Radial Hilbert transform with the spatially variable half-wave plate

We analyzed the point spread function (PSF) of the typical 4f optical image processing system by use of a spatially variable half-wave plate as the spatial filter and found that the PSF is an elementary vector beam. Theoretical analysis and real experiments show that the optical system can be used for a radially symmetric Hilbert transform that permits two-dimensional edge enhancement as the spiral phase plate. This kind of radial Hilbert transform is useful for image processing because it can enhance the edges of an input image selectively by exerting a polarization analyzer before the output plane. The optical system also can be used for generation of vector beams with arbitrary array and shape in real time conveniently.