Particle Sizes from Sectional Data

Abstract

We propose a new statistical method for obtaining information about particle size distributions from sectional data without specific assumptions about particle shape. The method utilizes recent advances in local stereology. We show how to estimate separately from sectional data the variance due to the local stereological estimation procedure and the variance due to the variability of particle sizes in the population. Methods for judging the difference between the distribution of estimated particle sizes and the distribution of true particle sizes are also provided.

Estimation of variance components of local stereological volume estimators: a pilot study

“…the intersection of K and any line passing through x is a line‐segment). Thus, the profile K ∩ ( x + L ) is also star‐shaped, and it is convenient to use polar coordinates in and to determine the estimators see Pawlas et al (2009). Here, ρ denotes the radial function, i.e.…”

“…We assume that the locations X i are given. In general, it would be possible to admit random locations and describe the particle model by means of stationary marked point processes (Stoyan et al, 1995), see also Pawlas & Jensen (2006) or Pawlas et al (2009).…”

“…It is imperative to be able to assess when such a procedure is justifiable. In a previous study (Pawlas et al , 2009), it was suggested to compare the estimates of the particle volume variance and the local estimator variance. If they are close to one another, we may expect that the estimated volume distribution will mimic the particle volume distribution.…”

“…The main aim of this paper is to investigate the situation when there are dependencies among the particles. We extend work of Pawlas et al (2009), where a sample of independent particles was considered. The dependency structure leads to the definition of variance estimators other than the standard sample variance.…”

Local Stereology of Extremes

“…In contrast to traditional sampling, where the system of particles is sectioned by an isotropic uniform probe, the local sample is representative for the particle population. This fact was exploited in Pawlas et al (2009) for the development of a statistical procedure for obtaining information about particle size distribution from central sections without specific assumptions about particle shape. The motivation for local stereology comes from the study of biological tissues.…”