We deal here with a specific type of linear filter in which the input is a spatially limited nonnegative intensity such as that which appears on the image plane of an optical system having a finite field of view. This intensity is passed through a mask, or reticle, having a transmittance with two-dimensional periodicity. The transmitted intensity is detected so as to produce a voltage output which varies with time as the mask is moved. For a discrete set of uniform translations of the mask, the filter codes linear combinations of the Fourier transform of the input at certain spatial frequencies into the Fourier coefficients of the output.
For certain input classes of images, characterized by dominant spatial frequencies, the reticle as well as its motion can be chosen in an optimum manner so as to “best distinguish” these classes by their outputs.
There is an interesting analogy between this linear filter and that of a crystalline material irradiated with a plane monochromatic X ray. The reticle has a two-dimensional primary and reciprocal lattice corresponding to the three-dimensional ones of a crystal. Just as certain positions of the crystal, specified by its reciprocal lattice, give rise to Bragg reflections, so do certain translations of the reticle, specified by its primary lattice, give rise to the “coded voltage waveforms.”
W. D. MONTGOMERY, "Periodic Reticle for Optical Classification," J. Opt. Soc. Am. 55, 1082-1085 (1965)