Discrete space models for self-similar random images

Images exhibiting statistical self-similarity are of interest in various areas of image processing such as textures and scene synthesis. In continuous-space, statistical self-similarity is defined through statistics invariant to spatial scaling. However, because of lack of discrete-space scaling operation, statistical self-similarity in discrete-space has been characterized by approaches such as increments of fractional Brownian motion rather than scaling. We address these two issues regarding self-similar random fields through the paper. We show that the current self-similarity definition for continuous-space is somewhat restrictive, and we offer a new self-similarity definition in continuous-space more general than the current one. Furthermore, we provide a new formalism for statistical self-similarity in discrete-space by defining a scaling operation for discrete-space images. Consequently, a wider class of self-similar random images can be synthesized for different applications in discrete-space. The paper presents theoretical development and synthesis examples.