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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

Email: adel.hamzekalaei@kit.edu

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

results.

Index Terms—microwave imaging, tomography, moisture de-

tection

I. INTRODUCTION

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

[1], 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 [2]. 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 [3].

In [4] 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

TOMOGRAPHY

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

2.

The Lippmann-Schwinger integral equation of the EM scat-

tering problem shown in Fig. 2, can be written in the following

form [5], [6]:

(1)

~

Esct

n(~rr, ~rt) = ~

Etot

n(~rr, ~rt)−~

Einc(~rr, ~rt)

=k2ZΩ

d~r 0¯

¯

G(n1)(~rr, ~r0)·O(~r0)~

Etot

n(~r0)

where ~

Etot

nis the total electric ﬁeld in layer n(n= 1,2,3),

~

Esct

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 ¯

¯

G(n1)(~rr, ~rt)

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

~

Etot

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 [5], [7]

~

Esct

n(~rr, ~rt) = k2ZΩ

d~r 0¯

¯

G(n1)(~rr, ~r 0)·O(~r 0)¯

¯

G(n1)(~r 0, ~rr)

(2)

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 [6], [8]

(3)

¯

¯

G(n1)(~r , ~rt)

=1

πZ+∞

−∞

˜

Tn(ky, kz)e−jkzn (z−zt)

kz1

e−jky(y−yt)dky

If z > ztand =(k2

n−k2

y)1/2<0.˜

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) [6]. The dispersion re-

lation in the layer lis expressed by kzl =qk2

l−k2

yl and

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-

space wavenumber.

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

y=ky+k0

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:

(4)

I(k00

y)≈ |˜

Tn(k00

y

2, kz)|2rπ

|An(ky, z, ω)|

e−j[k00

znz6˜

Tn(k00

y,k00

zn,tl)+ k00

y

2(yt−yr)−π

4]

where k00

zn =q4k2

n−k00

yand

(5)An(k00

y, z, ω) = k2

n

k3

zn

z+∂2

∂k2

y

[6˜

Tn]|k00

y

2

Where 6denotes the phase and tlis the thickness of the layer

l(l= 1,2). The term k00

y

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 [4].

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

(6)

Where ˜

Esct(k00

y, ω)is the spatial Fourier transform of the

received scattered ﬁled, and

˜

Esct(k00

y, ω) = Z+∞

−∞

Esct(yt, ω)e−jk00

yytdyt.(7)

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 [9], 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.

IV. CONCLUSION

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

z(m)

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

y(m)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

z(m)

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

y(m)

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 4. Corresponding normalized (Top) UDT (Bottom) MUDT imaging

results.

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.

ACKNOWLEDGMENT

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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