Measuring the geometric properties of various brain structures is of interest to a diverse group of researchers. For example, the volume of the hippocampus (a brain structure involved in memory) decreases in patients with Alzheimer's disease, and the ratio of the surface area of the left and right planum temporale (a brain region involved in understanding language) appears to be altered in patients with schizophrenia.

In the past few years, there have been many advances in automated and semi-automated methods for the measurement of various geometric properties of brain structures. However, strictly manual methods for measuring the properties remain important for two reasons. First, they remain the "gold standard" in most cases for validating the results of automated methods. Second, some of these manual methods are startlingly ef.cient. For example, if an experiment devolves to measuring brain volume with a coef.cient of error of 5%, the argument can be made that a skilled neuroanatomist could actually complete this measurement in little more time than it would require to check an automated segmentation.

Stereology is a group of techniques for measuring geometric properties of objects which involves ideas from integral geometry and geometric probability. These techniques, commonly used by microscopists, have been little applied to medical imaging studies.

For example, the figure shows a typical application of the stereological method of vertical sections. Here, the surface area of the object is estimated by counting the number of intersections of the surface of the brain with a collection of cycloidal arcs. Under appropriate sampling conditions, it can be shown that the surface area of the object is proportional to the number of intersections of the arcs with the brain surface in a series of cross-sections. In this case, simply counting intersections would thus suf.ce to perform an experiment, or to check an automated method for measuring surface area such as one derived from extracting the complicated manifold of the gray matter/cerebrospinal interface, approximating it by triangles, and summing the area of the triangles.