Fourier Slice Photography

by Ren Ng

Unlike the 2D definition of image, light field more expressively measures the light radiance along all possible 4D rays. This notion was initially suggested to synthesize new view of non-Lambertian scenes by ray tracing, yet in this paper, the author is interested in employing this tool to model photographic imaging.

The core of this work was the Fourier Slice Photography Theorem, developed from a generalization of Bracewell’s Fourier Slice Theorem. The latter addresses that the Fourier Transform (FT) of a signal’s integral projection corresponds to a sheared slice in its FT. As the imaging process can be seen as a sheared integral of the light field, in the Fourier domain, a photograph formed with full aperture corresponds to a 2D slice in the 4D light field. This discovery speeds up digital refocusing from O(n⁴) to O(n²logn) and improves precision because Fourier Transform has a fast implementation and avoids the numerical loss in integral. The advantage of the new algorithm was also validated by experiments.

To show the utility of the theorem, the author goes further to provide in close-form the performance limit of plenoptic cameras of finite aperture. Preliminarily quantitative analysis was only possible for the simplest pinhole model as the assumption of aperture complicates the problem by low-passing the light field. The author tackles this difficulty by working in the Fourier domain. Firstly he shows that the 4D convolution of the light field yields 2D convolution of photographs focused at a variety of depths. Applying the Fourier Slice Photography Theorem to this statement, he concludes that a band-limit assumption on plenoptic cameras would degrade its performance in digital refocusing, and the amount of degradation increases linearly with the directional resolution of the sampled light field.

This paper gives an in-depth insight to the relationship between a light field and its photographic images, and pioneers theoretical analysis of optical designs in the Computational Photography community. It also inspires the invention of new light field cameras, e.g. Raskar’s design of dappled photography. Although the author only focused on aberration-free lens models, the Theorem of Fourier Slice Photography can be applied to a broader range of optical systems.