Wigner Distributions and
How They Relate to the Light Field

Zhengyun Zhang Marc Levoy
Proc. IEEE International Conference on Computational Photography, April 2009

Winner of Best Paper award

Abstract

In wave optics, the Wigner distribution and its Fourier dual, the ambiguity function, are important tools in optical system simulation and analysis. The light field fulfills a similar role in the computer graphics community. In this paper, we establish that the light field as it is used in computer graphics is equivalent to a smoothed Wigner distribution and that these are equivalent to the raw Wigner distribution under a geometric optics approximation. Using this insight, we then explore two recent contributions: Fourier slice photography in computer graphics and wavefront coding in optics, and we examine the similarity between explanations of them using Wigner distributions and explanations of them using light fields. Understanding this long-suspected equivalence may lead to additional insights and the productive exchange of ideas between the two fields.

Examples of coherence and Wigner distributions Similarities between the light field and Wigner representations of the cubic phase plate system
A single coherent plane wave approaches two slits I and II in (a) while plane waves blue (left) and green (right), incoherent relative to each other, hit slit I and II, respectively, in (b). The mutual intensities for the two cases can be seen in (c) and (d), the Wigner distributions in (e) and (f), and the ambiguity functions in (g) and (h). In all cases, darker equals greater magnitude. The presence of cross terms caused by interference between the two slits in the left column give rise to diamonds A and C in the mutual intensity, band B in the Wigner distribution and the sidebands A and C in the ambiguity function. The periodic features in the Wigner distribution and ambiguity function are aliasing artifacts due to discrete sampling in the numerical simulation.
The impulse response, taken at z = 4f, of the cubic phase plate system, viewed as a Wigner distribution (a), an ambiguity function (b), a light field (c), and its Fourier transform (d). Horizontal (red dashed) and tilted (blue) slices (e) through the ambiguity function are very similar in magnitude. The same is true for slices (f) through the Fourier transform of the light field. These results were generated numerically and thus only simulate a finitely large cubic phase plate.

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