Wigner Distributions and
How They Relate to the Light Field
Proc. IEEE International Conference on Computational Photography, April 2009
Winner of Best Paper award
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.
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.