Journal of Microscopy, Vol. 173, Pt 1, January 1994, pp. 67-72. Received 30 October 1 9 92; accepted 28 September 1 9 9 3 Some remarks on the accuracy of surface area estimation using the spatial grid K. SANDAU* & U. HAHN lnstitut fur Angewandte Mathematik und Statistik. Universitat Hohenheim. D-70593 Stuttgart, Germany Key words. Counting rule, covariogram, spatial grid, stereology, surface area estimation, variance estimation. Summary (1993) estimated the external surface area of a lung with A set of three line grids in three orthogonal directions is called a spatial grid. This spatial grid can be used for surface area estimation by counting the number of intersection points of a surface with the grid lines. If direction and localization of the spatial grid are suitably randomized, the expectation of this number is proportional to the surface area of interest. The method was especially developed for cases where the surface to be measured is embedded in a medium, which is the usual case in microscopical applications, and where a stack of serial optical sections of the surface is available. The paper presents an improvement of an earlier version of the counting rule for intersection points. Furthermore, if the direction of sectioning is not uniform random, a bias results. This bias is calculated for a disc as a perfectly anisotropic object. A generalization of the estimator is considered by introducing a weighted mean instead of the usual arithmetic mean. The variance due to the randomized direction is investigated depending on the weights, and the minimum of this variance is derived. The relationship between the covariograrn and the variance of the surface area estimated with the spatial grid is considered. 1. Introduction The area A of a surface S in space can be estimated using a spatial grid of lines if a stack of serial sections of the surface is available. This method is described in Sandau (1987) and will be called the 'spatial grid'. The idea is to estimate the projected area of the surface in three orthogonal directions using a grid of paraIlel lines for each direction. To achieve this aim the sections must be perfectly registered; this can be realized, for example, with a confocal scanning laser microscope (CSLM; Howard & Sandau, 1992) or with a computer tomograph (CT; Pache et al.. 1993). Pache et al. 'Present address: FB Mathernatik und Naturwissenschaften. FH Darmstadt. Schofferstr. 1-3. D-64295 Darmstadt. Germany. 0 1994 The Royal Microscopical Society the 'spatial grid' using CT images and with the method of vertical sections (Baddeley et al., 1986) using images obtained with a light microscope. The randomization concepts of these two methods are very different and therefore the kind of application, particularly the method for obtaining the sections, usually determines which technique is preferable. First, we will give a short introduction to the spatial grid method. We denote the projected area of S onto a plane vertical to the direction 1 by pl (S). The projected area is counted in multiplicity, which means that the area of each part of the projection is added as often as it is the projection of different parts of S. This is especially important if the surface is not convex. To exclude pathological cases such as, for example, fractals, the surface S IR3 is assumed throughout this work to be the finite union of ranges of continuously differentiable functions of simply connected compact two-dimensional (2-D) regions. If the direction of the projection is uniform random in 3-D space, the expectation of the projected area (counted in multiplicity) is half of the surface area. This follows from a theorem of Cauchy (Santalo, 1976). Taking three directions x, y and z, which are orthogonal and the result of a suitable joint randomization procedure (Sandau. 198 7), then the sum of the three projection areas leads to the unbiased estimator In the following the joint randomization procedure is called the complete randomization of the directions. It can be carried out by randomizing the z-direction uniformly in space as described by Mattfeldt (1990). The x-direction is then randomized in a plane vertical to the z-direction, and the y-direction is orthogonal to the x- and z-directions (cf. Sandau, 1987). Now the three projection areas can be estimated using three line grids of parallel lines. The arrangement of these line grids can be imagined as follows.