Homogeneity, Autocorrelation and Anisotropy in Patterns

Abstract

The interpretation of the spatial distribution of a phenomenon can only be done by evaluating both the global (large scale trend or values in each point in the space) and local scale effects due to the interaction of each point with its neighboring points [7]. The absence of these local effects make the values of the phenomenon vary depending on the place. In other words, the values observed in a window change systematically, hence, there does not exist any spatial dependence among values and the process is heterogeneous or non-stationary. On the contrary, if the existence of local effects is detected, the process is spatially homogeneous or stationary. Spatial dependence is a particular case of homogeneity. In images whose elements show spatial correlation, it is a fact that the existence of a concrete value of the phenomenon makes it more likely that this value should occur in near places. The existing statistics for determining the existence of spatial dependence among elements of an image are very varied [1][12], and they include non-spatial techniques, such as ANOVA, error terms in spatial autoregresive systems, Moran’s I, Geary’s C, Getis’s G*, variograms, correlograms….[6]. All of them determine the existence or absence of autocorrelation in an image, but they do not provide spatial information about the direction in which it is manifested, that is, the directional trend of the variation of parameters defining the thematic characteristics of the image. In this paper we propose a method, based on first and second-order bivariate circular statistics, that will allow us to study the spatial variability of the phenomenon, to establish the existence of spatial dependence in an image and to calculate the direction in which this appears, by using a parametric procedure based on the standard and confidence ellipses for normal samples. A study about the flow of seawater is presented as an example of the potential use of the proposed method.