The Sizes of Spheres from Profiles in a Thin Slice I. Opaque Spheres

Abstract

Thin slices through specimens am made into slides for use in microscopy. If a specimen consists of opaque spherical particles in a transparent medium, there will be seen through the slice circular profiles of particles and sections through particles. For a random slice. the size distributios of these profiles can be related to the size dktribution of the population of spheres. The extensive literature dealing with this relationship iS surveyed.An important generalization of praotiad importanoe is made with the introduction of a reaolution interval which excludes .from observation those profilea that are too large or too small. How this affects the relationship between the profile and sphere size distributions is considered, and some special cases are used as illustrations.In Part I1 (COLEHAN, 1983) the corresponding results for the case of transparent spheres in an opaque specimen are given.

Unfolding Sphere Size Distributions with a Density Estimator Based on Tikhonov Regularization

“…The relation between both distributions has been studied and discussed in detail (see, e.g., [19,[21][22][23]). Assuming…”

“…(22) has been solved with a standard algorithm for finding roots. Within the optimization, the quantity has been evaluated using the approximations introduced above:…”

Incidence and lattice calculus with applications to stochastic geometry and image analysis

“…(4) holds, namely The adjunction is unique, since any adjunction satisfying (4) must be of the form (5)- (6) The standard example is X"' Y =-X n Yi= 0 for subsets X, Y of an arbitrary space.…”

“…In the related field of mathematical morphology [21] recent work [12,17,18,19,22] suggests that the partial order structure is more natural, and enables one to harness the theory of complete lattices [3,4].…”

The Sizes of Spheres from Profiles in a Thin Slice II. Transparent Spheres

“…The formulae incorporate resolution constraints which cause various profile size ranges to be unobservable. Approximate solutions are obtained under specific resolution constraints when the slice is very thin.The corresponding case of opaque spheres in a transparent material was the subject of Part I (COLEMAN, 1982).…”