An optical study of the fractional Fourier transforms for a regular fractal pattern

摘要

In this paper we have successfully applied optical diffraction method to the study of fractional Fourier transforms of a regular fractal pattern―a snowflake. Fractals may be divided into two kinds: the 1st kind is called regular fractal which may be generated artificially according to a set of mathematical rules and it exhibits strict self-similarity; while the 2nd kind is called irregular fractal in which strict self-similarity is replaced by statistical self-similarity. We have obtained fractional Fourier transforms of a snowflake by optical method as an example of regular fractal. Different orders of fractional Fourier transforms are obtained. These are compared with ordinary Fourier transform of the same pattern. These experimental results are discussed in the light of the fractional Fourier transform for pattern recognition. These results may be readily extended to the case of irregular fractals which closely correspond to most of natural patterns.