Stereological estimation of covariance using linear dipole probes

Classical stereology is capable of quantifying the total amount or 'density' of a geometrical feature from sampled information, but gives no information about the local spatial arrangement of the feature. However, stereological methods also exist for quantifying the 'local' spatial architecture of a 3D microstructure from sampled information. These methods are capable of quantifying, in a statistical manner, the spatial interaction in a structure over a range of distances. One of the key quantities used in a second-order analysis of a volumetric feature is the set covariance. Previous applications of covariance analysis have been 'model-based' and relied upon computerized image analysis. In this paper we describe a new 'design-based' manual method, known as linear dipole probes, that is suitable for estimating covariance from microscopic images. The approach is illustrated in practice on vertically sectioned lung tissue. We find that only relatively sparse sampling per animal is required to obtain estimates of covariance that have low inter-animal variability.