  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 in transmission 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.
