Fourier Slice Photography (2005)

One can build a camera that captures the angle of incoming light as well as its brightness at each pixel. This is called "4D" photography, or now more commonly called Light Field photography (like the Lytro camera). This captures light at "all" depths of focus. They present a method to use the fourier transform to evalute and quantify the quality of this light field by mapping the angle resolution to the quality of digital focus that can be produced from the light field. The cameras work by capturing a narrow slice of angles at many pixels, which determines how accurately they know each light ray making up the scene. Light computes a Fourier transform as it propogates - 1 focal distance past where an image is focused, you will find the light representing the spatial frequency of the image (i.e, its Fourier transform). By capturing many angles, you hope to capture and unmix the Fourier transformed representations of an image at many focal distances.

You may remember the Lytro camera (https://en.m.wikipedia.org/wiki/Lytro), that company's founder was the author of this paper. The camera works in a unique way, with a special lens construct. The full 4D light field is captured by the camera (vs a conventional 2d image), which allows you to view the resulting image with any arbitrary focus.

I had a chance to play with one once. While it's a fascinating toy, the bizarre form factor, and proprietary data format, seem to have effectively doomed the product.