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 speciic surface area when it is stationary as well as anisotropic by rst performing a three-dimensional radial average and then taking the rst 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 speciic 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 rst 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 speciic 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 speciic surface area by averaging measurements on three orthogonal cross sections. One important general observation for porous media is that the surface area from nearly at 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 at surfaces as well.