Artiﬁcial Neural Networks for Quantitative Microwave Breast Imaging

M. Ambrosanio

, S. Franceschini

, F. Baselice

and V. Pascazio

Department of Engineering, University of Naples Parthenope, Naples, Italy

Keywords:

Microwave Imaging, Inverse Scattering, Artiﬁcial Neural Network, MIMO Systems, Biomedical Imaging.

Abstract:

This paper is focused on the use of artiﬁcial neural networks (ANNs) for biomedical microwave imaging of

breast tissues in the framework of advanced breast cancer imaging techniques. The proposed scheme processes

the scattered ﬁeld collected at receivers locations of a multiview-multistatic system and aims at providing an

estimate of the morphological and dielectric features of the breast tissues, which represents a strongly non-

linear scenario with several challenging aspects. In order to train the network, a simulated data set has been

created by implementing the forward problem and an automatic randomly-shaped breast proﬁle generator

based on the statistical distribution of complex permittivity of breast biological tissues was developed. Some

numerical tests were carried out to evaluate the performance of the proposed method and, in conclusion, we

found that the use of ANNs for quantitative biomedical imaging purposes seems to be very promising.

1 INTRODUCTION

Inverse scattering (IS) techniques represent a valuable

imaging modality for several applications in which

a non-destructive testing is required (Massa et al.,

2005; Persico et al., 2018; Ambrosanio and Pascazio,

2015), especially for biomedical diagnostics (Bevac-

qua et al., 2019; Ambrosanio et al., 2016). The capa-

bility of these approaches to retrieve physical as well

as geometrical properties of the objects under test lo-

cated in an inaccessible domain by exploiting electro-

magnetic waves makes them very attractive.

In order to detect inhomogeneities in a medium,

the scattered ﬁeld related to these targets is collected

and processed in a coherent fashion. Nevertheless,

the intrinsic ill-posedness and strong non-linearity of

the inverse problem at hand still represent a big issue

(Colton and Kress, 2012; Isernia et al., 1997). Classi-

cal approaches exploit some linear and nonlinear ap-

proximations to handle the non-linearity issue, such

as the iterative Born method, the distorted Born iter-

ative method (Ahsan et al., 2018) and others (Bevac-

qua and Isernia, 2018; Estatico et al., 2016). There-

fore, by minimising a proper functional, the mismatch

between estimated and measured data is evaluated at

each step of the iterative procedure in order to provide

https://orcid.org/0000-0003-3669-8183

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a recovery of the unknown object.

Unfortunately, these approaches are time-

consuming and computationally expensive, and thus

not suitable for real-time applications. However,

some non-iterative methods are available to provide

reconstructed images in a fast fashion, but they are

still not accurate in the recoveries, especially if strong

scatterers are present in the region of interest.

In this framework, some recent methodologies

based on artiﬁcial neural networks (ANNs) and more

in general on machine learning may be very bene-

ﬁcial to face the drawbacks related to classical ap-

proaches (Lucas et al., 2018). Recently, machine

learning has attracted attention with interesting results

for image classiﬁcation and segmentation, but ANNs

have proven to provide good results also in case of ill-

posed inverse problems (Caorsi and Gamba, 1999).

In this paper, we propose an approach based on

neural networks for the quantitative biomedical imag-

ing of breast proﬁles via a direct inversion scheme.

Thus, the output of the network consists in an estimate

of the complex permittivity proﬁle given the scattered

ﬁeld as input.

2 MATHEMATICAL

BACKGROUND

For the sake of simplicity, a bounded and simply-

connected investigation domain Ω is considered in

204

Ambrosanio, M., Franceschini, S., Baselice, F. and Pascazio, V.

Artiﬁcial Neural Networks for Quantitative Microwave Breast Imaging.

DOI: 10.5220/0009172802040208

In Proceedings of the 13th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2020) - Volume 2: BIOIMAGING, pages 204-208

ISBN: 978-989-758-398-8; ISSN: 2184-4305

𝜖

𝑠

𝒓

𝜖

𝑠

𝒓

𝑇𝑥

𝑅𝑥

Figure 1: Sketch of the multiview-multistatic imaging sys-

tem for the non-invasive testing of the imaging domain Ω.

The antennas are locate on a measurement curve Γ.

a homogeneous background medium whose electro-

magnetic features (ε

, σ

) or an estimate of theirs

are assumed to be known a priori. All the scatter-

ers, as well as the background medium, are assumed

to have a constant magnetic permeability equal to

= 4π · 10

−7

H/m.

In the considered scattering experiments, an im-

pinging time-harmonic wave illuminates the objects

of interest at a certain frequency by a transmitting an-

tenna, and the corresponding scattered ﬁeld generated

by the interaction of the incident ﬁeld with the targets

is collected by some receivers located on a measuring

curve which surrounds the imaging domain. In or-

der to simplify the mathematical formulation, a scalar

two-dimensional (2D) scenario will be considered in

the following.

The incident ﬁelds are modelled as transverse-

magnetic (TM) polarised with respect to z axis which

represents the symmetry direction and all the scatter-

ers located in the domain Ω are assumed to have a

constant section along this axis, as shown in Fig. 1.

Under these hypotheses and by omitting the time fac-

tor e

jωt

, the scattering problem can be stated as a 2D

scalar equation, known as electric ﬁeld integral equa-

tion (EFIE), i.e. (Colton and Kress, 2012):

, r

, ω) =

= k

Ω

G(r

− r

, ω)χ(r

, ω)E

, r

, ω)dr

= A

[χE

], r

∈ Ω, r

, r

∈ Γ, (1)

with:

χ(r

r, ω) =

r, ω)

(ω)

− 1, (2)

being the contrast function relating the electric prop-

erties of the objects inside the imaging domain ε

to those of the homogeneous hosting medium ε

the working frequency ω, k

is the wavenumber in

the background medium, ε

r, ω) = ε

r) − j

ωε

and

(ω) = ε

− j

ωε

are the complex permittivities of

the targets and background medium, respectively. Fi-

nally, E

, E

are the total and scattered electric ﬁelds

and G(r

− r

, ω) is the Green’s function for the case

of homogeneous background and observation points

located along the measurement line. Under this as-

sumption, the relationship between contrast and elec-

tric ﬁeld becomes quite easy since it is a convolution;

therefore, fast Fourier transform (FFT) codes can be

adopted for its evaluation (Isernia et al., 1997).

The aforementioned problem, whose model is

depicted in Eq. (1), aims at retrieving the un-

known contrast function χ(·) from the measurements

of the scattered ﬁeld collected at receivers loca-

tions and for different incident angles E

(·). Thus,

the proposed framework represents a multiple-input-

multiple-output (MIMO) system, commonly referred

to as multiview-multistatic. As most inverse prob-

lems, the attempt of retrieving the contrast function

χ(r

r, ω), or equivalently the complex permittivity of

the targets ε

r, ω), from the measurement samples

represents an ill-posed problem which needs proper

regularisation strategies to obtain reliable solutions

(Colton and Kress, 2012). Moreover, the problem

at hand is also strongly nonlinear, and the degree of

non-linearity (DNL) of the considered integral model

depends on the electromagnetic and geometrical fea-

tures of the targets embedded in the scattering region.

Therefore, the higher the DNL, the harder the prob-

lem at hand and thus the difﬁculty of solving the in-

verse scattering problem as well as its computational

burden.

In order to face these drawbacks and provide an

efﬁcient, almost real-time imaging strategy also in

complicated, strongly non-linear scenarios, in the fol-

lowing a machine-learning-based approach is pro-

posed.

3 ARTIFICIAL NEURAL

NETWORK FOR

QUANTITATIVE IMAGING

The reconstruction of the inner part of an unknown

object from scattered ﬁeld measurements is computa-

tionally expensive in both time and memory require-

ments. Thus, there is a strong interest in the devel-

opment of online techniques for quantitative imag-

ing purposes, whose reconstructions are obtained in

a short time after the acquisition.

Artiﬁcial Neural Networks for Quantitative Microwave Breast Imaging

205

Fully-

connected

Layer

Fully-

connected

Layer

Fully-

connected

Layer

Retrieved

Prole

Scaered

Field

Acial Neural Network

Figure 2: Architecture of the proposed network. This direct inversion scheme has the samples of the scattered ﬁeld as input

and provides an estimate of unknown complex permittivity proﬁle maps as output.

The use of ANNs for imaging purposes goes

back till to the nineties for simple imaging scenar-

ios (Caorsi and Gamba, 1999), but nowadays has be-

come more and more attractive due to the improve-

ment in the computational power of modern technol-

ogy as well as to the innovative network architec-

tures proposed in the scientiﬁc literature. Most arti-

cles propose the use of machine learning either for the

imaging of simple scenarios or as a complementary

strategy in the inversion procedure for regularisation

and super-resolution issues (Shah and Moghaddam,

2017)–(Ashtari et al., 2010).

In this framework, ANNs based on multilayer per-

ceptrons could be very promising for online imaging

purposes. Firstly, they act as universal function ap-

proximators, and secondly they prove to be robust in

presence of noise and fast, since after a training step

they are able to implement a direct mapping between

data and unknowns without any analysis of the phys-

ical rules associated with these data.

Due to these interesting capabilities, they have

been exploited for remote sensing (Vitale et al., 2019;

Aghababaee et al., 2013) as well as for inverse scatter-

ing applications. Most of the research articles focus

on the use of machine learning techniques in order to

ﬁnd a more stable solution, i.e. as an efﬁcient regular-

isation (Shah and Moghaddam, 2017)-(Ashtari et al.,

2010), as well as a hybrid strategy with some analyti-

cal information.

A critical issue in the use of ANNs resides in the

choice of a properly large data set for the training of

the network, since it is fundamental for the estima-

tion of its weights. After an initial training proce-

dure, which represents the bottleneck of this kind of

approaches due to the required computational burden,

a direct mapping between data (i.e., the scattered ﬁeld

samples) and unknowns (i.e., the geometrical and di-

electric features of the targets) can be obtained, which

speeds up the imaging procedure considerably.

In this manuscript, the authors want to propose an

ANN architecture in order to provide a quantitative

online imaging of the electric properties of female

breast tissues starting from measures of the scattered

ﬁeld. The universal approximation theorem (Hornik

et al., 1990) states that any arbitrary nonlinear func-

tion can be approximated via a proper network archi-

tecture. Based on it, in this manuscript a three-layer

network is proposed. Each hidden layer combines all

of the features (local information) learned by the pre-

vious layers across the image to identify the largest

patterns. Fig. 2 provides a sketch of the considered

ANN architecture.

4 NUMERICAL BREAST

PHANTOMS GENERATION

The selection of a proper training data set is of rel-

evant importance for the learning procedure since

the choice of the weights involved in the network is

strongly related to the considered pairs in the data

set. As a matter of fact, large pairs of scattered data

and reference proﬁles are required to build a data set

which is relevant in order to obtain good recovery per-

formance. To this aim, a numerical 2D randomly-

shaped breast proﬁle generator has been exploited in

order to create the reference proﬁles, and the forward

problem has been implemented in order to create the

related scattered data, obtaining the training data set

required by the network.

Due to the relatively simple geometry of the breast

shape, and since the biological tissues can be mainly

grouped into ﬁbro-glandular, transitional and adipose

tissues (Lazebnik et al., 2007), the authors proposed

an automatic numerical breast generator which allows

to obtain ellipsoidal-shaped phantoms with a variable

percentage of ﬁbro-glandular internal tissue. The skin

thickness is modelled as a uniform random variable in

the range [1.5, 2.5] mm whose dielectric permittivity

is equal to 36 and conductivity to 0.86 S/m. Regard-

ing the complex permittivity of the breast inner tis-

sues, the statistical distributions reported in (Lazebnik

et al., 2007) have been considered.

In order to model the spatial variability of the

BIOIMAGING 2020 - 7th International Conference on Bioimaging

206

(a) (b) (c)

(d) (e) (f)

Figure 3: Numerical results (a),(d) real and imaginary parts of the reference complex permittivity, respectively; (b),(e):

retrieved proﬁles via a classical non-linear approach (distorted-Born iterative method) and (c),(f) via the proposed neural

network architecture.

ﬁbro-glandular tissue, a random-shape generation

based on a universal multi-fractal random ﬁeld gen-

erator proposed in (Schertzer and Lovejoy, 1989) has

been adopted. The proﬁle generator can be controlled

via setting three different parameters which govern

the level of sparsity of the ﬁbro-glandular tissue as

well as the ruggedness or smoothness of the proﬁle.

5 NUMERICAL RESULTS

The scattered ﬁeld related to the numerical breast

phantoms was evaluated via a method of moments

(MoM) forward solver. This information was ex-

ploited for the training phase of the network. To test

the performance of the proposed ANN, 50.000 breast

proﬁles were generated and split into training (80%),

testing (15%) and validation (5%) data sets.

The ANN architecture proposed in Section 3 was

trained by employing the stochastic gradient descent

algorithm with momentum which updates the net-

work parameters by taking small steps in the direction

of the negative gradient of the loss. The default values

of the initial weights belong to a Gaussian distribution

with zero-mean and standard deviation equal to 0.01,

and initial bias equal to zero. Finally, a regularisation

term for the weights of the loss function is added to

reduce the overﬁtting.

Regarding the geometry of the problem at hand,

the investigated area is 15 cm

. The matching ﬂuid

employed as background medium is lossless with

r,b

= 15 in order to maximise the matching with the

skin layer and, thus, the amount of ﬁeld reaching the

breast internal tissues (Catapano et al., 2010).

The operating frequency is ﬁxed at 1 GHz and the

circle on which the receivers and transmitters are lo-

cated has radius equal to 22.5 cm. Regarding the

number of antennas, thirty elements acting as trans-

mitters/receivers in a multiview-multistatic fashion

have been assumed. Reconstruction results related to

one case are reported in Fig. 3 in comparison with

classical distorted-Born iterative method (DBIM) es-

timation.

6 CONCLUSION

In the this paper, a novel and computationally fast ap-

proach based on ANNs for the quantitative microwave

imaging of breast tissues has been presented. A pre-

liminary performance assessment was proposed in a

Artiﬁcial Neural Networks for Quantitative Microwave Breast Imaging

207

simpliﬁed 2D scenario which can be easily gener-

alised to the more complete and realistic case of three-

dimensional breast.

In the framework of imaging techniques,

microwave-based tomographic breast imaging may

represent a valid alternative or a complementary

medical exam, since it is safe compared to the

standard mammography and less expensive rather

than magnetic resonance imaging.

For the generation of the training data set, a

randomly-shaped breast proﬁle generator has been

proposed whose tissues electric parameters were se-

lected according to proper statistical distributions as

reported in the scientiﬁc literature (Lazebnik et al.,

2007). Regarding the network design, a three fully-

connected layers network architecture was proposed

and compared with a classical inversion scheme

(DBIM). It is worth to underline the capability of the

proposed approach to retrieve the imaginary part of

complex permittivity with a good accuracy compared

with classical approaches, as well as the capability of

correctly estimating the thickness of the skin layer.

Future work will focus on testing new network ar-

chitectures and on the proper design of the training

data set.

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