Improved estimation of the pair correlation function of random sets

The texture of binary spatial structures can be characterized by second-order methods of spatial statistics. The pair correlation function, which describes the structure in terms of spatial correlation as a function of distance, is of central importance in this context. Conventionally, the pair correlation function of stationary and isotropic random sets is estimated as the ratio of the covariance to the square of volume fraction of the phase of interest. In the present paper, an improved estimator of the pair correlation function is presented, where the covariance is divided by the square of a distance-adapted estimator of volume fraction. The new estimator is explained mathematically and applied to simulated images of the Boolean model and to microscopic images from neoplastic and non-neoplastic human glandular tissues. It leads to a considerable reduction of bias and variance of estimated pair correlation functions, in particular for large distances.