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Super Resolved Imaging in Wigner-Based Phase Space 203

6.3.4 Wavelength Multiplexing
In this case we will perform wavelength averaging (so again we do not
have to assume the spatial randomality of the encoding and decoding
masks), so Eq. (6.18) becomes

x

U R ( x ) = U( ) rect
x
x

∗

× G( − ,p( ,t), ,t)G (− x + ,p( ,t), ,t) d
x
x
x

×d d x
x
(6.29)
since

∗
G( − ,p( ,t), ,t)G (− x + ,p( ,t), ,t) d ≈ ( x − )
x
x
x
x
(6.30)
We obtain once again as the final expression for the reconstructed
spectrum

x

U R ( x ) = rect d x U( ) ( x − ) d = x · U( x )
x
x
x
x
(6.31)
Note that in this case the meaning of the encoding mask is that
every spatial pixel of the input object is “painted” with a different
color. The decoding mask is identical to the encoding one, and it is
picking out, from the blurred image, the right color in every high-
resolution spatial location. The realization of such a mask can be
straightforward by placing a chromatic spatially varying filter (e.g.,
chromatic absorption filter) in front of the input object which is being
illuminated by a white light source. Otherwise this may also be re-
alized by illuminating a dispersion grating with a white light source
while this grating is positioned before the object, and thus the input
object will be illuminated with the dispersed colors.
6.3.5 Gray-Level Multiplexing
Instead of using the domains previously discussed, for the coding
of the spatial degrees of freedom, one may use the gray-level or the
dynamic range domain as well. We assume that we have a priori
information that the dynamic range of the input object is limited. Once
again we attach a gray-level coding mask to the input object. Thus,
prior to the blurring due to the low-resolution imaging, we attach
a different transmission value to each pixel of the input, while the
M
ratio between every one of those values is 2 , where M is the a priori
known and limited number of bits spanning the dynamic range of the