1

Induced-Current Learning Method for NonlinearReconstructions in Electrical Impedance

TomographyZhun Wei and Xudong Chen

Abstract—Electrical impedance tomography (EIT) is an attrac-tive technique that aims to reconstruct the unknown electricalproperty in a domain from the surface electrical measurements.In this work, the induced-current learning method (ICLM) isproposed to solve nonlinear electrical impedance tomography(EIT) problems. Specifically, the cascaded end-to-end convo-lutional neural network (CEE-CNN) architecture is designedto implement the ICLM. The CEE-CNN greatly decreases thenonlinearities in EIT problems by designing a combined objectivefunction and introducing multiple labels. A noticeable character-istic of the proposed CNN scheme is that the input parameters arechosen as both induced contrast current (ICC) and the updatedelectrical field from a spectral analysis and the output is chosenas ICC, which is fundamentally different from prevailing CNNschemes. Further, several skip connections are introduced to focuson learning only the unknown part of ICC. ICLM is verified withboth numerical and experimental tests on typical EIT problems,and it is found that ICLM is able to solve typical EIT problems inless than 1 second with high image qualities. More importantly, itis also highly robust to measurement noises and modeling errors,such as inaccurate boundary data.

Index Terms—Electrical impedance tomography, deep learn-ing, convolutional neural network.

I. INTRODUCTION

AS a non-invasive imaging technique, electrical impedancetomography (EIT) aims to recover property of materials,such as conductivity, in a domain of interest (DOI) fromthe surface electrical measurements [1]–[3]. Due to its non-invasive, cheap, and non-radiative properties, EIT is widelyused in medical applications, such as monitoring lung function[4], [5], detecting breast cancer [6], and imaging brain activity[7]. In medical EIT, small alternating currents are injectedinto the body from the attached electrodes on skin, andelectrical voltages are collected, which are used to reconstructconductivity of tissues or organs in the body.

Nevertheless, solving EIT problems, i.e., recovering con-ductivities from the outside measurements, is challenging dueto its nonlinearity and ill-posedness. On one hand, the ill-posedness means that, if a small perturbation or error occursin measured data, it may cause a much larger error in thereconstructed results. In practice, the measured data in EITare indeed always contaminated by noises and modelingerrors, such as inaccurate boundary data [8], [9]. On theother hand, the nonlinear property of EIT problems means

Department of Electrical and Computer Engineering, National University ofSingapore, Singapore 117583 (Zhun Wei: weizhun1010@gmail.com, XudongChen: elechenx@nus.edu.sg).

Fig. 1. Illustration of EIT problem with a chest-shaped domain: Currentsare injected from electrodes attached on the boundary ∂Ω, and voltages arecollected from the electrodes. The goal of EIT problems is to reconstruct theproperty of materials in a domain of interest (DOI) D.

that the unknowns (conductivity in DOI) follow a nonlinearrelationship with the measured data (voltages on the surface),which dramatically increases difficulty of the reconstructionproblem.

Regularizations are usually used in iterative algorithms toincrease the stability of the reconstructions in EIT problem-s, such as total variation based algorithms [10], boundaryelement shape-based method [11], variationally constrainednumerical method [12], and basis-expansion subspace-basedoptimization method [13]. To reduce dimensionality of theproblem, shape-based approach, such as level set methods[14], [15], has also been used to solve EIT problems, whereit reformulates the problem of the conductivity reconstructionas an inverse problem for a special geometrical representationof embedded objects. In addition, to increase the speed ofreconstructions, some non-iterative inversion methods providequick solutions by making some approximations, such asfactorization method [16], Calderóns method [17], and D-barmethod [18], [19].

Thanks to the rapid evolution in the field of artificialintelligence, pixel-based learning approaches have providedimpressive results on inverse problems, such as signal denois-ing [20], ill-posed linear inverse problems [21], deconvolution[22], interpolation [23], [24], and nonlinear inverse scatteringproblems [25]. Recently, learning approaches, such as deep D-bar [26], accelerated structure-aware sparse Bayesian learning[2], and dominant current deep learning scheme (DC-DLS)[13], have also been proposed to solve EIT problems, whereconductivity is estimated based on U-net in deep D-bar andDC-DLS. In deep D-bar, simulated data with “noisy” bound-aries of the inclusions are used to train the U-net, and thetrained network is tested on experimental data with inclusionspresented, where the inputs of the network are calculated byD-bar method non-iteratively. Different with deep D-bar, DC-

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DLS learns a relationship between ground-truth conductivityand the conductivity that is obtained from induced contrastcurrent (ICC) by a spectral analysis in the natural pixelbases. Nevertheless, in DC-DLS, the inputs of the networkare obtained by an iterative process, and consequently it ismore time-consuming than the deep D-bar.

In this paper, we introduce a induced-current learningmethod (ICLM) to solve EIT problems, where no iterativepre-processing is needed. ICLM is based on a cascaded end-to-end CNN (CEE-CNN) with an analytical solution as inputs.The proposed ICLM fundamentally differs from the deep D-bar and DC-DLS methods since ICLM turns our attentionfrom directly regressing conductivity to regressing a secondarysource, i.e., ICC, where the inputs of CEE-CNN are ICC andelectrical field, both being vectorial parameters. In addition,multiple labels and skip connections are adopted to further im-prove the performance of ICLM. The proposed ICLM is testedby using both numerical and experimental data and the resultshows that it is able to solve typical EIT problems fast (in lessthan 1 second) with high image quality. More importantly, it isfound that ICLM is highly robust to measurement noises andmodeling errors in EIT problems, such as inaccurate boundarydata.

The original contributions of the paper consist of five aspect-s: (1), Different from the prevailing EIT methods that choosethe default scalar potential as the primary parameter, this paperchooses as the primary parameters both the vectorial electricfield and the new concept of induced contrast current. (2),The paper has transformed the EIT to a form that resemblesthe inverse scattering problem, which enables many usefultools and concepts of inverse scattering problem to be usedfor solving EIT after minor modifications. (3), The ICLMmodel has been applied to EIT for the first time, wherereal-valued Poisson equation is involved, which is differentfrom the ICLM proposed in inverse scattering problem [27]where complex-valued scalar wave equation is involved. (4),A more practical case of inaccurate boundary position isconsidered in the proposed machine learning, whereas thecounterpart is not considered in the previous work in inversescattering problem [27]. (5), We have tested the performanceof the proposed learning model in several public database andour reconstruction results can be chosen as a benchmark forresearchers in EIT community to compare with.

In this work, the matrix expression X and vector expressionX are used to denote a discretized parameter X . The complexconjugate and conjugate transpose are denoted by the super-scripts ∗ and H , respectively.

II. THEORY

A. Standard EIT formulation

A typical two-dimensional chest-shaped domain Ω in Fig. 1is considered as an example to present the method. Followingthe setup in [13], the conductivity of the background in domainΩ is σ0(r). Some inclusions with conductivity of σ(r) arepresented in a domain of interest (DOI) interior to Ω, wherethe DOI is denoted as the dashed line and labeled as D inFig. 1. A total number of Ni excitations of current are injected

from Nr electrodes attached on the boundary ∂Ω, which aredenoted as e1, e2, ..., eNr .

The section is a quick review of the forward solver proposedin [13]. Under the gap model, we formulate the Neumannboundary problem of EIT as follows [28]:

∇ · [σ(r)∇µ(r)] = 0 r ∈ Ω, (1)

with boundary conditions

σ0(r)∂µ(r)

∂ν= Jq/|eq| r ∈ eq, q = 1, 2, ..., Nr, (2)

and

σ0(r)∂µ(r)

∂ν= 0 r ∈ ∂Ω

⋂r /∈ eq, q = 1, 2, ..., Nr. (3)

In the above equations, the domain electrical potential inΩ and outer normal direction of its boundary are denotedas µ and ν, respectively. The length and current of the qthelectrode are denoted as |eq| and Jq , respectively. Besides the

above equations, two conditions includingNr∑q=1

Jq = 0 and

Nr∑q=1

∫eqµds = 0 are given to ensure conservation of charge

and uniqueness of the solution. In the following text of thiswork, a homogeneous background σ0 is used for the simplicityof expression, though our proposed inversion model applies toboth homogeneous and inhomogeneous backgrounds.

B. Integral format of EIT

Based on Green’s Theorem [29], it is easy to obtain the so-lution of the above equations. The µ(r) satisfies the followingintegral equation,

µ(r) = µ0(r) +

∫D

−∇′G(r, r′) · {[σ(r′)− σ0]∇′µ(r′)}dr′,(4)

where µ0(r) is the voltage when there is no inclusion pre-sented in the domain Ω. Here, G(r, r′) satisfies the followingdifferential equations:

∇ · [σ0∇G(r, r′)] = −δ(r − r′), (5)

with boundary conditions:

σ0∂G

∂ν= − 1|Lt|

r ∈ eq, q = 1, 2, ..., Nr, (6)

and

σ0∂G

∂ν= 0 r ∈ ∂Ω

⋂r /∈ eq, q = 1, 2, ..., Nr, (7)

Here, δ(r− r′) is the Dirac delta function and |Lt| is the totallength of all electrodes.

If we take gradient on both side of (4), the following self-consistent equation is obtained:

Et = E0(r) +

∫D

−∇{∇′G(r, r′) · [(σ(r′)− σ0)Et(r′)]}dr′

(8)where electric field Et = −∇µ and E0 = −∇µ0.

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Fig. 2. The CEE-CNN architecture in ICLM for EIT problems: J+x and J+y denotes the x- and y- components of J

+, respectively, and J+ consists ofJ+p from all the current injections. Similarly, E

t,+x and E

t,+y denotes x- and y- components of E

t,+, respectively, and Et,+ consists of Et,+p from all thecurrent injections. L1, L2, L3, Lc denote the loss function of the 1th stage, 2nd stage, 3th stage, and the final combined loss function, respectively [27].

C. Discretization format

The pulse basis and delta testing functions are used inmethod of moment [30] to discretize the DOI, and a totalnumber of M subunits are obtained with the central locationsr1, r2, . . . , rM . The polarization tensor ξn at nth subunit isdefined as

ξn = An[σ(rn)− σ0]I2, (9)

where An, σ(rn), and I2 are area of the nth subunit, conduc-tivity at the nth subunit, and two-dimensional identity matrix,respectively. The polarization tensor ξn relates the ICC J(rn)with the total electric field E

t

p(rn) in the nth subunit as

J(rn) = ξn · Et(rn). (10)

According to (8), the total electric field Et

p(rn) for thepth injection of current satisfies the following self-consistentequation

Et

p(rm) = E0

p(rm) +

M∑n=1

GD(rm, rn) · ξn · Et

p(rn), (11)

where E0

p(rm) is the electric field in the background andp = 1, 2, . . . , Ni. GD(rm, rn) is the Green’s function matrixcharacterized by GD(r, r′) · d = −∇[∇′G(r, r′) · d] with ddenoting an arbitrary dipole.

If we write (11) in a vectorized version as:

Et

p = E0

p +GD · Jp, (12)

then the vectorized version of (10) is written as

Jp = ξ · [E0

p +GD · Jp], (13)

where Jp is a 2M -dimensional vector

Jp = [Jxp (r1), J

xp (r2), ..., J

xp (rM ), J

yp (r1), J

yp (r2), ..., J

yp (rM )]

T

(14)Here, Jxp (rM ) and J

yp (rM ) are x and y components of

ICC at rM for the pth injection of current, respectively.Considering both x and y components, GD can be writtenas a 2M × 2M matrix with the form [Gxx, Gxy;Gyx, Gyy].

Here, Gxx(m,n) is calculated as the x component of electricfield at the location rm from a unit x-oriented dipole atthe location rn. Gxy , Gyx and Gyy are defined followingthe definition of Gxx(m,n). In addition, ξ is written asa 2M × 2M diagonal matrix here, where the diagonal el-ements are [ξ1, ξ2, ..., ξm, ..., ξM , ξ1, ξ2, ..., ξM ] with ξm =Am[σ(rm)− σ0] based on (9).

Based on (4), the boundary differential voltage V (r∂Ω) =µ− µ0 is obtained as:

V (r∂Ω) =

∫D

−∇′G(r∂Ω, r′) · {[σ(r′)− σ0]∇′µ(r′)}dr′

(15)with the boundary location r∂Ω. Similarly, V (r∂Ω) is dis-cretizated as V p:

V p = G∂ · Jp, (16)

In (16), G∂(r∂Ω, r′) is written as a Nr × 2M matrix withthe form of [G

x

∂ , Gy

∂ ]. Here, Gx

∂(q, n) can be calculated as thepotential on the boundary node at rq due to a unit x-orientedat rn and G

y

∂(q, n) is calculated in a similar way.In the above forward solver [13], (13) and (16) are two

critical equations referred to as state and data equations,respectively. It is important to note from (13) and (16) thatthe unknowns (ξ) follow a nonlinear relationship with themeasurements (V p), and this nonlinear property in EIT isdifferent with the concept of nonlinear activation function inthe neural network.

III. METHOD

A. Two general approaches to EIT

The first approach is the traditional objective functionapproach. If we simply denote Ψ as the operator of solvingboundary voltages in EIT, the forward solver is described asa nonlinear function:

V p = Ψp(σ) (17)

Here, σ denotes vectorized version of conductivity in DOI.The optimization problem in traditional method is usually

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formulated as:

argminσ

:

Ni∑p=1

f(Ψp(σ), V p) + g(σ). (18)

In (18), f denotes a measure of mismatch, such as a squareof l2 norm ||Ψp(σ) − Vp||2. To increase the stability of thesolution, regularization g(σ) is also added. The EIT problem(18) is nonlinear and iterative optimization algorithms areusually used to solve (18) like those in [10]–[13], [31].

The second approach is the learning approach, where atraining dataset usually contains a total number of Mt pairsof the ground-truth conductivity and their corresponding col-lected voltages, {σm, V m}Mtm=1. In both deep D-bar and DC-DLS, the inputs are changed from the measurement domain tothe conductivity domain D to simplify the learning process,where an approximate solution of conductivity is firstly ob-tained through pre-processing. Specifically, in deep D-bar, aparametric reconstruction algorithm Rbl , is constructed:

Rbl = argminRθ,θ

:

Mt∑m=1

f(Rθ(σbm), σm) + g(θ). (19)

where Rθ represents the neural network and θ represents theweights in Rθ that are learned in the training process. σbm isobtained from D-bar method and used as input of the network.Here, f is the measure of mismatch, and g(θ) is the regularizerto avoid overfitting [32].

In DC-DLS, a multiple-channel scheme is used to utilizethe information from all current injections, and the parametricreconstruction algorithm in DC-DLS Rdl , is constructed:

Rdl = argminRθ,θ

:

Mt∑m=1

Ni∑p=1

f(Rθ(σdm,p), σm) + g(θ), (20)

where σdm,p is the conductivity obtained from the dominantcurrent part of ICC through an iterative processing for the pthcurrent injection [13].

B. The proposed ICLM implemented by CEE-CNN

The previous work in EIT [33] has shown that choosinginduced contrast current has great advantages in solving EITproblems. This motives us to propose the induced currentlearning method (ICLM).

1) Inputs and output: The input of ICLM is an analyticalsolution derived from the data equation (16). Specifically, weconduct a singular value decomposition (SVD) on G∂ , and thefollowing equation is obtained:

G∂ =∑

mumσmν

Hm. (21)

with G∂ · νm = σmum and σ1 ≥ σ2 ... ≥ σ2M ≥ 0.Considering the orthogonality of the singular vector, the majorpart of ICC J

+

p spanned by the first L dominant singular valuesis analytically obtained from the data equation (16):

J+

p =

L∑j=1

µHj · V pσj

νj , (22)

Fig. 3. An example of profiles in random-ellipse dataset consisting of fourrandom ellipses, where locations and radii of all the ellipses are also randomlydistributed in prescribed ranges.

The obtained J+

p is robust to noise since the first L singularvalues are larger than the remaining ones. The method ofchoosing the value of L can be found in [34]. Calculating J

+

p

using (22) only needs the large singular values and vectors,and consequently a thin SVD on G∂ is sufficient, of whichthe associated computational cost is much smaller than that ofa full SVD. With J

+

p , an updated electrical field Et,+

p is alsoobtained following (12):

Et,+

p = E0

p +GD · J+

p (23)

In ICLM, both the major part of ICC J+

p and the updatedelectrical field E

t,+

p are chosen as the input into the neuralnetwork to regress the ICC Jp. Here, E

t,+

p is also chosen asthe input because it provides additional information neededfor the regression, such as GD and E

0

p, which decreases thedifficulties of the learning process.

As shown in Fig. 2, we separate the x and y componentsof both J

+

p and Et,+

p into different input channels and thereare a total number of 4Ni input channels if Ni injectionsof currents are considered. The output of ICLM has 2Nichannels, consisting of the x and y components of the ICC,Jp, for each injection of current.

2) Cascaded end-to-end CNN: First, we introduce amultiple-label scheme. As mentioned above, our final goal isto regress Jp from the inputs. However, due to the highlynonlinear properties in EIT problems, directly regression ischallenging. Partially inspired by the basis-expansion subspaceoptimization method in [35], multiple labels are set in the inter-mediate layers of the network to guide the learning gradually.The multiple labels eventually reduces the nonlinearity of theproblem, which further alleviates the learning task of neuralnetwork.

In this work, the proposed ICLM consists of S stages, wherethe labels corresponding to the sth stage J

ls

p is formulated as:

Jls

p = FH· {Ms ◦ [F · (Jp − J

+

p )]}+ J+

p . (24)

The notation ◦ is the element-wise matrix product and F isthe Fourier transform matrix. A low-frequency mask Ms isused in (24) at the sth stage, where the low frequency parts,i.e., elements of the central βs × βs blocks, are set as oneand other elements are zero. βs gradually increases to addhigh-frequency parts of ICC into the labels.

Considering the inputs and multiple labels in ICLM, theparametric reconstruction algorithm (RIl ) is described as:

RIl = argminRθ,θ

:

Mt∑m=1

Ni∑p=1

f(Rθ(J+

p , Et,+

p ), Jl1

p , Jl2

p , ..., Jp)+g(θ),

(25)

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Fig. 4. The losses of training and validation data vary with epoches for (a)DC-DLSa and (b) ICLM. The loss functions for DC-DLSa and ICLM aredefined based on conductivity and ICC, respectively. Circles in both (a) and(b) represent early-stopping points.

Specifically, we have chosen a cascaded end-to-end CNN toimplement (25), where multiple CNN structures are cascadedwith multiple labels (In Fig. 2, we have three CNNs with thesame structure). In some prevailing iterative algorithms fornonlinear inverse problems, such as [34], the minor part ofsecondary sources is chosen as a set of unknowns to acceleratethe convergence speed. Inspired by this idea, ICLM is designedto focus on learning only the minor part of ICC instead of ICCitself, which is achieved by adding skip connections to enforcethe major part of ICC to be bypassed to the output layers, asshown in Fig. 2 [27].

In CEE-CNN, instead of training each stage separately, acombined loss function is defined. Specifically, we define theloss function at the sth stage Ls as:

Ls =

Mo∑e=1

(Jos

e − Jls

e )2. (26)

Here, Mo is the total number of elements in the output Jos

or label J ls at the sth stage. The combined loss function Lcis defined as

Lc =

S∑s=1

(αs · Ls). (27)

with αs being the weighting coefficients at each stage.3) Calculation of conductivity: In the testing process, the

direct output of CNN is the estimated ICC Jo

(Jo=J

o3in Fig.

2). With Jo, the total electrical field E

t,o

p for the pth injectionis calculated as

Et,o

p = Eo

p +GD · Jo

p, (28)

An analytical solution of the polarization tensor ξ can bedirectly obtained with E

o

p and Jo

p by combining all injections:

ξo(n, n) =

Ni∑p=1{Jop(n) · [E

t,o

p (n)]∗ + J

o

p(n′) · [Et,op (n′)]∗}

Ni∑p=1{|Et,op (n)|2 + |E

t,o

p (n′)|2}

,

(29)where n′ = n + M . With the polarization tensor ξ, conduc-tivities are easily obtained following (9).

IV. NUMERICAL RESULTS

In this section, we evaluate the performance of two fastlearning approaches, and one traditional iterative algorithm.

Fig. 5. An example of the x component (upper row) and y component (lowerrow) of the induced contract current (ICC) for: the input, output, ground truthof the proposed ICLM when the DOI is excited by the 1st current patterns(Constant coefficient multiplied based on Section IV.A ).

Fig. 6. Results with 5% Gaussian noises (SNR=26 dB) from DC-DLSa,iterative method (SOM), and the proposed ICLM. The “Input” results arecalculated by (30) with p = 1.

i.e., subspace-based optimization method (SOM) [33]–[35],on EIT problems. The first one is the proposed ICLM. Thesecond one is called DC-DLSa, where we simply replace theinputs of DC-DLS proposed in [13] with the analytical solutioncalculated from the inputs of ICLM (J

+

p and Et,+p ). Note that

no iterative preprocessing is needed in DC-DLSa. Specifically,similar to (29), the polarization tensor ξ

afor the pth injection

is calculated by:

ξa

p(n, n) =J

+

p (n) · [Et,+

p (n)]∗ + J

+

p (n′) · [Et,+p (n′)]∗

|Et,+p (n)|2 + |Et,+

p (n′)|2

,

(30)and the conductivity is obtained following (9), which are usedas the inputs of DC-DLSa.

A. Implementation Details1) EIT configuration: In this section, we present the results

of reconstructed conductivity using ICLM and DC-DLSa fromthe simulated voltages, which is collected from Nr = 32electrodes attached on the boundary of a chest shape domain Ωin Fig. 1. The perimeter of the boundary ∂Ω is 106.4 cm, andthe electrodes has a width of we = 2.5 cm. The conductivityof the background is σ0 = 0.424 S/m. Trigonometric currentpatterns are injected from the electrodes, and there are a totalnumber of Ni = 16 current patterns:

J2t−1q = J0 cos(tθq) (31)

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TABLE ISSIMS FOR THE DIFFERENT NUMERICAL EXAMPLES: THREE NOISE CONTAMINATED EXAMPLES INCLUDING 5% (SNR=26 DB), 10% (SNR=20 DB),

AND 15% (SNR=16 DB) ARE CONSIDERED. “P1”, “P2”, AND “P3” REPRESENT PHANTOM 1, PHANTOM 2, PHANTOM 3, RESPECTIVELY.

5% (P1) 5% (P2) 5% (P3) 10% (P1) 15% (P1) Round Error (P1) Ellipse Error (P1)DC-DLSa 0.88 0.90 0.89 0.87 0.81 0.84 0.81

SOM (Iterative) 0.8 0.81 0.79 0.79 0.77 0.79 0.78The proposed ICLM 0.9 0.89 0.90 0.90 0.87 0.88 0.86

TABLE IIAVERAGE SSIMS AND TIME FOR 100 RANDOM PROFILES IN RED FOR 5% (SNR=26 DB), 10% (SNR=20 DB), AND 15% (SNR=16 DB).

5% (100 tests) 10% (100 tests) 15% (100 tests) Average Time (s)DC-DLSa 0.86 0.82 0.78 0.5

SOM (Iterative) 0.81 0.78 0.77 51The proposed ICLM 0.88 0.87 0.86 0.5

Fig. 7. Illustration of inaccurate boundary data: correct chest-shapedboundary (line), boundary with round error (dash line) and ellipse error (dotline), where Phantom 1 is presented as the inclusion.

andJ2tq = J0 sin(tθq), (32)

Here, the magnitude J0 = 0.125 mA/cm, angle θq = 2πq/Nr,t = 1, 2, . . . , Ni/2, and q = 1, 2, . . . , Nr.

The DOI D is an ellipse with long and short radii of18 cm and 12 cm, respectively. We discretize the DOI intoM = 1739 subunits, and each subunit has a dimension of0.625 cm× 0.625 cm. Voltages are collected at all electrodeswith the size of Nr · Ni, and Gaussian noise is added tothe voltages before reconstructions. In this work, structuralsimilarity index (SSIM) [36] are used to quantitatively evaluatethe image qualities in the reconstructed results.

2) Training and testing: To reconstruct “heart and lung”phantoms in EIT problems, we synthetically generate random-ellipse dataset (RED) adopted in [13], which consists of fourrandomly distributed ellipses with random conductivity andsizes as depicted in Fig. 3. Specifically, four ellipses arerandomly generated, where two of them are used to modellungs (E1 and E2), one is used to model heart (E3), andthe last ellipse (E4) models any deformation or pathologypresented in lung. In testing, the network trained with REDis used to reconstruct “heart and lung” phantoms with bothnumerical and experimental data.

3) Parameters Setup: In the training process, we employ2000 randomly generated RED profiles, where 1900 of themare used as training dataset and the other 100 profiles are usedas validation dataset.

Following parameter settings in [13], the radii of E1 andE2 are randomly generated in ranges of 6-10 cm and 4-7cm for vertical and horizontal directions, respectively. Theconductivity of both ellipses is randomly assigned in the range0.1 S/m-0.3 S/m. For E3, the radii are randomly generatedbetween 2 cm and 6 cm with random conductivity between

Fig. 8. Reconstructions with 15% noise (SNR=16 dB) (the 1st row) andinaccurate boundary data by DC-DLSa, the proposed ICLM, and SOM: Inreconstructions with inaccurate boundary data (the 2nd and 3rd rows), 5%Gaussian noises (SNR=26 dB) are further added to the collected voltages.

0.6 S/m and 1 S/m. For the last ellipse (E4), the rangeof radii is the same as that of E3, and the conductivity israndomly obtained in the range of 0 S/m-1 S/m. In addition,the locations of all the ellipses are also randomly distributedin prescribed ranges, where E4 is randomly distributed in anarbitrary position of E1 and E2 to model possible pathologiesor deformations in any part of lungs.

We set various hyperparameters as follows: the momentumin the training process as 0.99, weight decay as 10−6, andlearning rate as decreasing logarithmically from 10−6 to 10−8.Further, three stages (S = 3) are used in this paper, wherethe corresponding low-frequency coefficients β1, β2, and β3are 10, 20, and 39 in (24) following the basis-expansioncoefficients in [13], respectively. The weighting coefficientsα1, α2, and α3 in (27) are 0.3, 0.3, and 0.4, respectively,and L = 15 is used following the suggestion of [25]. Forthe proposed ICLM, electrical field Et,+p is multiplied by aconstant coefficient (the ratio between the maximum of J

+

p

and Et,+p ) and then is used as the input of the network, whichis to balance the scale of J

+

p and Et,+p .

We set a maximum of 25 epoches in the training processand an “early stopping” strategy is also applied to mitigate theeffects of overfitting. Specifically, as shown in Fig. 4, we stop

7

the training for both DC-DLSa and ICLM at the position wherethere is no obvious decrease in the validation loss (marked outby circles), and it takes about 4 hours for both of the methods.In all reconstructions, MatConvNet toolbox [37] is used toimplement the training and testing with a server (Intel XeonCPU and 128 GB RAM).

B. Numerical validations

In the first example, we test DC-DLSa, the proposed ICLM,and iterative SOM on three different “heart and lung” phan-toms, of which the profiles are referred from human thoraxand the conductivity is randomly chosen from the rangesintroduced in previous section. Before each reconstruction,5% Gaussian noises (SNR=26 dB) are added in the collectedvoltages. It is found from Fig. 6 that both DC-DLSa and ICLMare able to reconstruct the profiles with satisfying results,whereas there are some small artifacts presented in the resultsof DC-DLSa and the reconstructed shapes are distorted inSOM. To further evaluate the performance of both methods,we also increase the noise level before reconstructing Phantom1 to 10% (SNR=20 dB) and 15% (SNR=16 dB), where theresults for 15% (SNR=16 dB) are presented in the first row ofFig. 8. To quantitatively compare the results, the SSIMs arealso presented in Table I.

Besides, all the three methods are used to reconstruct 100random profiles in RED for 5% (SNR=26 dB), 10% (SNR=20dB), and 15% (SNR=16 dB) noises. The average SSIMs andtime for all the tests are also presented in Table II. It isseen that, compared with DC-DLSa, the proposed ICLM haslarger advantages for a higher noise level, and the two learningapproaches (DC-DLSa and ICLM) have a much faster speedcompared with SOM.

C. Effects of inaccurate boundary data

Besides noises, the received voltages are easily contaminat-ed by modelling errors in practical EIT measurements, suchas inaccurate boundary data. In the second example, to furtherverify the robustness of ICLM, we reconstruct conductivityfrom voltages simulated with inaccurate boundaries. As pre-sented in Fig. 7, voltages are simulated from domains withwith round boundary error (dash line) and elliptical boundaryerror (dot line), and 5% Gaussian noises (SNR=26 dB) arealso added to the collected voltages. During reconstruction,we are unaware of the perturbation of boundaries and still usethe information of unperturbed boundary, referred to as thecorrect chest-shaped boundary.

In last two rows of Fig. 8, we present the reconstruct-ed results using voltages obtained from the two inaccurateboundary. It is seen that ICLM obtains satisfying results withround boundary error, and the shape of inclusions are stillsuccessfully reconstructed even for the case with ellipticalboundary error. On contrary, for DC-DLSa, despite of thecorrect reconstructed positions of heart and lungs, apparentdeformations are shown in the shape of reconstructions forboth boundary errors. To quantitatively compare the perfor-mance, we also present the SSIMs for each reconstruction withinaccurate boundary data in Table I, where ICLM displays

obvious advantage over the DC-DLSa in those challengingcases.

V. EXPERIMENTAL RESULTS

To further validate the proposed methods, we test DC-DLSa, the proposed ICLM, and iterative SOM with publicexperimental data from two different measurement systems.

In the first experimental test, data from KIT4 (KuopioImpedance Tomography) EIT system [38] are used: A simpleschematic of ground-truth profiles is shown in Fig. 9 for Case#1 and Case #2, where conductive and resistive targets arepresented in the tank filled with saline with the measuredconductivity of 0.03 S/m. The tank is circular cylinder shapewith the radius of Rt = 14 cm, and a total number of Ni = 16adjacent current injections were applied with an amplitude of2 mA, where each electrode has a width of 7 cm and widthof 2.5 cm.

For both DC-DLSa and ICLM, the DOI is a circle with aradius of 0.95Rt, which is divided into M = 1696 subunitswith dimensions 0.571 cm × 0.571 cm for each subunit. Forthe profiles in Case #1 and Case #2, we simply use randomcircular profiles in the training process: We randomly generateone to three circular inclusions in DOI with random radiivarying from 1 cm to 6 cm, where interlapping of circles isallowed. Following the settings in [26], we assign the valueof conductivity in the range of [0.05, 0.12] S/m and [0.005,0.015] S/m to model “conductive” and “resistive”, respectively.In the first two rows of Fig. 9, the reconstructed results forboth DC-DLSa and ICLM are presented. It is seen that bothmethods are able to reconstruct the positions of inclusions,whereas ICLM achieves better performance than DC-DLSafor the more challenging case #2.

In the second experimental test, a set of widely usedexperimental data from [18] is considered: The voltage iscollected on a saline bath tank with inclusions consistingof agar heart and lungs presented, where the saline has theconductivity of 424 mS/m. As depicted in Case # 3 of Fig. 9,the circular tank has a radius Rt = 15 cm with 32 electrodes ofsize 1.6 cm hight and 2.5 cm width, and Ni = 32 trigonometriccurrent patterns are applied with an amplitude of 0.2 mA on32 electrodes.

In reconstructions, the DOI is still a circle with a radius of0.95Rt and it is discretized into M = 1696 subunits with adimension of 0.612 cm × 0.612 cm for each subunit. In thetraining process, we randomly generate 2000 RED profilesto train the network (1900 for training purpose and 100 forvalidation purpose). Following the settings in Case #1 andCase #2, we only use the voltage data from the first 16current patterns in the reconstructions. In Fig. 9, we present thereconstructed results from the measured voltages using DC-DLSa, the proposed ICLM, and iterative SOM. It is found thatall the methods obtain satisfying results in terms of the numberof objects and their positions. However, ICLM outperforms theother two methods in terms of image qualities.

To quantitatively compare the performance of differentmethods on the experimental tests, we also calculate the SSIMsfor all the cases based on the approximated ground truth given

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Fig. 9. Testing with experimental data by DC-DLSa, iterative method(SOM), and the proposed ICLM: For Case #1 and Case #2, conductiveand resistive targets are reconstructed from the data measured on KuopioImpedance Tomography EIT system [38]. Random circular inclusions are usedto train the network in DC-DLSa and ICLM. For Case # 3, the agar heartand lung phantom contained in a saline filled tank are considered [18], wherethe network is trained with RED.

in [18], [38]. Specifically, the SSIMs of reconstructions for theCase #1, Case #2, and Case #3 are 0.95, 0.88, 0.75 for DL-DLSa, 0.96, 0.91, 0.81 for the proposed ICLM, and 0.95, 0.91,and 0.71 for the iterative SOM, respectively. It is seen that theproposed ICLM quantitatively outperforms DL-DLSa in morechallenging Case #2 and Case #3.

VI. DISCUSSION

In the above results, we have compared the proposed methodwith both an existing learning approach (DC-DLSa) and iter-ative algorithm (SOM). We first comment on the comparisonbetween learning approach and iteration algorithm. We findthat both the learning approaches are much faster than theiterative method. On the other hand, for the experimentaltests, available experimental data are often insufficient fortraining neural networks, and one possible solution is touse numerical data for training. However, as seen from theresults in the previous section, the advantages of the learningapproaches degrade when testing on practical data using thenetwork trained with numerical data, where we see that thereconstructed results of DC-DLSa are even worse than SOMfor Case #2. Compared to DC-DLSa and SOM, the obviousadvantage of the proposed method is demonstrated in challeng-ing reconstruction scenarios, such as high noise contaminationand inaccurate boundary data. The robustness of the proposedalgorithm to inaccurate boundary shape is highly desired inclinical applications since the practical boundary can hardlybe accurate. Further, in practical case, the chest boundariesof different patients differ a lot. As a data-driven approach,the proposed deep learning method has potential to furtheralleviate the inaccurate boundary problem by incorporatingbig-data information of patients in the training data.

As for limitations or disadvantages of the proposed method,since the ICLM learns ICC instead of directly learning con-ductivity, both the inputs and outputs has x and y compo-nents corresponding to current-excitations. Therefore, thereare more unknown parameters to be regressed from the net-work compared with previous learning approaches. Despitethe aforementioned disadvantage, due to its nearly real-time

reconstruction and robustness to both noise contaminationsand inaccurate boundary data, the proposed ICLM has greatpotential for clinical applications, such as monitoring healthcondition of lungs. In addition, the proposed method hasalready obtained satisfying reconstruction results just usingrandom profiles in the training process, and we believe theusage of the dataset from the practical scenarios will definitelyfurther improve the performance of the proposed method.

VII. CONCLUSION

An induced-current learning method (ICLM) is proposedto solve nonlinear electrical impedance tomography (EIT)problems. Different with the prevailing deep learning schemesthat directly learns conductivity from an approximated solution(described by (19) or (20)), the proposed ICLM learns asecondary source, i.e., ICC, from the major part of ICC andthe updated electrical field (described by (25)). Specifically,the inputs of ICLM is analytically solved from a spectralanalysis of data equation (16) and, during the training process,the network learns the underlying physics of electromagneticinteractions within the DOI described by state equation (13).The CEE-CNN architecture is designed to implement ICLM,where the nonlinearities in EIT problems are decreased byintroducing a cascaded structure with multiple labels anda combined objective function. Skip connections are alsoadded in CEE-CNN to enforce the network to learn onlythe minor part of ICC. It was demonstrated by numericaland experimental data that the proposed ICLM improves theimaging qualities and robustness to noises and inaccurateboundary data, compared with the other popular deep learningapproach. It is expected that the proposed ICLM will findits application in fast, high-quality, and stable imaging inEIT. Further, as discussed in [9], the proposed frameworkcan be easily extended to estimate the absolute conductivitydistribution and solve three dimensional EIT problems.

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