Fourier transforms of measure-valued images, self-similarity and the inverse problem

摘要

After recalling the notion of a complete metric space (Y.d_Y) of measure-valued images over a base (or pixel) space X, we define a complete metric space (F.d_F) of Fourier transforms of elements μ∈ Y. We also show that a fractal transform T : Y→Y induces a mapping M on the space F. The action of M on an element U ∈ F is to produce a linear combination of frequency-expanded copies of M. Furthermore, if Tis contractive in Y, then M is contractive on F: as expected, the fixed point U of M is the Fourier transform of μ∈ Y.