1. bookVolume 13 (2022): Edition 1 (January 2022)
Détails du magazine
License
Format
Magazine
eISSN
1891-5469
Première parution
01 Jan 2010
Périodicité
1 fois par an
Langues
Anglais
Accès libre

A high accuracy voltage approximation model based on object-oriented sensitivity matrix estimation (OO-SME model) in electrical impedance tomography

Publié en ligne: 08 Jan 2023
Volume & Edition: Volume 13 (2022) - Edition 1 (January 2022)
Pages: 106 - 115
Reçu: 22 Nov 2022
Détails du magazine
License
Format
Magazine
eISSN
1891-5469
Première parution
01 Jan 2010
Périodicité
1 fois par an
Langues
Anglais
Introduction

Electrical impedance tomography (EIT) is a promising visualization technique with low hardware cost, which reconstructs the conductivity change inside a domain from the voltage change on the boundary [1]. EIT has the potential to evaluate the effect of electrical muscle stimulation (EMS) on human muscles [2,3], where EMS is being expected to replace physical exercises of human body in the future. However, the EIT image has low accuracy [4], which restricts the accurate reconstruction of muscle compartments and the quantitative evaluation of EMS effect [5,6]. To increase the reliability of quantitative evaluation of EMS, the reconstruction accuracy of muscle compartments needs to be improved.

EIT reconstructs the image by matching the measured voltage change ΔU with an approximated voltage change uσ) resulting from the conductivity change Δσ, in which the approximation error e between ΔU and uσ) causes the low accuracy of image reconstruction. Generally, uσ) is formulated as the product of a sensitivity matrix and the conductivity change based on a linear model [7]. With conductivity σ changing from the background-field σb to the object-field σo = σbσ, the voltage U changes from Ub to Uo = UbU accordingly, and the nonlinear function U = f(σ) mapping σ to U changes from fb(σb) to fo(σo). Replacing operator fo with fb and using the Taylor formula on fb(σo) the result uσ) is formulated as JbΔσ [8], where Jb is the Jacobian matrix of fb(σb). For simplicity, Jb is replaced by the sensitivity matrix Sb, which is calculated from a linearized form of fb(σb) [9]. As e between ΔU and uσ) is non-negligible, the reconstructed conductivity change Δσ* has low accuracy, which influences the further quantitative evaluation [4].

To reduce e between ΔU and uσ), two nonlinear models were modified from the linear model to approximate uσ) by optimizing Sb. The first nonlinear model is the so-called “sensitivity updating model” [10], in which Sb is updated. In detail, a conductivity σb* as the new background-field is used to calculate the new sensitivity matrix Sb*. The second nonlinear model is the so-called “second-order sensitivity model” [11], in which the Hessian matrix of fb(σb) is estimated as Sb to compensate for Sb. In detail, the mth row of Sb is represented by the pivots of [Sb]mT[Sb]m, where [Sb]m is the mth row of Sb. However, the reduction of e by optimizing Sb in the two nonlinear models is insufficient. Analytically, with σ changing from σb to σo, the operator f changes from fb to fo accordingly. The change of f has an inevitable influence on uσ), in which the influence reflects on σb and Δσ simultaneously. To eliminate e thoroughly, it is necessary to consider the influence of change of f on uσ) completely.

Under these circumstances, a new voltage approximation model for conductivity reconstruction is proposed, which is called the “object-oriented sensitivity matrix estimation model (OO-SME model)”. The OO-SME model is derived by linearizing U = f(σ) as the product of a sensitivity matrix S and the conductivity σ. Thus, two linear equations are formulated simultaneously on σb as Ub = Sbσb and on σo as Uo = Soσo, where Sb and So are the sensitivity matrices corresponding to σb and σo respectively. Therefore, uσ) resulting from Δσ is related to σb, Δσ, Sb, and ΔS in the OO-SME model, where ΔS is the sensitivity matrix change from Sb to So. In the OO-SME model, e between ΔU and uσ) is eliminated, thus, an image with higher accuracy can be reconstructed.

The objectives of this study are (1) to propose the OO-SME model for conductivity reconstruction with high accuracy, (2) to reconstruct the lean meat in meat sample accurately as a mimic reconstruction of muscle compartment, and (3) to evaluate the mass of lean meat quantitatively from the reconstruction.

Conductivity reconstruction with OO-SME model
OO-SME model for voltage change approximation based on object-oriented sensitivity matrix estimation

The approximation error e between approximated voltage change uσ) and measured voltage change ΔU is (1),

e=ΔUu(Δσ) $$e=\text{ }\!\!\Delta\!\!\text{ }\mathbf{U}-u(\text{ }\!\!\Delta\!\!\text{ }\sigma )$$

where ΔU is the measured voltage change from background-field with conductivity σb to object-field with conductivity σo = σbσ, uσ) is the approximated voltage change resulting from the conductivity change Δσ. In the OO-SME model, uσ) is approximated as (2),

u(Δσ)=SbΔσ+ΔSσb+ΔSΔσ $$u(\text{ }\!\!\Delta\!\!\text{ }\sigma )={{\mathbf{S}}^{b}}\text{ }\!\!\Delta\!\!\text{ }\sigma +\text{ }\!\!\Delta\!\!\text{ }\mathbf{S}{{\sigma }^{b}}+\text{ }\!\!\Delta\!\!\text{ }\mathbf{S}\text{ }\!\!\Delta\!\!\text{ }\sigma $$

where Sb is the sensitivity matrix calculated from σb, ΔS is the sensitivity matrix change from Sb to So, where So is the sensitivity matrix calculated from σo. Sb and So are calculated by (3a) and (3b),

[ Sb ]m,e=1IΩe[ φmc(σb)φmv(σb) ]e $${{\left[ {{\mathbf{S}}^{b}} \right]}_{m,e}}=\frac{1}{I}\int_{{{\text{ }\!\!\Omega\!\!\text{ }}_{e}}}{{}}\left[ \nabla {{\varphi }_{m-c}}\left( {{\sigma }^{b}} \right)\cdot \nabla {{\varphi }_{m-v}}\left( {{\sigma }^{b}} \right) \right]\text{d}{{\text{ }\!\!\Omega\!\!\text{ }}_{e}}$$ [ So ]m,e=1IΩe[ φmc(σo)φmv(σo) ]e $${{\left[ {{\mathbf{S}}^{o}} \right]}_{m,e}}=\frac{1}{I}\int_{{{\text{ }\!\!\Omega\!\!\text{ }}_{e}}}{{}}\left[ \nabla {{\varphi }_{m-c}}\left( {{\sigma }^{o}} \right)\cdot \nabla {{\varphi }_{m-v}}\left( {{\sigma }^{o}} \right) \right]\text{d}{{\text{ }\!\!\Omega\!\!\text{ }}_{e}}$$

where e is the element index, Ωe is the pixel of eth element, φm-c(•) and φm-v(•) are the field potentials relevant to mth electrode-combination, m-c and m-v represent the current stimulating with current-stimulation pair and voltage-measurement pair respectively [9]. I is the current amplitude. ΔS is calculated from Sb and So by (4),

ΔS=SoSb $$\text{ }\!\!\Delta\!\!\text{ }\mathbf{S}={{\mathbf{S}}^{o}}-{{\mathbf{S}}^{b}}$$

and used to optimize uσ) by (2). In practice, an estimated conductivity σo* of the object-field is used to calculate the estimated sensitivity matrix So*, from which an estimated sensitivity matrix change ΔS* is calculated to replace ΔS.

The OO-SME model is derived by linearizing U = f(σ) as a product of the sensitivity matrix S and the conductivity σ as (5) based on the divergence theorem [9],

[U]M×1=[S]M×N[σ]N×1 $${{[\mathbf{U}]}_{M\times 1}}={{[\mathbf{S}]}_{M\times N}}{{[\sigma ]}_{N\times 1}}$$

where U is the voltage from the boundary with different electrode-combination, σ is the discretized conductivity in the domain, S is the sensitivity matrix. M is the total number of electrode-combinations, N is the total number of elements. Based on (5), the voltage Ub from the conductivity σb is formulated as (6-a),

[ Ub ]M×1=[ Sb ]M×N[ σb ]N×1 $${{\left[ {{\mathbf{U}}^{b}} \right]}_{M\times 1}}={{\left[ {{\mathbf{S}}^{b}} \right]}_{M\times N}}{{\left[ {{\sigma }^{b}} \right]}_{N\times 1}}$$

the voltage Uo from the conductivity σo is formulated as (6-b).

[ Uo ]M×1=[ So ]M×N[ σo ]N×1 $${{\left[ {{\mathbf{U}}^{o}} \right]}_{M\times 1}}={{\left[ {{\mathbf{S}}^{o}} \right]}_{M\times N}}{{\left[ {{\sigma }^{o}} \right]}_{N\times 1}}$$

Replacing Uo with UbU, So with SbS, and σo with σbσ in (6-b) and subtracting (6-a) from (6-b) gets the expression of ΔU from Ub to Uo, which is used to formulate uσ) in the OO-SME model in (2).

Compared to the existing linear and nonlinear models, uσ) in the OO-SME model is optimized by considering the influence of ΔS completely. As a result, e between uσ) and ΔU is significantly eliminated. Thus, Δσ is expected to be reconstructed with a higher accuracy.

Conductivity reconstruction with OO-SME model

Fig. 1 shows the flowchart of conductivity reconstruction with the OO-SME model, which includes two steps. In the first step, an initial conductivity change Δσinit* is reconstructed from Sb based on the linear model. In the second step, an updated conductivity change Δσupdt* is reconstructed from So* based on the OO-SME model.

Fig. 1

Flowchart of conductivity reconstruction with the OO-SME model.

In the 1st step of reconstructing Δσinit*, Sb is calculated from σb by (3a), uσ) is formulated based on the linear model as (7).

u(Δσ)=SbΔσ $$u(\text{ }\!\!\Delta\!\!\text{ }\sigma )={{\mathbf{S}}^{b}}\text{ }\!\!\Delta\!\!\text{ }\sigma $$

Δσinit* is reconstructed by matching ΔU with uσ).

In the 2nd step of reconstructing Δσupdt*, the process includes three parts. Firstly, estimate the conductivity of object-field as σo* = σb + Δσinit*, and calculate the estimated sensitivity matrix So* from σo* by (3b). Secondly, calculate the estimated sensitivity matrix change ΔS* by (4), and formulate u*σ) based on the OO-SME model as (8).

u(Δσ)=SbΔσ+ΔSσb+ΔSΔσ $${{u}^{*}}(\text{ }\!\!\Delta\!\!\text{ }\sigma )={{\mathbf{S}}^{b}}\text{ }\!\!\Delta\!\!\text{ }\sigma +\text{ }\!\!\Delta\!\!\text{ }{{\mathbf{S}}^{*}}{{\sigma }^{b}}+\text{ }\!\!\Delta\!\!\text{ }{{\mathbf{S}}^{*}}\text{ }\!\!\Delta\!\!\text{ }\sigma $$

Thirdly, reconstruct Δσupdt* by matching ΔU with u*σ). By using the OO-SME model, Δσupdt* is output with a higher accuracy than Δσinit*.

To stabilize the ill-posed conductivity reconstruction model, the matching between ΔU with uσ) and u*σ) are replaced by minimizing a regularized least square error function as (9a) and (9b) respectively [12],

Δσini=min{ ΔUu(Δσ)22+μ2R(Δσ)22 } $$\text{ }\!\!\Delta\!\!\text{ }{{\sigma }^{ini{{*}^{*}}}}=min\left\{ \parallel \text{ }\!\!\Delta\!\!\text{ }\mathbf{U}-u(\text{ }\!\!\Delta\!\!\text{ }\sigma )\parallel _{2}^{2}+{{\mu }^{2}}\parallel R(\text{ }\!\!\Delta\!\!\text{ }\sigma )\parallel _{2}^{2} \right\}$$ Δσupdt=min{ ΔUu(Δσ)22+μ2R(Δσ)22 } $$\text{ }\!\!\Delta\!\!\text{ }{{\sigma }^{updt*}}=min\left\{ \text{ }\!\!\Delta\!\!\text{ }\mathbf{U}-{{u}^{*}}(\text{ }\!\!\Delta\!\!\text{ }\sigma )_{2}^{2}+{{\mu }^{2}}\parallel R(\text{ }\!\!\Delta\!\!\text{ }\sigma )\parallel _{2}^{2} \right\}$$

where ‖Rσ)‖ is the regularization term, and μ is the regularization factor. Iterative methods such as the steepest descent method (SDM) [13], the Gauss-Newton method (GN) [14], and the conjugate gradient method (CG) [15] have been used to minimize (9a) and (9b). In this study, CG is used because of its lower computational cost and the faster convergence speed.

Conductivity reconstruction by simulation
Voltages and sensitivity matrices of background-field and object-field

Fig. 2 shows a mesh, the conductivity of the background-field, and the object-fields in the simulation. Fig. 2(a) is a 2D mesh with the following parameters: diameter d = 100 mm, nodes number P = 1958, elements number N = 3767, electrodes number L = 16. Fig. 2(b) is a background-field with conductivity σb. Fig. 2(c) – (f) are four object-fields with conductivity σo = σb + Δσ. The positions of objects are defined by parameters a, b, c, d, and e, where a = 0.32d, b = 0.27d, c = 0.34d, and e = 0.18d. The magnitude of σb and σo are 0.021 S/m and 0.267 S/m, respectively, thus, the magnitude of Δσ is 0.246 S/m. In the simulation, a random conductivity noise δσ with a magnitude of 20% of Δσ is added to a partial of 40% of the elements to approximate the inconsistency of conductivity. The electrode-combination for current-stimulation and voltage-measurement is chosen as quasi-adjacent pattern [16] because of its higher signal-to-noise ratio (SNR) than the adjacent pattern [17]. The amplitude of current stimulation is I = 1 mA.

Fig. 2

Mesh, conductivity of background- and object-fields

To obtain the voltages Ub and Uo, and the sensitivity matrices Sb and So of the background-field and object-field in the simulation, an elliptical partial differential equation with Neumann boundary is solved by the finite element method (FEM) to calculate the potential distribution [18]. The governing equation of quasi-static electric field and boundary condition of current stimulation are expressed by (10) and (11) [19],

[σφ(σ)]=0 $$\nabla \cdot [\sigma \nabla \varphi (\sigma )]=0$$ { Γσφn=IΓEll=1,2,Lφn|Γ=0ΓΓ/l=1LEl $$\left\{ \begin{matrix}\int_{\text{ }\!\!\Gamma\!\!\text{ }}{{}}\sigma \frac{\partial \varphi }{\partial n}\cdot \text{d }\!\!\Gamma\!\!\text{ }=I\ \ \ \ \ \ \ \ \ \text{ }\!\!\Gamma\!\!\text{ }\in {{E}_{l}}\quad l=1,2,\cdots L \\{{\left. \frac{\partial \varphi }{\partial n} \right|}_{\text{ }\!\!\Gamma\!\!\text{ }}}=\mathbf{0}\ \ \ \ \ \ \ \ \ \text{ }\!\!\Gamma\!\!\text{ }\in \text{ }\!\!\Gamma\!\!\text{ }/\bigcup\limits_{l=1}^{L}{{}}{{E}_{l}} \\\end{matrix} \right.$$

where σ is the conductivity, φ(σ) is the field potential, Ω is the field domain, Γ is the boundary of Ω, dΓ is the outwards area element vector, n is the outwards normal vector on Γ, I is the current amplitude, and El is the boundary of the lth electrode.

The solution of φ(σ) corresponding to the discretized background-field and object-field are represented by φ(σb) and φ(σo). Ub and Uo are extracted from φ(σb) and φ(σo), which are the column vectors with size M×1. Sb and So are calculated from φ(σb) and φ(σo), which are two matrices with size M×N.

Regularization matrix and regularization factor

Regularization is used in (9a) and (9b) to stabilize the conductivity reconstruction [20], in which regularization matrix ‖Rσ)‖ and regularization factor μ are needed.

Different ‖Rσ)‖ provides different form of penalty. The Tikhonov [21] and Noser [22] regularizations are suggested to penalize the finiteness of conductivity on individual element. The Laplace [23] and total variation (TV) [24] regularizations are suggested to consider the continuity of conductivity on adjacent elements. A hybrid regularization is used to balance the finiteness and continuity simultaneously [25]. In this study, a hybrid regularization with Noser and Laplace is used. The regularization term is calculated as (12).

R(σ)22=0.5 RNoser (σ) 22+0.5 RLaplace (σ) 22 $$\parallel R(\sigma )\parallel _{2}^{2}=0.5\left\| {{R}_{\text{Noser }\!\!~\!\!\text{ }}}(\sigma ) \right\|_{2}^{2}+0.5\left\| {{R}_{\text{Laplace }\!\!~\!\!\text{ }}}(\sigma ) \right\|_{2}^{2}$$

The Noser regularization term of σ is calculated by (13),

RNoser (σ) 22=e=1Nωeσe2 $$\left\| {{R}_{\text{Noser }\!\!~\!\!\text{ }}}(\sigma ) \right\|_{2}^{2}=\underset{e=1}{\overset{N}{\mathop{\sum }}}\,{{\omega }_{e}}\sigma _{e}^{2}$$

where ωe is the square sum of eth column of Sb, and σe is the eth element of σ. The Laplace regularization term of σ is calculated by (14),

RLaplace (σ) 22=e=1N[ σe(σeσl)+σe(σeσm)+σe(σeσn) ] $$\left\| {{R}_{\text{Laplace }\!\!~\!\!\text{ }}}(\sigma ) \right\|_{2}^{2}=\underset{e=1}{\overset{N}{\mathop{\sum }}}\,\left[ {{\sigma }_{e}}\left( {{\sigma }_{e}}-{{\sigma }_{l}} \right)+{{\sigma }_{e}}\left( {{\sigma }_{e}}-{{\sigma }_{m}} \right)+{{\sigma }_{e}}\left( {{\sigma }_{e}}-{{\sigma }_{n}} \right) \right]$$

where l, m, n, and e are element indices, lth, mth, and nth elements are adjacent to the eth element.

Regularization factor μ plays crucial role in stabilizing the conductivity reconstruction from noisy voltage ΔU* due to the condition number of the sensitivity matrix is high [26], where the influence of noise is inhibited by choosing μ properly [27]. In this study, a Gaussian white noise δUb is added to ΔU to generate ΔU* as (15),

ΔU=ΔU+δUb $$\text{ }\!\!\Delta\!\!\text{ }{{\mathbf{U}}^{*}}=\text{ }\!\!\Delta\!\!\text{ }\mathbf{U}+\delta {{\mathbf{U}}^{b}}$$

where δUb is quantified by Ub from the simulation and SNR from the experiment. SNR is defined as (16),

SNR=avg(20log10UbδUb) $$SNR=avg\left( 20{{\log }_{10}}\frac{{{\mathbf{U}}^{{{b}^{*}}}}}{\delta {{\mathbf{U}}^{{{b}^{*}}}}} \right)$$

where Ub* is the voltage from the experiment, δUb* is the estimated noise from Ub* based on reciprocity. The magnitude of δUb is controlled by (17).

max{ δUb }=10(SNR/20)max{ Ub } $$\max \left\{ \delta {{\mathbf{U}}^{b}} \right\}={{10}^{-(SNR/20)}}\max \left\{ {{\mathbf{U}}^{b}} \right\}$$

μ is analyzed by the truncated singular value decomposition (TSVD) as (18) [28],

Δσ=k=1KvkukTσk[ Sb ]TΔU+k=1KvkukTσk[ Sb ]TδUb $$\text{ }\!\!\Delta\!\!\text{ }{{\sigma }^{*}}=\underset{k=1}{\overset{K}{\mathop{\sum }}}\,\frac{{{\mathbf{v}}_{k}}\mathbf{u}_{k}^{T}}{{{\sigma }_{k}}}{{\left[ {{\mathbf{S}}^{b}} \right]}^{T}}\text{ }\!\!\Delta\!\!\text{ }\mathbf{U}+\underset{k=1}{\overset{K}{\mathop{\sum }}}\,\frac{{{\mathbf{v}}_{k}}\mathbf{u}_{k}^{T}}{{{\sigma }_{k}}}{{\left[ {{\mathbf{S}}^{b}} \right]}^{T}}\delta {{\mathbf{U}}^{b}}$$

where Δσ* is the reconstruction of Δσ, σk is the kth singular value of Sb, uk and vk are the left and right orthonormal vectors of Sb corresponding to σk, and K is the rank of Sb. ΔU is the voltage change from Ub to Uo, δUb is a perturbation of Ub. To ensure stable reconstruction, μ is chosen by guaranteeing the second term is smaller than the first term in (18). Thus, μ for stable reconstruction satisfies (19),

σ1μ2max{ UbδUb }10(SNR/20) $$\frac{{{\sigma }_{1}}}{{{\mu }^{2}}}\le \max \left\{ \frac{{{\mathbf{U}}^{b}}}{\delta {{\mathbf{U}}^{b}}} \right\}\approx {{10}^{(SNR/20)}}$$

where σ1 is the maximum singular value of Sb.

Voltage approximation

uσ) is different in different conductivity reconstruction models. uσ) in the linear model is formulated by (7). In the sensitivity updating model, uσ) is updated by an updated sensitivity matrix Sb* that is calculated from a new background-field σb*, which is formulated as (20).

uupdt(Δσ)=SbΔσ $${{u}^{updt}}(\text{ }\!\!\Delta\!\!\text{ }\sigma )={{\mathbf{S}}^{b*}}\text{ }\!\!\Delta\!\!\text{ }\sigma $$

In the second-order sensitivity model, uσ) is compensated by an estimated second-order sensitivity matrix Sb, which is formulated as (21).

ucomp(Δσ)=(Sb+Sb)Δσ $${{u}^{comp~}}(\text{ }\!\!\Delta\!\!\text{ }\sigma )=\left( {{\mathbf{S}}^{b}}+{{\mathbf{S}}^{b\dagger }} \right)\text{ }\!\!\Delta\!\!\text{ }\sigma $$

In the OO-SME model, uσ) is approximated based on (2), in practice, it is formulated as (8). In the forthcoming subsection, the influence of uσ) among the existing models and the OO-SME model will be compared by Δσ*.

Evaluation of reconstructed conductivity

The relative accuracy (RA) between Δσ* and Δσ is defined to verify the high accuracy of the OO-SME model for conductivity reconstruction. RA is defined as (22),

RA=1Noi=1No| ΔσiΔσi |/| Δσi | $$RA=\frac{1}{{{N}^{o}}}\underset{i=1}{\overset{{{N}^{o}}}{\mathop{\sum }}}\,\left| \text{ }\!\!\Delta\!\!\text{ }\sigma _{i}^{*}-\text{ }\!\!\Delta\!\!\text{ }{{\sigma }_{i}} \right|/\left| \text{ }\!\!\Delta\!\!\text{ }{{\sigma }_{i}} \right|$$

where Δσ* iand Δσi are the ith component of Δσ* and Δσ respectively as well belong to the object.

Simulation result

Fig. 3 shows the voltage change ΔU* of different objects in the simulation. ΔU* is divided into 16 loops, and in each loop it has 13 measurements, where the value varies periodically with the electrode-combinations. Fig. 3 indicates that the conductivity change in the domain is reflected reliably by the voltage change on the boundary.

Fig. 3

Voltage changes of different objects in the simulation.

Fig. 4 shows the reconstructed conductivity Δσ* based on the linear model, two nonlinear models, and the OO-SME model in the simulation. Fig. 4(a) shows four object-fields with different objects. Fig. 4(b), (c), and (d) show Δσ* based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 4(e) shows Δσ* based on the OO-SME model. Compared to the objects in Fig. 4(a), Δσ* in Fig. 4(b), (c), and (d) displays the position of the object only, in which the quantitative values of Δσ* are not reconstructed accurately. In contrast, besides the accurate position information, the magnitude of Δσ is reliably reconstructed by Δσ* in Fig. 4(e). The comparison in Fig. 4 indicates that the conductivity reconstructed based on the OO-SME model has a higher accuracy than the existing models.

Fig. 4

Reconstructed conductivity based on different conductivity reconstruction models in the simulation. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.

Fig. 5 shows RA of reconstructed conductivity Δσ* in Fig. 4. On average, RA with the linear model is 9.39%. RA with the OO-SME model is 83.98%. RA with the sensitivity updating model is 31.30% and RA with the second-order sensitivity model is 24.61%. Compared to the linear model, RA with the sensitivity updating model and the second-order sensitivity model increased slightly by 21.91% and 15.24% respectively. RA with the OO-SME model increased significantly by 73.59%. The comparisons in Fig. 5 show that the reconstructed conductivity Δσ* with the OO-SME model has a higher accuracy to evaluate the ideal conductivity change Δσ than the existing models.

Fig. 5

Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the simulation.

Conductivity reconstruction by experiment
Experimental setup

Fig. 6 shows the experimental setup of an EIT system, which consists of 4 parts, a personal computer (PC), an impedance analyzer, a digital multiplexer, and an EIT sensor. The impedance analyzer is IM3570 made by Hioki. The multiplexer is made based on Arduino, which has 16 channels to switch on and off between different electrode-combinations for current-stimulation and voltage-measurement. The sensor is a polylactic acid-made circular tank printed with a 3D printer. The diameter of the tank is d = 100 mm. The 16 electrodes made of stainless screw are mounted along the circumference of the tank evenly.

Fig. 6

Experimental setup of EIT system

Experimental method

As shown in Fig. 6, the PC controls the signals to trigger the impedance analyzer and switch on and off the channels on the multiplexer. The impedance analyzer generates a current signal on two output channels (HC and LC) to stimulate the target and measures the voltage signal from the target via two input channels (HP and LP), from which the impedance of the target is calculated. The multiplexer chooses 4 of 16 channels to stimulate the current and measure the voltage, the electrode-combinations for current-stimulation and voltage-measurement in the experiment are the same as in the simulation.

The experiment is conducted as follow. At first, the impedance from the background-field and the object-field are measured. Then, the voltages of background-field Ub* and object-field Uo* are extracted, and the voltage change ΔU* from Ub* to Uo* is calculated. At last, Δσ* is reconstructed by matching ΔU* with uσ) based on different conductivity reconstruction models.

Experimental condition

In the experiment, the meat sample from pig rump was used. The background-field is fat. The object-field is a lean meat mass enclosed by fat. The conductivity of fat and lean meat are σf = 0.021 S/m and σm = 0.267 S/m respectively at f = 100 Hz [29]. The lean meat masses with different shapes and sizes are measured, which have the same dimension as the objects in the simulation. The amplitude of current stimulation is I = 1mA. The current frequency is f = 100 Hz.

Experimental results

Fig. 7 shows the voltage change ΔU* of different lean meat masses in the experiment. ΔU* is divided into 16 loops, and in each loop it has 13 measurements, where the value varies periodically with the changing of electrode-combinations. Fig. 7 indicates that the conductivity change in the meat sample is able to be measured reliably by the voltage change on the boundary.

Fig. 7

Voltage changes of different objects in the experiment.

Fig. 8 shows the reconstructed conductivity Δσ* of lean meat mass based on the linear model, two nonlinear models, and the OO-SME model. Fig. 8(a) shows four object-fields of lean meat masses. Fig. 8(b), (c), and (d) show Δσ* based on the linear model, sensitivity updating model, and second-order sensitivity model, respectively. Fig. 8(e) shows Δσ* based on the OO-SME model. Compared to the lean meat masses in Fig. 8(a), Δσ* in Fig. 8(b), (c) and (d) display the position of lean meat mass only, in which the quantitative value of Δσ* is not accurate. In contrast, reconstructed Δσ* in Fig. 8(e) has a high agreement with ideal Δσ. The comparison in Fig. 8 indicates that the proposed OO-SME model reconstructs the lean meat mass in meat sample accurately. Furthermore, the reconstruction can be used to evaluate the mass of lean meat quantitatively.

Fig. 8

Reconstructed conductivity based on different conductivity reconstruction models in the experiment. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.

Fig. 9 shows RA of reconstructed conductivity Δσ* in Fig. 8. On average, RA with the linear model is 7.74%. RA with the OO-SME model is 54.60%. RA with the sensitivity updating model is 34.45% and RA with the second-order sensitivity model is 24.62%. Compared to the linear model, RA with the sensitivity updating model and the second-order sensitivity model increased slightly by 26.71% and 16.88% respectively. RA with the OO-SME model increased significantly by 46.86%. In conclusion, images in EIT reconstructed based on the OO-SME model have a higher accuracy to quantify the conductivity change in lean meat mass.

Fig. 9

Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the experiment.

Discussion
Approximation error of the OO-SME model

Due to the approximation error e between ΔU* and uσ), reconstructed conductivity Δσ* has low accuracy. Fig. 10 shows the comparison between ΔU* and uσ) from different conductivity reconstruction models in the simulation. Fig. 10(a), (b), (c) and (d) are corresponding to four different objects, respectively. The comparison shows that e from the linear model and the second-order sensitivity model are obvious. The relative ratio of e to ΔU* reaches up to 622% in the linear model and 477% in the second-order sensitivity model on average. Comparing to the linear model and second-order sensitivity model, even though e is significantly reduced in the sensitivity updating model, the relative ratio of e is still around 50% on average. In contrast, e from the OO-SME model is sufficiently eliminated, the relative ratio of e is as low as 2.4% on average. The comparison in Fig. 10 indicates that non-negligible approximation error e in the existing models is significantly reduced in the OO-SME model.

Fig. 10

Comparison between ΔU* and u(Δσ) based on different conductivity reconstruction models in the simulation.

The approximation error e is caused by inaccurate sensitivity matrix estimation in different conductivity reconstruction models. In the linear model for approximation of uσ) as (7), the simplification of omitting the change of f and replacing Jb with Sb leads to non-ignorable e. Consequently, Δσ* of different objects in Fig. 4(b) have similar distributions, and the magnitude of Δσ* does not match with Δσ. In the sensitivity updating model for approximation of uσ) as (20), e is expected to be reduced by replacing Sb with Sb*. Even though Sb* from σb* has a higher accuracy compared to Sb, it is still not accurate enough to reduce e obviously. Therefore, as shown in Fig. 4(c), the accuracy of Δσ* is slightly improved. In the second-order sensitivity model for approximation of uσ) as (21), e is expected to be reduced by considering the estimated second-order sensitivity Sb. Since Sb is much smaller than Sb, leaving the contribution of Sb on uσ) is limited. Thus, e is slightly reduced compared to the linear model. As a result, as shown in Fig. 4(d), the accuracy of Δσ* has not been improved obviously. However, different from the derivation in the linear model based on the Taylor formula, uσ) in the OO-SME model is derived as (2) based on the divergence theorem, in which the influence of ΔS on uσ) is completely considered. Therefore, e is eliminated in the OO-SME model and the accuracy of Δσ* is significantly improved compared to the existing models. As shown in Fig. 4(e), Δσ* based on the OO-SME model obtained the object shape and size accurately.

Contribution of sensitivity change ΔS to uσ)

As expressed by (2), uσ) based on the OO-SME model contains three components, all of which have non-ignorable contributions to uσ). Fig. 11 shows the three components of uσ) from different objects in the simulation. On average, the ratio of SbΔσ to ΔU is around -6.34, the ratio of Δb to ΔU is around 1.18, and the ratio of ΔSΔσ to ΔU is around 6.16. The comparison implies two facts. Firstly, the nonlinear operator f acting on σ changes from fb to fo with σ changes from σb to σo; the contribution of change of f to uσ) should be considered by sensitivity matrix change ΔS. Secondly, the contribution of ΔS to uσ) is expressed as ΔS(σbσ), which consists of two parts; one is ΔSΔσ implying that ΔS induced by Δσ influences the operator f acting on Δσ; another is Δb implying that the influence of ΔS on σb has to be considered to approximate uσ).

Fig. 11

Comparison of components of u(Δσ) with different objects in the simulation.

Omitting the change of f in the linear model led to a non-negligible e. The reduction of e from the optimized sensitivity matrix Sb* or Sb + Sb in the two nonlinear models is limited since the contribution of ΔS corresponding to the change of f is not properly considered. In the OO-SME model, e is eliminated by completely considering the contribution of ΔS to uσ), including the components relevant to σb and Δσ simultaneously.

Comparison of sensitivity matrices

Fig. 12 shows the comparison of sensitivity matrix with different conductivity reconstruction models in the simulation, where the sensitivity of each element from all electrode-combinations are collected. Fig. 12(a) shows the conductivity of object-fields. Fig. 12(b), (c) and (d) show Sb in the linear model, Sb* in the sensitivity updating model and Sb + Sb in the second-order sensitivity model, respectively. Fig. 12(e) shows So* in the OO-SME model. The comparison indicates that the sensitivity matrix change ΔS is necessary to be considered. Compared to the Sb from a homogeneous background-field and the Sb + Sb that is optimized from Sb directly, Sb* contains information of the object but it is not accurate. In contrast, So* has high accuracy, from which the object information is reliably detected. The comparison in Fig. 12 indicates that the accuracy of the sensitivity matrix in the OO-SME model is significantly improved, from which the estimated sensitivity matrix change ΔS* is calculated to optimize uσ).

Fig. 12

Comparison of sensitivity based on different conductivity reconstruction models in the simulation. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb in second-order sensitivity model; (e) So* in OO-SME model.

Fig. 13 shows the comparison of the sensitivity matrix in the experiment, where the sensitivity of each element from all electrode-combinations are collected. Fig. 13(a) shows the object-field. Fig. 13(b), (c), and (d) show Sb in the linear model, Sb* in the sensitivity updating model and Sb + Sb in the second-order sensitivity model respectively. Fig. 13(e) shows So* in the OO-SME model. Similar as in Fig. 12, So* in the OO-SME model has a higher accuracy compared to the existing models, from which ΔS* is estimated accurately to optimize uσ).

Fig. 13

Comparison of sensitivity based on different conductivity reconstruction models in the experiment. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb in second-order sensitivity model; (e) So* in OO-SME model.

Quantitative evaluation of measurement object by the OO-SME model

The conductivity reconstruction based on the OO-SME model has high accuracy to evaluate the measurement object quantitatively, which improves the reliability of EIT application in the biomedical field, such as evaluation of effect of EMS on muscle compartments. In this study, the lean meat mass enclosed by fat is accurately reconstructed by the proposed OO-SME model, the relative accuracy RA reaches up 83.98% in the simulation and 54.60% in the experiment, respectively. Compared to the RA of 34.45% in the simulation and 31.30% in the experiment from the sensitivity updating model, the reliability of reconstruction from the OO-SME model is significantly increased. Thus, the OO-SME model can be used for quantitative evaluation of measurement objects in the biomedical fields of EIT applications.

Conclusions

The approximation error in the OO-SME model proposed in this study is eliminated compared to the existing models. The reconstructed conductivity from the OO-SME model has higher accuracy to reflect the shape and size of measurement object.

The lean meat mass in meat sample is accurately reconstructed by the OO-SME model, from which the lean meat mass could be quantitative evaluated.

The relative accuracy of lean meat mass from the reconstructed conductivity based on the OO-SME model reaches up to 83.98% in the simulation and 54.60% in the experiment. The reconstruction has a higher reliability to evaluate the lean meat mass quantitatively.

Fig. 1

Flowchart of conductivity reconstruction with the OO-SME model.
Flowchart of conductivity reconstruction with the OO-SME model.

Fig. 2

Mesh, conductivity of background- and object-fields
Mesh, conductivity of background- and object-fields

Fig. 3

Voltage changes of different objects in the simulation.
Voltage changes of different objects in the simulation.

Fig. 4

Reconstructed conductivity based on different conductivity reconstruction models in the simulation. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.
Reconstructed conductivity based on different conductivity reconstruction models in the simulation. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.

Fig. 5

Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the simulation.
Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the simulation.

Fig. 6

Experimental setup of EIT system
Experimental setup of EIT system

Fig. 7

Voltage changes of different objects in the experiment.
Voltage changes of different objects in the experiment.

Fig. 8

Reconstructed conductivity based on different conductivity reconstruction models in the experiment. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.
Reconstructed conductivity based on different conductivity reconstruction models in the experiment. (a) Object-fields; (b) Linear model; (c) Sensitivity updating model; (d) Second-order sensitivity model; (e) OO-SME model.

Fig. 9

Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the experiment.
Comparison of RA of reconstructed conductivity based on the linear model, sensitivity updating model, second-order sensitivity model, and OO-SME model in the experiment.

Fig. 10

Comparison between ΔU* and u(Δσ) based on different conductivity reconstruction models in the simulation.
Comparison between ΔU* and u(Δσ) based on different conductivity reconstruction models in the simulation.

Fig. 11

Comparison of components of u(Δσ) with different objects in the simulation.
Comparison of components of u(Δσ) with different objects in the simulation.

Fig. 12

Comparison of sensitivity based on different conductivity reconstruction models in the simulation. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb† in second-order sensitivity model; (e) So* in OO-SME model.
Comparison of sensitivity based on different conductivity reconstruction models in the simulation. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb† in second-order sensitivity model; (e) So* in OO-SME model.

Fig. 13

Comparison of sensitivity based on different conductivity reconstruction models in the experiment. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb† in second-order sensitivity model; (e) So* in OO-SME model.
Comparison of sensitivity based on different conductivity reconstruction models in the experiment. (a) Object-field; (b) Sb in linear model; (c) Sb* in sensitivity updating model; (d) Sb + Sb† in second-order sensitivity model; (e) So* in OO-SME model.

Brown BH. Medical impedance tomography and process impedance tomography : a brief review. Meas Sci Technol. 2001;12:991-6. https://doi.org/10.1088/0957-0233/12/8/301Brown BH Medical impedance tomography and process impedance tomography : a brief review Meas Sci Technol 200112991 6 https://doi.org/10.1088/0957-0233/12/8/30110.1088/0957-0233/12/8/301Search in Google Scholar

Sun B, Baidillah MR, Darma PN, Shirai T, Narita K, Takei M. Evaluation of the effectiveness of electrical muscle stimulation on human calf muscles via frequency difference electrical impedance tomography. Physiol Meas. 2021;42(3):35008. https://doi.org/10.1088/1361-6579/abe9ffSun B Baidillah MR Darma PN Shirai T Narita K Takei M Evaluation of the effectiveness of electrical muscle stimulation on human calf muscles via frequency difference electrical impedance tomography Physiol Meas 202142335008 https://doi.org/10.1088/1361-6579/abe9ff10.1088/1361-6579/abe9ff33631732Search in Google Scholar

Sun B, Darma PN, Shirai T, Narita K, Takei M. Electrical-tomographic imaging of physiological-induced conductive response in calf muscle compartments during voltage intensity change of electrical muscle stimulation (vic-EMS). Physiol Meas. 2021;42(9). https://doi.org/10.1088/1361-6579/ac2265Sun B Darma PN Shirai T Narita K Takei M Electrical-tomographic imaging of physiological-induced conductive response in calf muscle compartments during voltage intensity change of electrical muscle stimulation (vic-EMS) Physiol Meas 2021429 https://doi.org/10.1088/1361-6579/ac226510.1088/1361-6579/ac226534467954Search in Google Scholar

Chitturi V, Farrukh N. Spatial resolution in electrical impedance tomography : A topical review. J Electr Bioimpedance. 2017;8:66-78. https://doi.org/10.5617/jeb.3350Chitturi V Farrukh N Spatial resolution in electrical impedance tomography : A topical review J Electr Bioimpedance 2017866 78 https://doi.org/10.5617/jeb.335010.5617/jeb.3350Search in Google Scholar

Darma PN, Baidillah MR, Sifuna MW, Takei M. Real-Time Dynamic Imaging Method for Flexible Boundary Sensor in Wearable Electrical Impedance Tomography. IEEE Sens J. 2020;20(16):9469-79. https://doi.org/10.1109/JSEN.2020.2987534Darma PN Baidillah MR Sifuna MW Takei M Real-Time Dynamic Imaging Method for Flexible Boundary Sensor in Wearable Electrical Impedance Tomography IEEE Sens J 202020169469 79 https://doi.org/10.1109/JSEN.2020.298753410.1109/JSEN.2020.2987534Search in Google Scholar

Darma PN, Takei M. High-Speed and Accurate Meat Composition Imaging by Mechanically-Flexible Electrical Impedance Tomography with k-Nearest Neighbor and Fuzzy k-Means Machine Learning Approaches. IEEE Access. 2021;9:38792-801. https://doi.org/10.1109/ACCESS.2021.3064315Darma PN Takei M High-Speed and Accurate Meat Composition Imaging by Mechanically-Flexible Electrical Impedance Tomography with k-Nearest Neighbor and Fuzzy k-Means Machine Learning Approaches IEEE Access 2021938792 801 https://doi.org/10.1109/ACCESS.2021.306431510.1109/ACCESS.2021.3064315Search in Google Scholar

Cui Z, Wang Q, Xue Q, Fan W, Zhang L, Cao Z, et al. A review on image reconstruction algorithms for electrical capacitance/ resistance tomography. Sens Rev. 2016;36(4):429-45. https://doi.org/10.1108/SR-01-2016-0027Cui Z Wang Q Xue Q Fan W Zhang L Cao Z et al A review on image reconstruction algorithms for electrical capacitance/ resistance tomography Sens Rev 2016364429 45 https://doi.org/10.1108/SR-01-2016-002710.1108/SR-01-2016-0027Search in Google Scholar

Kim BS, Kim KY. Resistivity imaging of binary mixture using weighted Landweber method in electrical impedance tomography. Flow Meas Instrum. 2017;53:39-48. https://doi.org/10.1016/j.flowmeasinst.2016.05.002Kim BS Kim KY Resistivity imaging of binary mixture using weighted Landweber method in electrical impedance tomography Flow Meas Instrum 20175339 48 https://doi.org/10.1016/j.flowmeasinst.2016.05.00210.1016/j.flowmeasinst.2016.05.002Search in Google Scholar

Brandstätter B. Jacobian Calculation for Electrical Impedance Tomography Based on the Reciprocity Principle. IEEE Trans Med Imaging. 2003;39(3):1309-12. https://doi.org/10.1109/TMAG.2003.810390Brandstätter B. Jacobian Calculation for Electrical Impedance Tomography Based on the Reciprocity Principle IEEE Trans Med Imaging 20033931309 12 https://doi.org/10.1109/TMAG.2003.81039010.1109/TMAG.2003.810390Search in Google Scholar

Zhang L. A Modified Landweber Iteration Algorithm using Updated Sensitivity Matrix for Electrical Impedance Tomography. Int J Adv Pervasive Ubiquitous Comput. 2013;5(1):17-29. https://doi.org/10.4018/japuc.2013010103Zhang L A Modified Landweber Iteration Algorithm using Updated Sensitivity Matrix for Electrical Impedance Tomography Int J Adv Pervasive Ubiquitous Comput 20135117 29 https://doi.org/10.4018/japuc.201301010310.4018/japuc.2013010103Search in Google Scholar

Ding M, Yue S, Li J, Wang Y, Wang H. Second-order sensitivity coefficient based electrical tomography imaging. Chem Eng Sci. 2019;199:40-9. https://doi.org/10.1016/j.ces.2019.01.020Ding M Yue S Li J Wang Y Wang H Second-order sensitivity coefficient based electrical tomography imaging Chem Eng Sci 201919940 9 https://doi.org/10.1016/j.ces.2019.01.02010.1016/j.ces.2019.01.020Search in Google Scholar

Kaipio JP, Kolehmainen V, Vauhkonen M, Somersalo E. Inverse problems with structural prior information. Inverse Probl. 1999;15(3):713-29. https://doi.org/10.1088/0266-5611/15/3/306Kaipio JP Kolehmainen V Vauhkonen M Somersalo E Inverse problems with structural prior information Inverse Probl 1999153713 29 https://doi.org/10.1088/0266-5611/15/3/30610.1088/0266-5611/15/3/306Search in Google Scholar

Wang H, Wang C, Yin W. A Pre-Iteration Method for the Inverse Problem in Electrical Impedance Tomography. IEEE Trans Instrum Meas. 2004;53(4):1093-6. https://doi.org/10.1109/TIM.2004.831180Wang H Wang C Yin W A Pre-Iteration Method for the Inverse Problem in Electrical Impedance Tomography IEEE Trans Instrum Meas 20045341093 6 https://doi.org/10.1109/TIM.2004.83118010.1109/TIM.2004.831180Search in Google Scholar

Yorkey TJ, Webster JG, Tompkins WJ. Algorthms for Impedance Tomography Electrcal. IEEE Trans Biomed Eng. 1987;BME-34(11):843-52. https://doi.org/10.1109/TBME.1987.326032Yorkey TJ Webster JG Tompkins WJ Algorthms for Impedance Tomography Electrcal IEEE Trans Biomed Eng 1987BME-3411843 52 https://doi.org/10.1109/TBME.1987.32603210.1109/TBME.1987.3260323692503Search in Google Scholar

Shewchuk JR. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Science (80- ) [Internet]. 1994; Available from: http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdfShewchuk JR An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Science (80- ) [Internet] 1994 Available from http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdfSearch in Google Scholar

Baidillah MR, Iman AAS, Sun Y, Takei M. Electrical Impedance Spectro-Tomography Based on Dielectric Relaxation Model. IEEE Sens J. 2017;17(24):8251-62. https://doi.org/10.1109/JSEN.2017.2710146Baidillah MR Iman AAS Sun Y Takei M Electrical Impedance Spectro-Tomography Based on Dielectric Relaxation Model IEEE Sens J 201717248251 62 https://doi.org/10.1109/JSEN.2017.271014610.1109/JSEN.2017.2710146Search in Google Scholar

Harikumar R, Prabu R, Raghavan S. Electrical Impedance Tomography ( EIT ) and Its Medical Applications : A Review. 2013;(4):193-8.Harikumar R Prabu R Raghavan S Electrical Impedance Tomography ( EIT ) and Its Medical Applications A Review 20134193 8Search in Google Scholar

Crabb MG. Convergence study of 2D forward problem of electrical impedance tomography with high-order finite elements. Inverse Probl Sci Eng. 2017;25(10):1397-422. https://doi.org/10.1080/17415977.2016.1255739Crabb MG Convergence study of 2D forward problem of electrical impedance tomography with high-order finite elements Inverse Probl Sci Eng 201725101397 422 https://doi.org/10.1080/17415977.2016.125573910.1080/17415977.2016.1255739Search in Google Scholar

Jehl M, Avery J, Malone E. A nonlinear approach to difference imaging in EIT ; assessment of the robustness in the presence of modelling errors. Inverse Probl. 2015;31(3):35012. http://dx.doi.org/10.1088/0266-5611/31/3/035012Jehl M Avery J Malone E A nonlinear approach to difference imaging in EIT ; assessment of the robustness in the presence of modelling errors Inverse Probl 201531335012 http://dx.doi.org/10.1088/0266-5611/31/3/03501210.1088/0266-5611/31/3/035012Search in Google Scholar

Peng L, Merkus H, Scarlett B. Using Regularization Methods for Image Reconstruction of Electrical Capacitance Tomography. Part Part Syst Charact. 2000;17:96-104. https://doi.org/10.1002/1521-4117(200010)17:3<96::AID-PPSC96>3.0.CO;2-8Peng L Merkus H Scarlett B Using Regularization Methods for Image Reconstruction of Electrical Capacitance Tomography Part Part Syst Charact 20001796 104 https://doi.org/10.1002/1521-4117(200010)17:3<96::AID-PPSC96>3.0.CO;2-810.1002/1521-4117(200010)17:3<96::AID-PPSC96>3.0.CO;2-8Search in Google Scholar

Vauhkonen M, Vadâsz D, Karjalainen PA, Somersalo E, Kaipio JP. Tikhonov regularization and prior information in electrical impedance tomography. IEEE Trans Med Imaging. 1998;17(2):285-93. https://doi.org/10.1109/42.700740Vauhkonen M Vadâsz D Karjalainen PA Somersalo E Kaipio JP Tikhonov regularization and prior information in electrical impedance tomography IEEE Trans Med Imaging 1998172285 93 https://doi.org/10.1109/42.70074010.1109/42.700740Search in Google Scholar

Cheney M, Engineering B, York N. NOSER : An Algorithm for Solving the Inverse Conductivity Problem. Int J Imaging Syst Technol. 1991;2(1990):66-75. https://doi.org/10.1002/ima.1850020203Cheney M Engineering B York N NOSER : An Algorithm for Solving the Inverse Conductivity Problem Int J Imaging Syst Technol 19912199066 75 https://doi.org/10.1002/ima.185002020310.1002/ima.1850020203Search in Google Scholar

Kang SI, Khambampati AK, Jeon MH, Kim BS, Kim KY. A sub-domain based regularization method with prior information for human thorax imaging using electrical impedance tomography. Meas Sci Technol. 2016;27:25703. https://doi.org/10.1088/0957-0233/27/2/025703Kang SI Khambampati AK Jeon MH Kim BS Kim KY A sub-domain based regularization method with prior information for human thorax imaging using electrical impedance tomography Meas Sci Technol 20162725703 https://doi.org/10.1088/0957-0233/27/2/02570310.1088/0957-0233/27/2/025703Search in Google Scholar

Borsic A, Graham BM, Adler A, Lionheart WRB. In vivo impedance imaging with total variation regularization. IEEE Trans Med Imaging. 2010;29(1):44-54. https://doi.org/10.1109/TMI.2009.2022540Borsic A Graham BM Adler A Lionheart WRB In vivo impedance imaging with total variation regularization IEEE Trans Med Imaging 201029144 54 https://doi.org/10.1109/TMI.2009.202254010.1109/TMI.2009.202254020051330Search in Google Scholar

Song X, Xu Y, Dong F. A hybrid regularization method combining Tikhonov with total variation for electrical resistance tomography. Flow Meas Instrum. 2015;1-9. https://doi.org/10.1016/j.flowmeasinst.2015.07.001Song X Xu Y Dong F A hybrid regularization method combining Tikhonov with total variation for electrical resistance tomography Flow Meas Instrum 20151 9 https://doi.org/10.1016/j.flowmeasinst.2015.07.00110.1016/j.flowmeasinst.2015.07.001Search in Google Scholar

Hu L, Wang H, Zhao B, Yang W. A hybrid reconstruction algorithm for electrical impedance tomography. Meas Sci Technol. 2007;18(3):813-8. https://doi.org/10.1088/0957-0233/18/3/033Hu L Wang H Zhao B Yang W A hybrid reconstruction algorithm for electrical impedance tomography Meas Sci Technol 2007183813 8 https://doi.org/10.1088/0957-0233/18/3/03310.1088/0957-0233/18/3/033Search in Google Scholar

Braun F, Proenc M, Sol J, Thiran J philippe, Adler A. A Versatile Noise Performance Metric for Electrical Impedance Tomography Algorithms. IEEE Trans Biomed Circuits Syst. 2017;64(10):2321-30. https://doi.org/10.1109/TBME.2017.2659540Braun F Proenc M Sol J Thiran J philippe Adler A A Versatile Noise Performance Metric for Electrical Impedance Tomography Algorithms IEEE Trans Biomed Circuits Syst 201764102321 30 https://doi.org/10.1109/TBME.2017.265954010.1109/TBME.2017.265954028141516Search in Google Scholar

Christian PER. Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM J Sci Comput. 1990;11(3):503-18. https://doi.org/10.1137/0911028Christian PER Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank SIAM J Sci Comput 1990113503 18 https://doi.org/10.1137/091102810.1137/0911028Search in Google Scholar

Gabriel S, Lau RW, Gabriel C. The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz. Phys Med Biol. 1996;41(11):2251-69. https://doi.org/10.1088/0031-9155/41/11/002Gabriel S Lau RW Gabriel C The dielectric properties of biological tissues: II Measurements in the frequency range 10 Hz to 20 GHz. Phys Med Biol 199641112251 69 https://doi.org/10.1088/0031-9155/41/11/00210.1088/0031-9155/41/11/0028938025Search in Google Scholar

Articles recommandés par Trend MD

Planifiez votre conférence à distance avec Sciendo