Use of Synthesized Fields in Microwave
Tomography Inversion
Nozhan Bayat, Puyan Mojabi, and Joe LoVetri
Electrical and Computer Engineering Department
University of Manitoba, Winnipeg, Manitoba, R3T 5V6, Canada
bayatn@myumanitoba.ca, Puyan.Mojabi@UManitoba.ca, and Joe.LoVetri@UManitoba.ca
Abstract—In an attempt to enhance microwave tomography
reconstruction, the inversion of a given scattering data set is
casted as the inversion of a new synthesized scattering data set.
To this end, synthesized incident fields having a focused beam
are created solely from the actual incident fields. It is speculated
that this method, as presented here, cannot synthesize sufficiently
focused beams from omnidirectional sources so as to enhance
inversion results.
I. I NTRODUCTION
The data collection step in microwave tomography (MWT)
relies on successive illumination of the object of interest
(OI) by some antennas, often operating in their near-field
zones, and then measuring the resulting scattered fields outside
the OI. We have numerically demonstrated that the use of
appropriate incident fields, often having a focused distribution,
can enhance the achievable accuracy and resolution [1], [2].
To practically achieve this, three different strategies might be
employed: use of (i) appropriate antennas, (ii) synthesized
incident fields, (iii) a combination thereof. Within the first
strategy, promising methods such as near-field plates [3] can be
utilized to achieve high levels of near-field focusing. The main
difficulty with using such methods is that the utilized antenna
should also satisfy some other criteria; e.g., being reasonably
small to allow sufficient sampling resolution, having sufficient
bandwidth to allow multiple-frequency data collection. As an
alternative, some existing antennas can be modified so as to
increase their near-field focusing abilities [4], [5]. The second
strategy, on the other hand, does not attempt to design new
antennas or modify the existing ones. It simply casts the actual
MWT problem into a new problem by synthetically creating
focused incident fields from the actual ones. The possibility of
using this strategy is the focus of this paper.
II. SYNTHESIZED FIELDS
Consider T antennas that are placed outside the imaging
domain D in a two-dimensional transverse magnetic time-
harmonic MWT system. The (calibrated) incident fields of
these antennas are assumed to have an omnidirectional dis-
tribution. Discretizing D into N cells, the discrete form of
the tth incident field distribution inside D will be a complex
vector of length N ; say, E
inc
t
∈ C
N
. Denoting the number of
receiving cites per transmitter by M , the scattered data due to
this incident field will then be stored in E
scat
t
∈ C
M
. We have
already shown that the use of “sufficiently” focused incident
field can enhance imaging results [2]. Now, let’s assume that
such desired focused incident field is represented by [2]
E
inc,des
t,m
= E
inc
t
⊙ [cos
m
ψ] (1)
where t is the index of the transmitter, ψ is the angle between
the tth antenna’s boresight axis and the line connecting each
cell within D to the antenna. The parameter m ∈ R
+
controls
the focusing level; the larger m, the larger focusing level. Also,
[cos
m
ψ] ∈ R
N
, and ⊙ denotes the Hadamard product of the
two vectors. The question that needs to be answered here is
whether or not we can create a synthesized incident field, say
E
inc
t,m
, from E
inc
i
(∀i) that is sufficiently close to E
inc,des
t,m
. To
this end, a linear combination of E
inc
i
(∀i) can be utilized as
E
inc
t,m
= L(E
inc
i
|α
m
t,i
)
T
i=1
α
m
t,i
E
inc
i
. (2)
The complex weighting coefficients α
m
t,i
are to be found by
minimizing the following L
2
-norm
α
m
t,i
= arg min
α
m
t,i
E
inc,des
t,m
−
T
i=1
α
m
t,i
E
inc
i
2
2
. (3)
Once α
m
t,i
are found for each transmitter, we can construct
T different synthesized incident fields based on (2), and also
calculate their corresponding synthesized scattered fields as
E
scat
t,m
= L(E
scat
i
|α
m
t,i
). The original MWT problem, which
was to reconstruct the OI’s dielectric profile based on the
knowledge of the actual incident and scattered fields, can
now be casted as a synthesized problem that aims to find the
same unknown but using the synthesized incident and scattered
fields. Based on the above discussion it can be concluded that
if the synthesized incident field is “sufficiently” focused, the
inversion of its corresponding synthesized scattering data set
outperform the inversion of the original data set.
III. RESULTS AND DISCUSSION
Consider the OI shown in Figure 1(a), which consists
of three circles. The separation between the top circles is
0.12λ where λ =0.1 m is the wavelength of operation. The
shortest distance between the top and bottom circles is 0.08λ.
The OI is successively irradiated by 24 antennas, equally
distributed on a circle of radius 0.1 m. The resulting scattered
fields are collected by all the antennas; thus, having 24
2
data
points. (3% noise is added to the data.) We now consider
reconstructing this target by Gauss-Newton inversion [6] of
three different data sets. First, let’s consider the data set that
has been collected by the use of omnidirectional incident
fields E
inc
i
(∀i). The distribution of this incident field in D
(discretized into 100 × 100 cells) corresponding to the 1st
transmitter is shown in Figure 2(a). Inversion of this data set,
shown in Figure 1(b), cannot resolve the bottom circle, and
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