Fast Infinitesimal Fourier Transform for Signal and Image Processing via Multiparametric and Fractional Fourier Transforms

The fractional Fourier transforms (FrFTs) is one-parametric family of unitary transformations {F} α=0. FrFTs found a lot of applications in signal and image processing. The identical and classical Fourier transformations are both the special cases of the FrFTs. They correspond to α = 0 (F = I) and α = π/2 (F = F), respectively. Up to now, the fractional Fourier spectra Fi = Fi {f} , i = 1, 2, ...,M , has been digitally computed using classical approach based on the fast discrete Fourier transform. This method maps the N samples of the original function f to the N samples of the set of spectra {F}Mi=1 , which requires MN (2 + log2N) multiplications and MN log2N additions. This paper develops a new numerical algorithm, which requires 2MN multiplications and 3MN additions and which is based on the infinitesimal Fourier transform.