Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

Mean glomerular volume has previously been estimated, using stereological techniques, specifically the point-sampled intercept (PSI), either from isotropic or from vertical sections. As glomeruli are approximately spherical structures, the same stereological technique was carried out on vertical and arbitrary sections to determine whether section orientation had any effect on mean glomerular volume estimation. Equine kidneys from 10 individuals were analysed using the PSI method of estimating volume-weighted mean glomerular volume (MGV); for each kidney, arbitrary and vertical sections were analysed. MGVs were not significantly different between arbitrary and vertical sections (P = 0.691) when analysing the data with the paired t test; when plotting MGV estimates from arbitrary sections against those from vertical sections the intercept was found not to be significantly different from zero (P > 0.8) and the slope of the regression line not to be significantly different from 1.0 (P > 0.4). For the estimation of MGV in equine kidneys using PSI, arbitrary sections may be used if it is not possible to use isotropic or vertical sections, but some caution must be exercised in the interpretation of results so gained.