On a link between kernel mean maps and Fraunhofer diffraction, with an application to super-resolution beyond the diffraction limit

TL;DR

This paper links Fourier optics and kernel mean maps to demonstrate that, under certain conditions, imaging with small apertures can be invertible, enabling super-resolution beyond the diffraction limit through a real-world experiment.

Contribution

It establishes a theoretical connection between kernel mean maps and Fraunhofer diffraction, enabling super-resolution beyond the diffraction limit.

Findings

Imaging with arbitrarily small apertures can be invertible under generic conditions.

Kernel mean map invertibility relates to Fraunhofer diffraction invertibility.

Experiment confirms super-resolution beyond Rayleigh limit.

Abstract

We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean map to provide a statement about the invertibility of Fraunhofer diffraction, showing that imaging processes with arbitrarily small apertures can in principle be invertible, i.e., do not lose information, provided the objects to be imaged satisfy a generic condition. A real world experiment shows that we can super-resolve beyond the Rayleigh limit.