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Super Resolved Imaging in Wigner-Based Phase Space 207

about time (or time-varied polarization) multiplexing, this spatial dis-
tribution is varied with time. Thus, in Fig. 6.7a the order of the different
colors is changed versus time. This indicates that the spatial distribu-
tion of the encoding mask is time-varying. The small spatial pixels
of the chart occupy a size of 3 in the spectral axis because in the
space domain the mask has pixels which are 3 times smaller than the
imaging resolution. To simplify our explanation, if we let x denote
by x the spatial resolution that corresponds to spectral bandwidth of
, then

1
x = (6.32)

When we have 3 times finer resolution of x/3, the spectral bandwidth
will be 3 times larger, or 3 , because the product of the spatial reso-
lution and the spectral bandwidth equals to 1 (a well-known property
of the Fourier transform).
In Fig. 6.7b we present the phase-space diagram of the signal that
has a bandwidth of 3 (3 times larger than the bandwidth that may
be transmitted through the aperture of the imaging lens).
In Fig. 6.7c we present the schematic sketch of the phase-space
diagram of the product of the random coding mask and the signal.
The phase-space distribution of the signal does not vary with time,
but the encoding mask does. The bandwidth of the product equals
6 since a well-known Fourier relation dictates that the product in
the space domain will be a convolution in the spectrum domain. A
convolution of two spectral functions having spectral width of 3
yields a result with width of 6 .
In Fig. 6.7d we show what happens when the high-resolution (and
time-varying) product distribution is passed through a size-limited
aperture. There is spatial blurring which reduces the spatial resolu-
tion (the thin rectangles became 3 times wider in the spatial axis and
3 times narrower in the spectral axis), and thus the various colors that
designated differentgray levelsof the spatial pixelsaremixedtogether
(which reduces their dimension in the spectral axis ). Note that the
area of each rectangle denotes a single degree of freedom, and this area
remains constant: increasing its dimension in the space domain re-
duces the dimension in the spectral axis while preserving the product.
The decoding process is described by Fig. 6.7e, and it includes mul-
tiplication of the captured information by the same high-resolution
and time-varying decoding spatial mask distribution and then per-
forming time averaging. The time averaging that is performed after
the multiplication will extract, from every high-resolution spatial de-
gree of freedom (that occupied spatial size of 3 ) the original value
while averaging to zero, the undesired information that was added to
it due to the spatial blurring.