  There has been a significant effort to design nanophotonic structures that
process images at the speed of light. A prototypical example is in edge
detection, where photonic-crystal-, metasurface-, and plasmon-based designs
have been proposed and in some cases experimentally demonstrated. In this work,
we show that multilayer optical interference coatings can achieve
visible-frequency edge detection with high numerical aperture, two-dimensional
image formation, and straightforward fabrication techniques, unique among all
nanophotonic approaches. We show that the conventional Laplacian-based
transmission spectrum may not be ideal once the scattering physics of real
designs is considered, and show that better performance can be attained with
alternative spatial filter functions. Our designs, comprising alternating
layers of Si and SiO$_2$ with total thicknesses of only $\approx 1{\rm \mu m}$,
demonstrate the possibility for optimized multilayer films to achieve
state-of-the-art edge detection, and, more broadly, analog optical
implementations of linear operators.
