Hemisphere Sizing from Linear Intercept Measurement

The stereological problem of unfolding the hemisphere radius distribution from the chord 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 chord 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.