Measuring homogeneity of planar point-patterns by using kurtosis

Kurtosis is generally associated with measurements of peakedness of a distribution. In this paper, we suggest a method where kurtosis can be used as a measure of homogeneity of any quantifiable property on a planar surface. A 2-dimensional, continuous and uniform distribution has kurtosis equal to 5.6. This value is also the limiting value for a discrete uniform distribution defined on a regular, rectangular grid when the number of grid points tend to infinity. Measurements of a planar surface, taken at regular grid points, are considered as realizations of random fields. These are associated with 2-dimensional random variables from which the value of kurtosis can be computed and used as a measure of the homogeneity of the field. A deviation from 5.6 indicates that the stochastic variable is not uniformly distributed and that the corresponding random field is not homogeneous. The model is applied on the spatial variation of the roughness on the surface of newsprint, an application where homogeneity is very important.