Multi‐dimensional perspectives on stereology

Publisher Website
Google Scholar
Abstract
SUMMARY: The structures of primary concern in Part A are varieties—the generalization in higher dimensions of smooth curves and surfaces. Fixed orientation random s‐sections (s‐dimensional flat sections) of a fixed t‐variety (t‐dimensional variety) in Rd (d‐dimensional space) are considered in Section 2. The t‐variety induces a density‐cum‐orientation distribution which is related to corresponding sectional quantities. The by now classical basic formula; of stereology are all special cases of the simple multi‐dimensional formula (2.16). Next (Section 3), statistics of the variety of intersection of several statistically homogeneous random varieties are related to the corresponding statistics of the parent varieties.Part B is concerned with the analysis by random s‐sections of fixed aggregates of t‐dimensional opaque particles (i.e. varieties for 1 ≤ t ≤ d — 1, domains for t = d) embedded within an opaque specimen. Fixed orientation random sections are considered in Section 5, and isotropically oriented ones in Sections 6–10. It is shown in Section 7 that the mean particle ‘caliper diameter’, a key quantity, may in theory be estimated by isotropic slice sectioning. The theory is particularly rich when the particles are convex, witness the arrays of useful formula; in Sections 8–10. Crofton's remarkable ‘second theorem’ comes into its own in Section 9, permitting simple estimates from isotropic test lines of mean area squared to mean perimeter (when d = 2) and mean volume squared to mean surface area (when d = 3); in fact, the formulæ of Sections 7–9 suggest estimates for both mean and variance of the areas (when d = 2) and volumes (when d = 3) of aggregates of embedded convex particles. Further results for aggregates of convex polytopes are given in Section 10. Certain general considerations and principles relating to estimation from random sections in the practical cases, i.e. d ≤ 3, are included in the final Section 11.Equations which are the specializations of multi‐dimensional formulæ to the practical cases are asterisked. A feature is the attention given to the proper probabilistic specification of random sections through the specimen, a matter that seems largely to have been ignored.

This publication has 10 references indexed in Scilit: