The 4f Fourier-Optics Processor

A lens placed one focal length from an object produces, one focal length beyond it, the exact 2D Fourier transform of the object's transmittance. Put a mask in that back focal plane and a second lens transforms back: that is the 4f coherent processor. Block the outer (high-frequency) part of the spectrum and the image is low-pass filtered, blurred; block the centre (the DC and low frequencies) and only edges survive, the Abbe-Porter experiment. With no mask the second transform inverts the first and the image is the object. The middle panel is $|\mathcal{F}\{t(x,y)\}|$ on a log scale with the filter drawn on it.