Two-dimensional affine generalized fractional Fourier transform

As the one-dimensional (1-D) Fourier transform can be extended into the 1-D fractional Fourier transform (FRFT), we can also generalize the two-dimensional (2-D) Fourier transform. Sahin et al. (see Appl. Opt., vol.37, no. 11, p.2130-41, 1998) have generalized the 2-D Fourier transform into the 2-D separable FRFT (which replaces each variable 1-D Fourier transform by the 1-D FRFT, respectively) and the 2-D separable canonical transform (further replaces FRFT by the canonical transform). Sahin et al., (see Appl. Opt., vol.31, no.23, p.5444-53, 1998), have also generalized it into the 2-D unseparable FRFT with four parameters. In this paper, we introduce the 1-D affine generalized fractional Fourier transform (AGFFT). It has even further extended the 2-D transforms described above. It is unseparable, and has, in total, ten degrees of freedom. We show that the 2-D AGFFT has many wonderful properties, such as the relations with the Wigner distribution, shifting-modulation operation, and the differentiation-multiplication operation. Although the 2-D AGFFT form seems very complex, in fact, the complexity of the implementation will not be more than the implementation of the 2-D separable FRFT. Besides, we also show that the 2-D AGFFT extends many of the applications for the 1-D FRFT, such as the filter design, optical system analysis, image processing, and pattern recognition and will be a very useful tool for 2-D signal processing.