped crack) limit
of an oblate spheroid, but holds for other types of flat surfaces as well.
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