Features of a two-dimensional random field: Number of degrees of freedom, correlation areas and correlation lengths

Abstract

One of the important features that characterize the physical random field or the image texture is Taylor's correlation length. However, there has not been presented a clear-cut theory as to the foundation of the correlation length from the viewpoint of probability theory or statistical theory. Another aspect is that the feature is one-dimensional, and, therefore, the application to the random field is limited to the isotropic random field. This paper defines the nth order statistical number of degrees of freedom as the equivalent number of uncorrelated points in the square finite region in the homogeneous random field. From the definition, the correlation area and the correlation length are derived. the correlation area is referred to as the average area of the region statistically affected by an arbitrary point on a two-dimensional field. the correlation length represents the effective distance between adjacent independent or uncorrelated points on the line parallel to the x- or y-axis. the first-order correlation length is the same as the conventional Taylor correlation length. the first-order correlation area is a natural extension of Taylor's correlation length to the two-dimensional random field. the correlation length along the principal axis, which is independent of the definition of the coordinate axis, is also derived. Those features are based on the theory of statistics and are expected to be useful for the analysis of the actual random physical field and the texture analysis of the image.