Discrete-Time continuous-dilation construction of linear scale-invariant systems and multi-dimensional self-similar signals

This dissertation presents novel models for purely discrete-time self-similar processes and scale- invariant systems. The results developed are based on the definition of a discrete-time scaling (dilation) operation through a mapping between discrete and continuous frequencies. It is shown that it is possible to have continuous scaling factors through this operation even though the signal itself is discrete-time. Both deterministic and stochastic discrete-time self-similar signals are studied. Conditions of existence for self-similar signals are provided. Construction of discrete-time linear scale-invariant (LSI) systems and white noise driven models of self-similar stochastic processes are discussed. It is shown that unlike continuous-time self-similar signals, a wide class of non-trivial discrete-time self-similar signals can be constructed through these models. The results obtained in the one-dimensional case are extended to multi-dimensional case. Constructions of discrete-space self-similar ran dom fields are shown to be potentially useful for the generation, modeling and analysis of multi-dimensional self-similar signals such as textures. Constructions of discrete-time and discrete-space self-similar signals presented in the dissertation provide potential tools for applications such as image segmentation and classification, pattern recognition, image compression, digital halftoning, computer vision, and computer graphics. The other aspect of the dissertation deals with the construction of discrete-time continuous-dilation wavelet transform and its existence condition, based on the defined discrete-time continuous-dilation scaling operator