Signal Processing - September 2016 - 113

and rect is the rectangular function. The phase function is
separable in the u and v spatial frequencies and has spatial
extent Wu and Wv along the respective axis. The constant c
represents the strength of the cubic phase. Figure 5 shows the
cubic phase element mounted on the antenna.
Figure 6 shows the measured PSFs for conventional imaging and imaging with a cubic phase. The width of the in-focus
PSF at 180 in is approximately 2 mm, which is consistent with
a 1 mm pixel width. Note that the response of the cubic phase
system is relatively unchanged, whereas the response of the
conventional system changes considerably. A postdetection
signal processing step is necessary to produce a well-defined
sharp response [23], [30], [31].
If we assume (3) represents a linear postdetection process,
we can implement T as a Wiener filter in Fourier space,
T (u, v) =

H * (u, v)
t N (u, v) ,
K -2 U
2
| H (u, v) | + t
U L (u, v)

(a)

(b)

Figure 5. The cubic phase element. (a) Side view and (b) front view of
the cubic phase element mounted on the antenna. (c) The detail of the
fabricated cubic phase element.

(5)

where H(u, v) is the optical transfer function associated with
the cubic phase element, the parameter K is a measure of the
t L and U
t N are the
signal-to-noise ratio, and the functions U
expected power spectra of the object and noise, respectively.
The optical transfer function is usually estimated from the
experimentally measured point responses. One can view the
estimated i p (x, y) as a diffraction limited response.
The extended object used in the experiments is represented
in Figure 7(a). Images of an extended object for conventional imaging system at 113, 146, and 180 in are shown in
Figure 8(a)-(c), respectively. Each image is represented by
41 × 51 measurements or pixels. The object size within the
image is a function of optical magnification. Note that the
conventional imaging system produces images with significant blurring. In contrast, even without signal processing,
the images produced with cubic phase element retain more
discernible characteristics of the object than the images
from the conventional system, as shown in Figure 8(d)-(f).
Figure 8(g)-(i) shows that postprocessing compensates for
the effect of the cubic phase element and retains frequency

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6. The measured PSFs for conventional imaging and imaging
with a cubic phase. PSFs for conventional system at (a) 113 in, (b) 146.5
in, and (c) 180 in. (d)-(f) PSFs for a system with cubic phase at the same
distances for (a)-(c).

1.0

12″

0.5

Metal Plate

Object

0.0

0.7

5″

-0.5

Liquid Nitrogen
Bath

0″

-1.0
-1.5

2.

5.

θε,max /2π

(c)

25
″

-2.0
100 120 140 160 180 200 220 240 260 280

(a)

do (in)

Figure 4. The maximum relative pupil phase error as a function of object
distance. The shaded region indicates a conventional depth of field.

(b)

Figure 7. (a) A representation of the extended object used to compare
conventional and cubic-phase imaging. (b) A schematic of object illumination [5].

IEEE SIgnal ProcESSIng MagazInE

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September 2016

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113
Table of Contents for the Digital Edition of Signal Processing - September 2016
