Abstract
	The uniform equiaxed grain structure of the examined system is suitable for a detailed metallographic analysis based on the grain size estimation methods recommended by the National Standards and closely imitating the approaches proposed by ASTM E-112 Standards. In particular, it supply a possibility to come the assumptions inherent to Standards with the recent results obtained by stochastic simulations of space-filling cell systems – random Voronoi tessellations – in the case of one real grain structure.
	The estimation of the grain size is presented for Cu-based composite prepared by powder metallurgy method of mechanical alloying. Microstructural data were obtained by manual analysis of planar and linear sections. Grain size estimation was carried out by using an improved approach based on an extensive database of spatial tessellations and combining both line intercept and planar profile counts together with profile area variance. 
	The volume fraction VV of the dispersed Al2O3 particles was estimated by the lattice point count. The both Jeffries and Heyn methods, namely profile and intercept counts, have been applied in order to obtain simultaneously the estimates of the profile intensity (density) NA and of the intercept intensity (density) NL. There is a fixed relation between NL and NA, namely NA  = cNL2, where c = 1/1.26 = 0.7937. The estimates of the grain size number G based on NL and NA are identical see Eq.1. Simultaneously, the chords lengths l have been directly measured: their mean length is [El] = 1/[NL], CV l is their coefficient of variation. There is a fixed relation between the spatial grain intensity NV and the planar feature intensities NL, NA, see Eq.2, (the underlying idea is that the system of profile chords is equivalent to the system of uniform random chords of a ball the volume of which is 1/NV ).
	The estimator of the mean profile area is [Ea] = 1/[NA]. Areas of profiles completely included within measuring window were estimated by the lattice point count. The coefficient of variation of such a size-weighted sample was used as a rough estimate of the coefficient of variation CV a of the whole induced planar tessellation. 
	A natural question whether there exists some spatial tessellation for which such assumptions are valid with a reasonable accuracy has been answered by Horálek [6]: it is the non-homogeneous Johnson-Mehl model with the rate of germ nucleation I = ?t, where t is time and ? is an arbitrary constant. The stochastic simulations carried out by Saxl and Ponížil [3] have shown that there are also several convex tessellations for which the ASTM assumptions are approximately valid. However, all such tessellations are far from being reasonably uniform as may be seen from the values of their coefficients of variation.