A Multistatic Uniform Diffraction Tomographic
Algorithm for Real-Time Moisture Detection
Adel Omrani∗, Guido Link∗, John Jelonnek∗†
∗Institute for Pulsed Power and Microwave Technology (IHM)
†Institute of Radio Frequency Engineering and Electronics (IHE)
Karlsruhe Institute of Technology, Karlsruhe, Germany
Abstract—To obtain the moisture distribution inside a polymer
foam after drying in a conveyor belt system in real time, a
Multistatic Uniform Diffraction Tomography (MUDT) imaging
algorithm is proposed. It is estimated that MUDT provides a
better spatial resolution than the original uniform diffraction
tomography (UDT). Additionally, it allows to resolve the distri-
bution of the different scatterers, which is shown by simulation
Index Terms—microwave imaging, tomography, moisture de-
Microwave drying is an emerging technology of increasing
interest. It is the preferred method of choice particularly
for polymer foam that has low thermal conductivity. Us-
ing an intelligent control of distributed microwave sources
, non-uniform moisture distribution could be much more
efﬁciently addressed. But, it requires the in-situ and non-
invasive measurements of the unknown moisture distribution
inside the material under test. A novel method for microwave
tomography (MWT) in real time is in preparation for moisture
detection of a polymer foam that will be used in combination
with a HEPHAISTOS microwave system, as shown in Fig. 1a.
This HEPHAISTOS microwave oven has a patented hexag-
onal cross-section . In our speciﬁc case, the overall length
and hexagon circumferential diameters of the oven are 4 m
and 1 m, respectively. The oven has a modular structure and
consists of three microwave modules of the same type, each 1
m in length and with six slotted waveguide antennas mounted.
Each of those antennas is fed by a 2 kW magnetron working at
the 2.45 GHz ISM frequency band. The total microwave power
installed in the system is about 36 kW. In addition, a 28 kW
convective heating system is installed to help the removal of
the water vapor out of the microwave-drying chamber. That
avoids the condensing of water on the cold metal walls of the
microwave oven. Another unique feature of the HEPHAISTOS
system is the conveyor belt as shown in Fig. 1, which allows
a continuous production process .
In  a Through Wall Imaging (TWI) method is proposed
that bases on the Uniform Diffraction Tomography (UDT)
method. The location of an object inside or behind the wall
is determined by the linear relationship between the object
function and the received signal in the spectral domain for the
Fig. 1. Microwave drying system with conveyor-belt.
multilayered media. Using a conveyor belt system complicates
the task to locate an object signiﬁcantly. Not only the real-
time image reconstruction is important but also the real-time
data acquisition is a critical task. Using only one antenna and
moving the antenna for data acquisition is not applicable in
fast data acquisition so the array of the antennas needs to
be ﬁxed. On the other side, the number of antennas cannot
be too high because, again, it takes time for data collection.
Under those conditions, i.e., 1) the low number of antennas,
and, 2) if two adjacent antennas are not in close vicinity
to each other, the UDT imaging algorithm fails to truly
reconstruct the location of the scatterers. The MUDT proposed
here does overcome this problem. It provides high-resolution
and real-time images through the foam by the same number
of transmitters and employing non-diagonal elements of the
scattering matrix. For simplicity, the algorithm is presented for
a 2-D problem. The paper is organized as follows: in Section
II the MUDT formulation and differences with UDT are
presented. In section III the simulation results are presented.
Concluding remarks are provided in section IV.
II. MULTISTATIC UNIFORM DIFFRACTION
A. The Forward Model
The 2-D conﬁguration of the multistatic microwave imaging
system is illustrated in Fig. 2. The array of antennas is located
in semi-inﬁnite free space and above the polymer foam with
variations of dielectric properties in the z-direction only. The
distance of the antenna from the top of the polymer foam is t1.
t2is the thickness of the polymer foam with the permittivity
The Lippmann-Schwinger integral equation of the EM scat-
tering problem shown in Fig. 2, can be written in the following
form , :
n(~rr, ~rt) = ~
nis the total electric ﬁeld in layer n(n= 1,2,3),
nrepresents the scattered ﬁeld due to the unknown irreg-
ularities in layer n, while ~
Einc denotes the incident electric
ﬁeld. O(~r 0) = (ˆ(~r 0)−b) is the object function. ˆ(~r 0)is
the proﬁle of the permittivity of the target and bdenotes
the permittivity of the background. Here, a time harmonic
ﬁeld is assumed. Hence, the complex time harmonic function
e−jωt can be eliminated from the equation. ωis the angular
frequency. In (1), the vectors ~rr= (yr, zr)and ~rt= (yt, zt)
represent observation and source points while ¯
is the background (multilayered media without any scatterer
inside) dyadic Green’s function (DGF). The superscript (n1)
denotes that the source point is located in layer 1and the
observation point is in layer n.
Under the ﬁrst-order Born approximation the total electric ﬁeld
ncan be replaced by the background electric ﬁeld of the
layer and due to the excitation by a line source the electric
ﬁeld can be replaced by the Green’s function and also using
the symmetry property of Green’s function. Equation 1can be
expressed as stated in , 
n(~rr, ~rt) = k2ZΩ
G(n1)(~rr, ~r 0)·O(~r 0)¯
G(n1)(~r 0, ~rr)
Furthermore, for writing the background Green’s function in
layer n, the contrast between the layers is assumed to be
small and the reﬂected wave from the layers is suppressed. So,
the Green’s function is modeled by the incident ﬁeld in that
layer. If the contrast for the permittivity between the different
layers is sufﬁciently large, this assumption leads to a late-time
shadow image in the layer.
The spectral representation of the Green’s function in the nth
layer when the line source is located in region 1is , 
G(n1)(~r , ~rt)
Tn(ky, kz)e−jkzn (z−zt)
If z > ztand =(k2
T(ky, kz) is the transmission
coefﬁcient in the nth layer and can be obtained by applying
the boundary conditions between layers for the transverse
magnetic ﬁeld in x-direction (TMx) . The dispersion re-
lation in the layer lis expressed by kzl =qk2
Fig. 2. The antenna array is located above a polymer foam. The ﬁrst and last
media are half space.
kl=k0√lis the wavenumber in layer lwhile k0is the free-
The above representation of the scattered ﬁeld, allows to ex-
tend the UDT to the multistatic case (transmitter and receiver
are not co-located) where the non-diagonal element of the
scattering matrix can be employed for the image reconstruction
which signiﬁcantly increases the resolution as a consequence.
B. MUDT Inverse Scattering
From (2) the object function can be determined. Substituting
(3) in (2) with the prime integrand for the second Green’s term,
changing variables to k00
yand using 2-D spatial
Fourier deﬁnition for the received signal an inner integral
is obtained. The stationary phase method can be applied to
evaluate that inner integral asymptotically for k0zas follows:
|An(ky, z, ω)|
y, z, ω) = k2
Where 6denotes the phase and tlis the thickness of the layer
l(l= 1,2). The term k00
2(yt−yr)in the phase of the inner
integral is the difference between UDT and MUDT which is
the consequence of considering the problem multistatic rather
than monostatic. In other words, if, in (4) the transmitter and
receiver are located at the same place (monostatic measure-
ment or yt=yr), the UDT formulation will be obtained .
However, the non-diagonal elements of the scattering matrix
cannot be used in the inversion scheme and this is the source of
the shadowing image and low resolution in the UDT compare
to the MUDT.
After a straightforward simpliﬁcation and using the Fourier
transform deﬁnition, the object function for the MUDT can
be obtained as follows
y, ω)is the spatial Fourier transform of the
received scattered ﬁled, and
y, ω) = Z+∞
III. MUDT SIMULATION RESULTS
The MUDT method is used to obtain the location of the
unknown scatterers inside the polymer foam and it is used to
compare the results with the UDT. The numerical scattered
ﬁeld is generated by the use of the time domain solver of
the commercial Software CST Studio Suiterfor a single
layer of the foam and 7x-band open-waveguide antennas
for multistatic transmitting/receiving the signal as shown in
Fig. 3. The distance between two adjacent antennae is 5cm.
Following , an antenna de-embedding is performed to relate
the scattering parameters (S-parameters) to the electric ﬁeld
for MUDT imaging. It is worth mentioning the permittivity
correlates with a certain amount of the moisture (Mn)based
on the wet basis, which is the percentage equivalent of the ratio
of the weight of the water to the weight of the wet foam. Here,
it is assumed that the permittivity of the scatterer is r= 2
which is equivalent to Mn≈40%. Moreover, additional to
the diagonal elements of the scattering matrix (DT and UDT),
the Si(i+1) (i= 1,2,...,6) is also used in the MUDT for
image reconstruction. Fig. 4(a) shows the reconstructed image
with UDT and Fig, 4(b) shows the reconstructed image with
MUDT. As can be seen from these two ﬁgures, with the
MUDT method, a good resolution is obtained. Furthermore,
the shadowing images with MUDT is signiﬁcantly reduced
compared to the UDT method.
A multistatic uniform diffraction tomography is proposed
to obtain the moisture distribution inside a polymer foam in a
Fig. 3. 2-D simulated imaging scenario.
0.0 0.1 0.2 0.3 0.4 0.5
0.0 0.1 0.2 0.3 0.4 0.5
Fig. 4. Corresponding normalized (Top) UDT (Bottom) MUDT imaging
running belt system in a real-time fashion. MUDT overcomes
low resolution caused by UDT, due to the monostatic nature
of this method, by using the non-diagonal element of the
scattering matrix rather than only the diagonal elements. The
result shows a signiﬁcant increase in the quality of the ﬁnal
image. In the next study, we obtain the moisture level inside
the foam after reconstruction of the location of that by deﬁning
an error function and a certain calibration.
This project has received funding from the European
Union’s Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie grant agreement No. 764902.
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