Multi‐dimensional perspectives on stereology

Abstract

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 formulae 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 I t I 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 formulae 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 formulae 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 2 3, are included in the final Section 11.Equations which are the specializations of multi-dimensional formulae 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. I N T R O D U C T I O NThe reader leafing through the Proceedings of the Second International Congress for Stereology (Elias, 1967) might well wonder whether the arrays of formulae he finds are part of some greater scheme. The answer is an emphatic 'yes', and it is the purpose of this paper to systematically develop at least part of this scheme, which is simply the d-dimensional generalization of the planar (d = 2) and spatial 181 R. E. Miles (d = 3) schemes. The formule obtained are those which would be applied by creatures inhabiting higher-dimensional spaces in inferring properties of opaque regions of their own space from, for example, a three-dimensional flat section like our own 'real' world. Many of the 'practical' cases, i.e. for d s 3, are included, being distinguished by an asterisk after the equation number. An excellent non-mathematical discussion of the role...

A comprehensive set of stereological formulae for embedded aggregates of not‐necessarily‐convex particles

“…'~ over j have random ranges, running from 1 to either n, or n , . For derivations of these formulae (except (9) and (14)), see Miles (1972) and Miles & Davy (1976). (9) corresponds to equivalent formulae in Jensen & Gundersen (1983), while (14) follows from formulae in Miles (1979); self-contained derivations are deferred to Section 6.…”

Early History of Geometric Probability and Stereology

“…In 1902 it was translated into French and it remained a classic until Robert Deltheil (1890-1972, a professor at the Toulouse University, published his book (Deltheil, 1926). The first and more extensive part of Czuber's monograph is devoted to geometric probability itself and it provides not only a detailed exposition of results achieved by French and British predecessors, supplemented with historical remarks and many exercises, but also new results and generalizations.…”

“…Nobody realized their potential practical utilization. The stereological formulas, simply deducible from Barbier's theorems, were rediscovered as late as the mid-20 th century by Sarkis Andreevich Saltykov (1945;1946;, Sergei Ivanovich Tomkeieff (1945) and others, 10 and they turned out to be part of the general multidimensional theory (Miles, 1972).…”

Stereology: A Historical Survey

“…The relevant books of Saltykov (1958), DeHoff and Rhines (1968), Underwood (1970), and references therein, illustrate this. Even Miles (1972) was mainly concerned with ratios. In the biological context, stereological equations were still derived with reference to a 'cube of material'.…”

“…8 are derived in the aforementioned integral geometry books. Miles (1972) gave generalized ratio estimators. General versions of Eq.…”