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Super Resolved Imaging in Wigner-Based Phase Space 213

blurring. The wavelength coding can be realized by using an optical
element performing spatial dispersion of colors which are used as
the color-space coding map (e.g., by a grating) or just by placing a
space-varying color transmission filter in the setup.
The object that we used for the simulation was the Gaussian signal
having the Wigner distribution presented in Fig. 6.10a. The Wigner
chart of the wavelength coded signal is presented in Fig. 6.11a. This
numerical simulation corresponds to the schematic sketch of Fig. 6.8a.
Since every spatial high-resolution pixel is coded with a different
wavelength, in the Wigner chart every such coded pixel is a nar-
row rectangle having spatial width of one pixel and maximal spectral
width of 3 . This narrow rectangle is shifted according to the spatial
position of the coded pixel.
The result of the reconstruction obtained after averaging over the
wavelength domain and using the same decoding color distribution
(i.e., realization of an inverse color-space mapping), yields the result
seen in Fig. 6.11b. We see that the obtained result is very similar to the
Wigner of the original nonblurred Wigner distribution presented in
Fig. 6.10a.

6.4 Conclusion
In this chapter we presented the usage of Wigner phase space for
the description and the understanding of the field of super resolu-
tion. We showed that the Wigner phase space is more than just a
heuristic representation. It is rather a chart that simplifies the under-
standing of the optical system and provides a representation that aids
the physical comprehension of the optical behavior of the imaging
system.
We focused in our description on five ways of performing super
resolution by using a priori knowledge on other domains into which
we could convert the spatial degrees of freedom such that they will not
be lost during transmission through the band-limited optical imaging
system. The five ways included code, time, polarization (which is
time-varying), wavelength, and gray-level multiplexing.
The additional domain used for the conversion of spatial degrees of
freedom generates a hyper phase space having more complete repre-
sentation than a conventional Wigner chart or a conventional phase-
space diagram.
In this chapter we presented both the schematic description of the
various super resolving approaches in the Wigner phase space and
the accurate numerical simulation of those approaches.