Fourier Series Expansion of Irregular Curves

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Abstract
Fourier theory provides an important approach to shape analyses; many methods for the analysis and synthesis of shapes use a description based on the expansion of a curve in Fourier series. Most of these methods have centered on modeling regular shapes, although irregular shapes defined by fractal functions have also been considered by using spectral synthesis. In this paper we propose a novel representation of irregular shapes based on Fourier analysis. We formulate a parametric description of irregular curves by using a geometric composition defined via Fourier expansion. This description allows us to model a wide variety of fractals which include not only fractal functions, but also fractals belonging to other families. The coefficients of the Fourier expansion can be parametrized in time in order to produce sequences of fractals useful for modeling chaotic dynamics. The aim of the novel characterization is to extend the potential of shape analyses based on Fourier theory by including a definition of irregular curves. The major advantage of this new approach is that it provides a way of studying geometric aspects useful for shape identification and extraction, such as symmetry and similarity as well as invariant features.

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