Optical implementation of the fractional Hilbert transform for two-dimensional objects

The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is pi-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.