## 1. Introduction

In microstructures containing dispersed particles (or voids), the ‘average’ particle size is an important microstructural parameter. Different size parameters and averaging procedures lead to different descriptors of average size, such as arithmetic average width ( DeHoff & Rhines 1968), surface area averaged width ( Gokhale, 1985), volume weighted mean volume ( Gundersen & Jensen, 1985), etc. Among these parameters, the arithmetic average width (further ‘arithmetic’ is omitted) is of significant practical importance for ensembles of convex particles. To most scientists, ‘average size’ implies average width. This parameter is built into a number of structure–property correlation models and microstructural evolution models. Therefore, it is of interest to develop stereological techniques for the estimation of the average width of an ensemble of particles dispersed in 3D space.

A tempting strategy to estimate the average width involves the following steps.

**1** Estimation of the total number of particles per unit volume *N*_{V} using the disector technique ( Sterio, 1984), which involves observations on a pair of parallel sectioning planes that are separated by a distance that is less than the smallest particle size.

**2** Estimation of the average number of particles intersected by an IUR sectioning plane of unit area, ¯*N*_{A}.

**3** The average width *D* is then accessible via the well-known stereological relationship ( DeHoff & Rhines, 1968)

Estimation of ¯*N*_{A} requires sampling the 3D structure using isotropic planes. The isotropic sampling is often inefficient, and in a number of cases it is not feasible. Therefore, it is of interest to develop stereological procedures that do not involve sampling the 3D microstructure using isotropic planes.

A number of microscopy techniques yield a projected image of a microstructure rather than 2D sections. For example, X-ray and other radiographic images, transmission microscopy images from thick slices, scanning electron microscopy images from nonplanar surfaces etc. represent projected images of a microstructure and not 2D sectioning planes. Furthermore, the microstructural information obtained from a number of 3D microscopy techniques such as confocal microscopy and tomography can be conveniently cast into projected image form. In the conventional microscopy of biological specimens, the observations on a set of thin focal planes at different depths can be utilized to generate projected image information (e.g. see McMillan *et al*., 1994 ). Alternatively, the test probes for the projected images can be laid directly onto the 3D microstructure for efficient stereological sampling. Therefore, it is of interest to develop stereological techniques to extract 3D microstructural properties from the projected images of microstructure.

Recently, efficient stereological techniques have been developed for quantitative characterization of 1D lineal microstructural features from projected images of vertical slices ( Gokhale, 1990, 1992, 1993) and from total vertical projections ( Cruz-Orive & Howard, 1991). These techniques have been successfully utilized to estimate the length density of cortical microvessels ( McMillan *et al*., 1994 ), the length density of capillaries in the heart ( Batra *et al*., 1993 , 1995), the length density of skeletal muscle fibres ( Artacho-Perula & Roldan-Vilalobos, 1995a, b), the total length of blood vessels from magnetic resonance images ( Roberts *et al*., 1991 ), the length of epidermal nerve fibres, neural dendrites ( Howard *et al*., 1992 , 1993), etc. Therefore, it is fruitful to develop unbiased and efficient stereological methods to estimate other important microstructural properties (such as the average width) from observations on projected images, which is the aim of the present paper.