Parametric Weighted Finite Automata and Multidimensional Dyadic Wavelets

Wavelets have found many applications in signal-analysis and as transform-functions in image compression, e.g. in JPEG 2000. This paper studies representations of well-known types of wavelets and their multidimensional variations in terms of finite-state devices called parametric weighted finite automata (PWFA). Since these PWFA can also simulate easily generalizations of iterated function systems, they provide a framework for fractal-type functions. PWFA are strictly more powerful than IFS and WFA. But also smooth functions like polynomials, sine, cosine etc. can be generated by small PWFA. Thus, we underline the fractal nature of the considered types of wavelets and open new perspectives on their representations and computation-algorithms. We provide convenient methods to display linear combinations of dilations and translations of one wavelet in a single compact automaton. These methods do not depend upon orthogonality or separability of the wavelet basis.