Signal Processing - September 2016 - 115

such as reducing noise. Figure 11 illustrates the real-time
Therefore, the system proposed in [4] samples discretely the
nature of the imager where a person behind a plywood is
complex wave-amplitude of a mmW signal and converts the
imaged at a video rate.
mmW signal to an optical one using electro-optical modulators while preserving the spatial distribution of samples using
an optical fiber array. The output of the fiber array is spatially
Compressive mmW imagers
Fourier transformed using a lens, and the resulting optical
Compressive sensing (CS) is an important tool that has shown
image is captured using an optical detector array or chargepromise in overcoming some of the common limitations
coupled device. One of the important features of this imager is
associated with mmW imaging. For example, a wide field of
that the imager volume does not scale with the aperture diamregard and high image frame rates are desired for many applieter, as the scale of the image-forming elements is fixed.
cations, like stand-off imaging of moving targets. Because
Another significant feature of this imager is its ability to
large-format arrays present cost challenges due to technologicontrol the relative phase of each receiving element in the
cal hurdles like availability of cost-effective and powerful
distributed aperture, which provides electronic control of the
source technology and sensitive, low-cost detectors, many
imager's PSF. It allows multidomain sensing by simultaneous
current systems use a single element or a small array in comand independent manipulation of both the
bination with a compound antenna system
Fourier and image planes of the system.
to scan a larger scene and build up an
Compressive sensing is
This is a unique capability that permits the
image (e.g., [35]-[38]). These architectures
an important tool that
change of the imager's PSF on-the-fly and
can present challenges to achieving the
has shown promise in
enables rapid sparse sampling of desired
video frame rates required for some imagtarget by electronically steering the beam in
ing applications [38]. As highlighted in this
overcoming some of
a manner similar to a phased array antenna.
section, mmW CS techniques have shown
the common limitations
Analysis of the imager as an incoherent
promise in overcoming issues like high
associated with
imaging system highlights the link between
frame-rate challenges by potentially reducmmW imaging.
element phase and PSF [7]. Mait et al. [7]
ing the overall number of scene measureshowed that, by modifying the aperture
ments needed to reconstruct an image.
phases of the hexagonal and nonredundant distributed aperCS can be viewed as a special case of computational imagture systems, a low-resolution analog image processing can be
ing in which partial or low-dimensional measurements are
performed. The simplest approach could be to take the differobtained by designing a specific sensing modality. In this case,
ence between two images of the same object captured using
the measurement matrix H in (1) has more columns than rows,
two different pupil functions,
and a nonlinear recovery algorithm is used to reconstruct the
f (x, y) = f+ (x, y) - f- (x, y),

(6)

where o(x, y) is the input object and
f+ (x, y) = o (x, y) * * h + (x, y) ,

(7)

f- (x, y) = o (x, y) * * h - (x, y) .

(8)

For example, one can construct a one-dimensional bandpass filter by manipulating the phase functions. To understand
this heuristically, one can model the corresponding PSFs as
d-functions
h + (x, y) = d (x, y),
h - (x, y) = 1 [d (x - x 0, y) + d (x + x 0, y)],
2

(a)

(e)

(9)
(10)

to approximate the composite transfer function H(u, v) as
H (u, v) = H + (u, v) - H - (u, v) = 1 - cos (2rx 0 u),

(b)

(c)

(11)

which filters low spatial frequencies and passes frequencies
centered at u = 1/x 0 . Such filtering can be useful for edge
detection (see Figure 10). Furthermore, [34] showed that
phase can also be used to do more complex signal processing

(d)

Figure 10. Edge detection performed using a nonredundant distributed
aperture. (a) Aperture phase and (b) corresponding image of an extended
object assuming zero aperture phase. (c) and (d) Same as in (a) and (b)
except with circular phase across the aperture. White represents 0-phase,
and black represents 2r-phase. Intermediate grey colors represent
phases between 0 and 2r. (e) The difference between (b) and (d) [7].

IEEE SIgnal ProcESSIng MagazInE

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September 2016

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