Autoregressive models of random fields on the circle

Abstract

At present, the vast majority of known image models are varieties of random fields defined on rectangular flat grids or grids of higher dimension. In some practical situations, images have a circular or radial-circular structure. For example, an image of a facies (a thin film of dried biological fluid), an eye, a biological cell, a flower, a slice of a tree trunk, etc. This circumstance requires the development of appropriate models of random fields describing images of this type. This paper proposes autoregressive models of homogeneous and inhomogeneous random fields defined on a circular spiral. The problem of correlation analysis of such models is considered. Examples of imitation of circular images are given. This paper proposes linear autoregressive models of homogeneous and inhomogeneous random fields defined on a circle. The samples of these fields are defined on a circular or spiral grid. The next field value is a linear combination of several previous values and a standard random variable. Expressions of the correlation function of these fields are given depending on the values of the autoregression parameters. A graphic representation of the correlation function on a circle and examples of simulated images are presented in the figures.