Page 220 - Phase Space Optics Fundamentals and Applications

Super Resolved Imaging in Wigner-Based Phase Space 201

while being imaged through a band-limited lens aperture. The input
object is a U.S. Air Force (USAF) resolution target. Its spatial blurring
is clear in Fig. 6.3a. However, after the addition of the random encod-
ing and decoding code multiplexing mask, one obtains the image of
Fig. 6.3b where the high-resolution features of the resolution target are
almost completely reconstructed. The resolution improvement here is
by more than a factor of 5.
Obviously the image of Fig. 6.3b is obtained after image processing
that includes some reduction of background noises.

6.3.2 Time Multiplexing
In this case we will perform time averaging (thus we do not have to
assume 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) dt
x
x
x
× d d x

x
(6.23)
Since we have a time-varying mask we can approximate that

∗

G( − ,p( ,t), ,t)G (− x + ,p( ,t), ,t) dt ≈ ( x − )
x
x
x
x
(6.24)
We obtain the final expression for the reconstructed spectrum:

x

U R ( x ) = rect d x U( ) ( x − )d = x · U( x )
x
x
x
x
(6.25)
Numerical demonstration of the time averaging SR approach may
be seen in Fig. 6.4. In Fig. 6.4a we present the blurred (lowpass) resolu-
tion target (USAF) and in Fig. 6.4b its reconstruction after the addition
of the time-varying encoding and decoding random mask and the
performing of the averaging operation in the time domain. In the sim-
ulation we averaged 800 images. One can clearly see the resolution
improvementdemonstratedinthissimulation.Theobtainedimprove-
ment factor is about 3.