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.

Figure 1. The 4f coherent optical processor: object, Fourier plane with the filter, and the filtered image. Method: in-line radix-2 2D FFT, frequency-domain mask, inverse FFT; intensity image.

object

filter

filter radius10

WHAT TO TRY

Switch the filter between low-pass, high-pass and none: blocking the high spatial frequencies blurs the image and rounds the edges, blocking the low ones leaves only the edges. The Fourier plane is where you cut.
Shrink the filter radius: the aperture passes fewer diffraction orders, and the reconstructed image loses ever more detail until only the coarsest features survive. The line profile shows the band-limiting and its ringing.
Change the object (grating, double slit, checker): its diffraction pattern in the back focal plane changes, and the filtered image is the inverse transform of whatever the mask lets through.