Hemisphere Sizing from Linear Intercept Measurement

Wilfried Gille

doi:10.3844/jmssp.2005.203.211

Abstract

The stereological problem of unfolding the hemisphere radius distribution from the length distribution is analyzed. Let a stationary isotropic process of hemispheres be given. The hemispheres have random diameters and are isotropically uniformly randomly orientated in space. A straight line probe yields a process of intercepts. The inverse problem of re-obtaining the size distribution of the hemispheres in terms of an experimental intercept length distribution is solved. The length distribution of a single hemisphere, known analytically, is approximated by piecewise polynomials in two intervals. The solution of the inverse problem is traced back to a simple recurrence equation. Numerical checks with exact and simulated data are performed to demonstrate the applicability. Data of chord length sampling, resulting from image analysis procedures, from scattering methods or from other appropriate physical apparatuses, are applicable.

Highlights

One of the fundamental problems in stereology is the derivation of the size distribution of geometric objects from partial information which is contained in the length distribution of linear intersections of the particles (Fig. 1)
The standard problem of tracing back the size distribution of hemispheres to linear intercept measurement occurs in different fields, involving specificities
The functions K (r) and γ (r) are assumed to be exclusively influenced by the sizes of the hemispheres. This is trivial in cases, if image material is available and exclusively chord lengths inside the. This project starts with the analysis of the relation between f (R), working function (WF) and Chord length distributions (CLDs), operating with the analytical expression γ H (r, R) for the single hemisphere of radius R (Fig. 2)

Summary

INTRODUCTION

One of the fundamental problems in stereology is the derivation of the size distribution of geometric objects (particles) from partial information which is contained in the length distribution of linear intersections of the particles (Fig. 1). The analytical expression for the CLD of the hemisphere with a fixed radius R , AH (l, R) , is known, Gille[10,14] (Fig. 2) The motivation of these calculations are experimental results, showing the existence of hemispherical micro-particles, Appendix C. The functions K (r) and γ (r) are assumed to be exclusively influenced by the sizes of the hemispheres This is trivial in cases, if image material is available and exclusively chord lengths inside the. This project starts with the analysis of the relation between f (R) , WF and CLD, operating with the analytical expression γ H (r, R) for the single hemisphere of radius R (Fig. 2).

Averaging over different radii

Application and stability

CONCLUSION