Self-similar random field models in discrete space

Self-similar random fields are of interest in various areas of image processing since they fit certain types of natural patterns and textures. Current treatments of self-similarity in continuous two-dimensional (2-D) space use a definition that is a direct extension of the one-dimensional definition, which requires invariance of the statistics of a random process to time scaling. Current discrete-space 2-D approaches do not consider scaling, but, instead, are based on ad hoc formulations, such as digitizing continuous random fields. In this paper, we show that the current statistical self-similarity definition in continuous space is restrictive and provide an alternative, more general definition. We also provide a formalism for discrete-space statistical self-similarity that relies on a new scaling operator for discrete images. Within the new framework, it is possible to synthesize a wider class of discrete-space self-similar random fields and texture images.