 Research
 Open Access
 Published:
Precise twodimensional Dbar reconstructions of human chest and phantom tank via sincconvolution algorithm
BioMedical Engineering OnLine volume 11, Article number: 34 (2012)
Abstract
Background
Electrical Impedance Tomography (EIT) is used as a fast clinical imaging technique for monitoring the health of the human organs such as lungs, heart, brain and breast. Each practical EIT reconstruction algorithm should be efficient enough in terms of convergence rate, and accuracy. The main objective of this study is to investigate the feasibility of precise empirical conductivity imaging using a sincconvolution algorithm in Dbar framework.
Methods
At the first step, synthetic and experimental data were used to compute an intermediate object named scattering transform. Next, this object was used in a twodimensional integral equation which was precisely and rapidly solved via sincconvolution algorithm to find the square root of the conductivity for each pixel of image. For the purpose of comparison, multigrid and NOSER algorithms were implemented under a similar setting. Quality of reconstructions of synthetic models was tested against GREIT approved quality measures. To validate the simulation results, reconstructions of a phantom chest and a human lung were used.
Results
Evaluation of synthetic reconstructions shows that the quality of sincconvolution reconstructions is considerably better than that of each of its competitors in terms of amplitude response, position error, ringing, resolution and shapedeformation. In addition, the results confirm nearexponential and linear convergence rates for sincconvolution and multigrid, respectively. Moreover, the least degree of relative errors and the most degree of truth were found in sincconvolution reconstructions from experimental phantom data. Reconstructions of clinical lung data show that the related physiological effect is well recovered by sincconvolution algorithm.
Conclusions
Parametric evaluation demonstrates the efficiency of sincconvolution to reconstruct accurate conductivity images from experimental data. Excellent results in phantom and clinical reconstructions using sincconvolution support parametric assessment results and suggest the sincconvolution to be used for precise clinical EIT applications.
Background
Electrical impedance tomography is a new noninvasive imaging technique in which the conductivity distribution inside a body is reconstructed via knowledge of injected current patterns and resulted induced voltages through finite number of electrodes placed on its surface [1]. This modality has many medical applications including monitoring heart and lung functions [2, 3], breast cancer detection [4] and diagnosis of pulmonary edema and diagnosis of the pulmonary embolus [5].
Reconstructing the conductivity images in EIT involves solving forward and inverse problems [1]. The solution of the forward problem is the potential distribution inside the body given the map of conductivity distribution. The inverse problem is to find the unknown conductivity map inside the body using finite sets of injected current patterns and measured voltages on the electrodes surrounding the body.
Algorithms for solving the forward problem of EIT use Finite Element Methods (FEM), Boundary Element Methods (BEM) and Finite Difference Methods (FDM)[1]. Existing approaches for solving the inverse problem of EIT include:

1.
Linearized iterative methods such as Calderon’s method [6], backprojection [7, 8] and NOSER [9], which are not able to reconstruct conductivity distributions with high variations [10].

2.
Nonlinear iterative methods such as equation error formulation [11], output least square [12], statistical inversion [13] and Newton–Raphson methods [14], which are accurate but suffer from the low convergence rate and high computational complexity [10].

3.
Layer stripping methods [15] which are sensitive to noise and are weak in reconstruction of nonsymmetric conductivities [10].

4.
Direct algorithms such as Dbar [16] and Block method [17] which solve the full nonlinear inverse problem without any iteration in the conductivity domain and do not require any intermediate estimation of the conductivity from a forward model. Block method gains considerably from the homogeneity of conductivity distribution for particles inside each block of the body [17]. The problem of high computational burden faced in this method can be resolved by the method of modified equations [18]. Recently, a noniterative linear inverse solution is introduced in [19] that raises the efficiency of this method via reduction in its computational complexity.
Dbar method is a new direct methodology, which was firstly introduced in the constructive proof of Nachman [16]. This method uses the properties of the Dbar operator of inverse scattering [20] to solve the full nonlinear inverse conductivity problem on the planar domains with two degrees of derivatives. An overview of this method is provided in the following section. The reader can refer to [16] for more details. Note that, the quality of the reconstructed conductivity images by the Dbar method is highly affected by approximate numerical solution to a weakly singular integral equation, named Dbar [21–23].
Concerning the efficiency of the solution to Dbar equation, two different numerical methods, namely product integrals (PI) and multigrid (MG) are considered. PIbased methods to solve Dbar equation require O(N^{6}) arithmetic computations on Npoint grids which is huge even for advanced ultrafast computers [21, 23, 24]. In addition, high error rates, reported in the reconstructed conductivity images of experimental phantoms using these methods [21] convinces the inefficiency of them for practical EIT.
The complexity and high rates of error of PIbased methods inspired the adaptation of MG methods [25] for solving Dbar integral equation. Although MG methods solve Dbar integral equation with a remarkable speed and decrease the computational burden from O(N^{6}) to O(N^{4} log N) incorporating Fast Fourier Transform (FFT), the convergence rate of these methods may not reach ultralinear levels [22]. Recently, Mueller [26] has employed MG solution of Dbar equation to reconstruct physical tank and human chest conductivity images. In addition to the presence of visual artifacts such as blurring, the position, size and orientation of the organs are not correctly reconstructed by MG.
These considerable drawbacks in aforementioned methods motivated us to present an effective computational algorithm based on sincconvolution method to solve Dbar equation with higher accuracy and lower computational burden [27]. But, for an EIT algorithm to be practically used, some numerical and experimental proficiency tests are required to show its actual efficiency [10].
The aim of this study is to assess the feasibility of empirical conductivity image reconstruction via sincconvolution algorithm in the Dbar framework of EIT. A regular EIT algorithm evaluation requires a standard test methodology which is followed by some experimental reconstructions. In this study, the approved parametric test methodology of [2] is used to evaluate sincconvolution algorithm based on the reconstructions of a specific synthetic model. The employed scenario is described subsequently. After parametric evaluation of the sincconvolution, two sets of boundary data are used to qualitatively asses the reconstructions of sincconvolution. Indeed, these experiments validate the parametric evaluations and show real potency of the sincconvolution for clinical EIT. For the purpose of comparison, two other algorithms including MG and NOSER are implemented.
The paper is organized as follows. In the immediately following section, steps of the Dbar algorithm of Nachman are reviewed. Next, the sinc–convolution algorithm for solving Dbar integral equation is described. After establishing synthetic models and explaining phantom and clinical measurements, computations of performance figures are described. The parametric evaluation results of sincconvolution, MG and NOSER are followed by their experimental reconstructions of a phantom tank and a human lung data.
Methods
The EIT problem on a twodimensional simply connected region Ω is modeled by the generalized Laplace equation as
where γ(.) and u(.) represent the conductivity of the domain and the electric potential, respectively. The Dirichlet boundary condition
represents the known voltage distribution, f, on the boundary of the Ω_{,} resulted from injecting a known current density, g, on the boundary that corresponds to Neumann boundary condition
Here, v denotes the outward normal on the boundary ∂Ω. The voltagetocurrent map takes the given voltage distribution f on the boundary to current density distribution g. This mapping is also called DirichlettoNeumann mapping and is denoted by Λγ in EIT literature [10].
Actually, the inverse conductivity problem as stated firstly by Calderon [6] is to uniquely determine the unknown conductivity distribution γ from the knowledge of Λγ. There have been extensive efforts to find and prove the uniqueness of the solution to this problem including the work of Nachman [16], BrownUhlmann [28] and recently Astala [29] for twodimensional inverse conductivity problem. All of these researches are based on the Dbar method of inverse scattering [30].
Methods: Dbar method of EIT
The essence of the Dbar method of EIT is to transform the conductivity equation to Schrödinger equation and use the Dbar approach of inverse scattering to solve the resulting equation. For more details about the theory, the reader is referred to [16]. Here, we only review Dbar equations from the constructive proof of Nachman [16] for solving inverse conductivity problem on a simply connected twodimensional region with two derivatives.
Change of the variable Ψ=γ^{1/2} μ and q=Δγ^{1/2}/γ^{1/2} and assuming that γ is a constant γ _{ best } in the neighborhood of the boundary transforms the conductivity equation (1) to Schrödinger equation in whole R^{2} [16]
Note that, in the Dbar method a point x=(x_{1} x _{2})∈Ω may be identified as a point x=x_{1}+ix_{2} where i^{2}=1 in complex plane. Also the complex parameter k=k_{1}+ik_{2}∈C may be identified as a point k=(k_{1},k_{2})∈R^{2}. Using the assumption that γ is a constant, γ _{ best } in the neighborhood of the boundary or equivalently q=0 outside the boundary, leads to another Schrödinger equation [16]
The key idea behind the proof of Nachman is that since two equations (4), (5) have same compact potentials q, the unique solution of equation (4) can be used to find the unique solution to equation (5). That is γ^{1/2}(x) = ψ(x, k), for x ∈ R^{2}. The unique solution ψ(x, k) to equation (4) is called exponentially growing solution which was first introduced by Faddeev [31]. This solution is asymptotic to e^{ikx} for large x or large k. Defining the function [16]
which is asymptotic to 1 and considering aforementioned key idea in the Nachman’s proof [16], the conductivity γ(x) can be computed as
In the constructive proof of Nachman, an intermediate nonephysical function named scattering transform of q(x) is defined as [16]
which plays an important role in relating the measurement data and the conductivity distribution γ(x). Note that, in equation (8), k¯ and x¯ are respectively the complex conjugates of k and x. By simplifying the equation (8), the scattering transform is related to the DirichlettoNeumann map using the formula [16]
Here, Λ _{ γ } denotes the voltagetocurrent density map when Ω has the conductivity distribution γ and Λ _{1} denotes the voltagetocurrent density map for homogenous conductivity γ = 1. Using the large x asymptotic behavior ψ(x, k)_{∂Ω } ≈ e^{ikx}, an approximation to scattering transform of equation (9), namely t^{exp}(k) is introduced [23] in the form
As shown in [32], as a regularization, the approximate computation of scattering transform t^{exp}(k) should be restricted to a disk of radius R in the complex plane and should be set to zero outside the disk. Therefore, the approximate scattering transform t _{ R } ^{exp}(k)is defined as a compactly support function by [23]
The ${t}_{{R}^{}}^{exp}\left(k\right)$approximation is used in some Dbar reconstructions using numerically simulated data [23, 24, 33], experimentally collected data on phantom tank [21] and human chest data [34].
It is shown by Nachman [16] that the connection between the scattering transform and the μ(x, k) is provided by Dbar equation as
where ${e}_{k}\left(x\right)=exp(i(xk+\overline{x}\overline{k})=exp(2i({x}_{1}{k}_{1}+{x}_{2}{k}_{2})\text{.}$ This equation has a unique solution that satisfies twodimensional singular Dbar integral equation [16]
In [27] a novel sincconvolution algorithm is introduced for solving Dbar equation of (13). This sincconvolution algorithm is based on using collocation to replace twodimensional Dbar convolution equation by a system of algebraic equations. Separation of variables in the proposed method allows elimination of the formulation of huge full matrices and therefore reduces the computational complexity drastically. In addition, sincconvolution method converges exponentially with a rate of $O\left({e}^{c\sqrt{N}}\right)$. An overview of this algorithm is presented in the following. The reader is referred to [27] for more detail.
Methods: numerical solution of Dbar equation via sincconvolution
Here, the iterative sincconvolution algorithm to solve the Dbar integral equation (13) is reviewed. The computational steps of sincconvolution algorithm are enlisted in Table 1. As a matter of fact, the sincconvolution method is used to replace the integral equation (13) by a system of algebraic equations.
Recall from the previous section that the support of scattering transform may be embedded in a disk of radius R. In the first step of the sincconvolution algorithm the required bounds of twodimensional convolution integral are determined as [ − 2R, 2R] × [ − 2R, 2R]. This provides the required knowledge to define the sincpoints via definition of regionrelated mapping functions in the next step of algorithm. In the second step of algorithm, the twodimensional convolution integral in the righthandside of equation (13) is decomposed into four twodimensional convolution integrals r _{ i }, i = 1, ., 4.
Third step of the sincconvolution algorithm forms the required matrices for iterative solution of the Dbar equation. In the fourth step, a special “Laplace transform” of the kernel of the Dbar equation should be computed. This transform is used in the iterative computations of the sincconvolution [27].
As clearly indicated in the fifth step of the sincconvolution algorithm in Table 1, the separationofvariables procedure of Table 2 is used to compute all four twodimensional convolution integrals r _{ i }, i = 1, ., 4. This feature of the sincconvolution allows computing a twodimensional convolution integral r _{ i }, i = 1, ., 4, by only some onedimensional vector operations.
Here, the algorithm for computing r _{2} is summarized and listed in Table 2. Note that, in the sincconvolution method, as fully explained in [27], the separation of the variables of all four twodimensional integrals in the Dbar equation may be done analogously.
Sum of these integrals reassembles the r matrix in the righthandside of the discreteform Dbar equation as:
Here, μ = μ _{ ij } _{ m×m } for m = M + N + 1 with elements μ _{ ij } = μ(z _{ i }, z _{ j }). That is, the elements of this matrix are actually the values of the solution at sinc points. The 1 on the right hand side of the equation (14) denotes a vector of size m^{2} of 1’s. The equation (14) is solved by means of an iterative solver such as GMRES [35]. It is worth noting that since GMRES can only work with reallinear operators, the real and imaginary parts of the solution matrix, μ, must be kept separate [35].
Methods: computational steps of Dbar reconstruction
To use both of the aforementioned datasets in the Dbar algorithm, the steps of the flowchart in Figure 1 must be followed. According to that flowchart, one may need to approximately compute the discrete form of the voltagetocurrent map from the finite measurement data and then approximately compute the scattering transforms.
Computing the discrete dirichlettoNeumann map
In this study, known patterns of current are injected through the electrodes surrounding the body and the induced voltages on the same set of the electrodes are measured. Hence, the primary step in reconstruction is to construct the discrete version of the voltagetocurrent density map in the form of a matrix from the injected current and measured voltage values. In this study, the method introduced by Isaacson in section 3 of [21] to construct the voltagetocurrent density matrix from the boundary measurements on a phantom chest is followed. This computational method is used in all experimental Dbar reconstructions such as [26, 34, 36]. The reader is referred to [21] for analytical derivation of this approximation. Here, we briefly summarize that to fix the notations. Let

L = the number of electrodes

A = the area of an electrode, which is uniform in this study

Δθ = the angle in radian between each electrode

r = radius of the circular domain (in this study the radius of the tank).
In our study, L1 trigonometric current patterns with amplitude M are used. The jth current pattern on the l th electrode is defined by [21]
Let t _{ l } ^{j} denote the vector of normalized currents ${t}^{j}=\frac{{T}^{j}}{\u2225{T}^{j}\u2225}$, where $\u2225{T}^{j}\u2225=\sqrt{\sum _{l=1}^{L}{\left({T}_{l}^{j}\right)}^{2}}$. Also let V _{ l } ^{j}denote the voltage measured on the l th electrode corresponding to j th current pattern T^{j} and normalized so that $\sum _{l=1}^{L}{V}_{l}^{j}=0\text{,}\phantom{\rule{0.24em}{0ex}}j=1,\dots ,L1$. Then, the voltages v^{j} that would result from the normalized current patterns are given by ${v}^{j}=\frac{{V}^{j}}{\u2225{T}^{j}\u2225}$.
Let the (u(.), w(.))_{ L } denote the discrete inner product defined by
Then the entries of the discrete NeumanntoDirichlet map R _{ γ,r } are approximated by [21]
Finally, by computing [21]
one can obtain the discrete approximation of the DirichlettoNeumann map Λ _{ γ }. Using the analytical method introduced in [21], the discrete currenttovoltage map R _{1,r } is approximated by the diagonal matrix
Similarly, the discrete approximation of the Λ _{1} is obtained by [21]
Finally, computing [21]
gives the discrete approximation to (Λ _{ γ } − Λ _{1}) .
Computing the scattering transform ${t}_{{R}^{}}^{exp}\left(k\right)$
The series formulation for scattering transform${t}_{{R}^{}}^{exp}$, firstly derived by Isaacson in [21] and used in practical implementations of the Dbar including [21, 26, 34, 36, 37], is also used in this study. The reader is referred to [21] for analytical derivation and exact formulation of this approximate computation of the scattering transform. For each point z of the grid defined in kplane, the approximated scattering transform is computed as [21]:
where
The method of computing γ _{ best }, the best constant conductivity fit to measured data, is found in Appendix A.
Reconstruction of the conductivity
To reconstruct the conductivity γ(x) at each point x in the xplane, first the Dbar equation of (13) is solved using the sincconvolution method with different discretization levels in kplane enlisted in the second column of Table 3 to find μ(x, k) and then the solution is evaluated to equation (7) at k = 0.
Methods: models
Two sets of synthetic data, resulted from simulated experiments were used to parametrically evaluate the efficiency of the sincconvolution based algorithm as well as other two methods. In addition, a dataset extracted from an EIT experiment on a phantom chest was used to validate the results of that assessment. Moreover, an EIT dataset measured on a human chest was used to illustrate the potency of the sincconvolution in clinical applications. Note that, in all simulations and experimental reconstructions complete electrode model (CEM) [38, 39] was used to represent the current density of electrodes. The meshing process was performed using NETGEN [40]. The type and number of mesh elements and nodes in forward and inverse solution of each simulation and experiment are enlisted in Tables 3 and 4, respectively. In each case, the forward problem mesh is finer than that used to solve the inverse problem. As a result, the forward problem is solved accurately; Meanwhile, this difference of meshes avoids the socalled “inverse crime” [10].
Simulated models
Chest model
A virtual chest phantom representing thoracic region of human body including two elliptical and one circular domain, respectively corresponding lungs and heart was used to evaluate the convergence rate of sincconvolution, MG and NOSER. The second column of Table 5 includes the conductivity values of objects inside this numerically simulated chest phantom, as depicted in Figure 2.
As shown in Figure 2, data collection was simulated by 32 finitesized boundary electrodes for current injection and voltage measurement like ACT3 system [42]. That is, 32 electrodes were arranged counter clockwise, with equally spaces on the boundary of a disk and the first electrode in the position of 3O’clock. The system could inject trigonometric current patterns [38, 43] and measure voltages on all 32 electrodes simultaneously. The magnitudes of the injected current patterns were chosen to 1 mA. The simulated boundary values, along with the conductivities of the second column of Table 5, were used to solve the forward problem represented by equations (1) and (2) via FEM and as a result extract the boundary voltages.
Rotating circular target
A numerical model including a circular target with a diameter equal to 0.05 of the diameter of its container tank was used to evaluate the accuracy of sincconvolution reconstructions via calculating some approved parameters. This model is introduced in [2] to evaluate the performance of EIT algorithms. In this model, the conductivity of the target is twice of the homogenous background conductivity.
Simulation data was generated from nine displacements of the target, starting from the medium center and progressing radially outward. The circular medium was surrounded by 16 electrodes. The amplitude of the injected current patterns was 1 mA. Simulated boundary values, were used to solve the forward problem represented by equation (1) and as a result extract the boundary voltages. In this study, to show the effect of measurement noise on the accuracy of undertest algorithms, a uniform noise with amplitude of 0.1 mA was added to resulted boundary data.
Experimental and clinical data
Chest phantom
A boundary dataset extracted from real measurements was acquired from the EIDORS [41] website (http://eidors3d.sourceforge.net/data_contrib/jn_chest_phantom/jn_chest_phantom.shtml). The dataset is gathered by J. Newell, and D. Isaacson [21] in an experiment on a phantom chest consisting of agar heart and lungs in saline tank of radius 15 cm with 32 equallyspaced boundary electrodes of size 1.6 cm height and 2.5 cm width. Figure 3 shows the configuration of this experimental phantom. The conductivity values of the objects and the saline are included in the third column of Table 5.
Neonate chest data
A clinical EIT dataset collected by Heinrich et.al using Gottingen GoeMF II device on a spontaneously breathing neonate [3] was found in EIDORS[44] website (http://eidors3d.sourceforge.net/data_contrib/ifneonatespontaneous/index.shtml).
This data set includes 220 frames of measured voltages on 16 electrodes using adjacent protocol. As shown in Figure 4, in this measurement the neonate had been lying in prone position with the head turn to the left.
Methods: Performance measures
Convergence rate versus grid size in kplane
Convergence rate (CR) versus grid size in kplane, is an important parameter showing the computational efficiency of EIT algorithms in Dbar framework. This calculation is motivated by [22] and calculated using reconstructions of synthetic thoracic region.
Let us denote the true conductivity as γ _{ true } and denote the approximate solution with a grid of size N _{ i }, i = 1, …, 5 in kplane as γ_{ i }. The supremum norm of the solution error may be defined as [22]:
Then, the convergence rate (CR) is defined as [22]:
Note that, to compare sincconvolution with other non Dbar algorithms such as NOSER, following performance measures are considered.
Accuracy measures versus target positions
Based on the approved test methodology introduced in [2], a scenario is arranged to parametrically evaluate sincconvolution algorithm. As described below, in this scenario the reconstructions of the rotating circular target are used to calculate a set of accuracy measures that describe the quality of reconstruction algorithms.
Preliminarily, a onefourth amplitude set γ_{q} is computed preliminarily based on reconstructions of circular target. This set contains all image pixels [γ]_{ i }, greater than onefourth of the maximum amplitude:
A onefourth threshold could guarantee to detect most of the visually significant effects in reconstructed conductivity images. The center of gravity of γ and γ_{q} are computed and the distances from the medium center to them are calculated as r _{ t } and r _{ q } respectively. Then the following performance measuring parameters are calculated.

Amplitude response (AR) measures the ratio of image pixel amplitude in the target to that in the reconstructed image. For a circular target of area A _{ t } with conductivity σ _{ t }in a medium with conductivity σ _{ r }[2]
$$AR=\frac{{\sum}_{k}{\left[\gamma \right]}_{k}}{{A}_{t}\left(\frac{{\sigma}_{t}{\sigma}_{r}}{{\sigma}_{r}}\right)}$$(27)
In this study, this parameter is normalized so that it AR = 1 for a circular target with $\left(\frac{{\sigma}_{t}}{{\sigma}_{r}}\right)=2$ in the center of medium.

Position error (PE) represents the extent to which reconstructed image truly represents the position of the circular target in the medium. This parameter is computed as [2]:
$$PE={r}_{t}{r}_{q}\text{.}$$(28) 
Ringing (RNG) measures the degree of opposite sign area surrounding the main reconstructed target area. For a circle C centered at center of gravity of γ_{q}, the ringing could be obtained by [2]:
$$RNG=\frac{{A}_{\mathit{out}}}{{A}_{\mathit{in}}}\text{.}$$(29) 
Resolution (RES) is a measure of the smallest visible object within the reconstructed image. This parameter is be defined as [2]:
$$RES=\sqrt{\frac{{A}_{q}}{{A}_{0}}}\text{,}$$(30)
where A _{ q } and A _{0} denote the number of pixels in γ_{q} and entire reconstructed image respectively.

Shape deformation (SD) measure quantifies the fraction of γ_{q} which did not fit within a circle of an equal area. This parameter is computed as [2]:
$$SD=\frac{\sum _{k\notin C}{\left[{\gamma}_{q}\right]}_{k}}{\sum _{k}{\left[{\gamma}_{q}\right]}_{k},}$$(31)
where C denotes a circle centered at COG of γ_{q} with an area equivalent to A _{ q }.
Results and discussion
All three methods were implemented within MATLAB and computations were performed in a Laptop with 2.4 GHZ CPU and 2 GB RAM. The methods were separately applied to the datasets extracted from aforementioned simulated and real models. To fairly compare the quality of reconstructed conductivity images, iteration parameters were set in a common range for all methods. In addition, samesize grids in kplane were used in implementation of sincconvolution and MG.
The following two steps were used to evaluate the quality of sincconvolution images.
First, the synthetic reconstructions were evaluated via efficiency parameters of the preceding section. Next, reconstructions of physical and clinical models were used to validate the parametric assessments.
Results of simulations
Convergence rate
The supremum of reconstruction errors and the required computation times for reconstructions of the synthetic chest phantom using MG and sincconvolution with different levels of discretization in kplane were measured according to equation (24) and then enlisted in third and fourth columns of Tables 6 and 7.
Comparing corresponding accuracies of the reconstruction methods, one can notice that in each case the accuracy of the sincconvolution method is much better than that of the MG, especially in reconstructions with large grids in kplane.
Next, for each discretization level in Tables 6 and 7, the corresponding CR values were computed using the corresponding accuracies according to equation (25), and then enlisted in the fifth column of Tables 6 and 7. Comparing the corresponding convergence rates of the reconstruction algorithms shows that while the sincconvolution method has a nearexponential convergence rate in reconstructing the conductivity distribution of the synthetic chest phantom, the MG method only converges with a linear rate, which is considered very slow. This result confirms the stated exponential convergence rate of sincconvolution [45] as well as the linear convergence rate of MG [22].
Moreover, observing the computation times of sincconvolution and MG in the fourth column of Tables 6 and 7, one may note that to obtain a low accuracy solution to the Dbar equation, the computational complexity of these two methods are approximately same, albeit, sincconvolution method performs a fraction of time better than MG. However, to obtain a high accuracy solution, MG performs very poor. For example, while the sincconvolution method converges to the approximate conductivity with accuracy of 10^{3} in 1871 seconds, the MG can only achieve low accuracies not better than 10^{2} in 3290 seconds, which is considered as a very poor performance. Now, it is predictable that to reconstruct higher resolution conductivity images in kplane, the performance of the sincconvolution would be finer than that of the MG.
Accuracy
The plots in Figures 5 illustrate different performance figures of each algorithm as functions of radial distance of the moving circular target from the medium center.

The amplitude response of all three methods increase from the center of medium toward the boundary. Remarkable oscillations appear in the amplitude response of MG and NOSER respectively when the target is in the midway point and closest point to the boundary. Despite these two methods, the amplitude response of sincconvolution is approximately uniform. This consistency guarantees that the same value of conductivities in different parts of the body contribute equally to the conductivity images produced by sincconvolution.

For position error, the plots show that when the target moves from the center to the boundary, the PE in MG, NOSER and sincconvolution increases from −0.3, 0.1 and −0.1 to 0.2, 0.2, and 0.5 respectively. It is clear that the variance of PE in sincconvolution curve is the closest one to zero. Therefore, the positions of objects are expected to be well recovered in the images reconstructed by sincconvolution algorithm.

The ringing plots indicate that for all three reconstruction algorithms, this artifact is increased as the target moves from the center of medium toward the boundary. The curves show that, for each position of the target, the maximum RNG is found in the image reconstructed by NOSER.

Resolution plots show that the resolution of the NOSER and sincconvolution are more uniform and considerably less than that of MG. It is clear that the RES of sincconvolution is fractionally lower than NOSER. Therefore, one may expect to observe most of the conductivity details in sincconvolution reconstructions.

Shape deformation plots show that the SD of the target in sincconvolution reconstructions is considerably less than that in images produced by each of other two algorithms. The optimum points for shape deformation in all three methods are near the boundary electrodes.
Aforementioned results evince the suitability of the sincconvolution algorithm for experimental impedance imaging. In the following, reconstruction of experimental phantom tank via sincconvolution is presented and compared with that of MG and NOSER.
Results of experiments
Chest phantom
Figure 6 illustrates reconstructions of the phantom tank using all three methods, derived on 64 × 64 grids in zplane. Note that, this experimental model is reconstructed by product integrals (PI) method in [21] and MG method in [37].
The relative errors in reconstructing heart and lung, using undertest methods are enlisted in the second and third columns of Table 8, respectively. For the purpose of comparison, same parameters for the reconstruction results in [21, 37] were computed and then enlisted in fourth and fifth rows of Table 8. It is clear that the relative errors in sincconvolution reconstructions are the least.
Let define degree of truth (DT) of reconstructions as:
where γ _{ rec } and γ respectively denote the reconstructed and true conductivity. For each reconstruction experiment in the first column of Table 8, the corresponding DT is computed using equation (32) and then enlisted in the fourth column of Table 8. Comparing DT values show that the range of the conductivity distribution of the chest phantom is well recovered in sincconvolution reconstruction.
It is clear that the representative results of this experiment in Figure 6, confirm the parametric results of Figure 5. The sincconvolution reconstruction contains a number of sensible features, as described below.

The overall size, position, and the orientation of the organs in the image produced by sincconvolution are more accurate than that in Figures 6(c) and 6(d) produced by MG and NOSER.

The sincconvolution image recovers the separation between the two lungs well while MG and NOSER images do not; MG algorithm overestimates that distance and NOSER underestimates it.

The distortion and blurring of the heart and lungs which are respectively evident in the MG and NOSER images are not appeared respectively in the sincconvolution image.

The degree of ringing artifact in sincconvolution image of Figure 6(b) is less than that in MG image of Figure 6(c) and NOSER image of Figure 6(d).
As can be seen, the representative results of this experiment agree very well with accuracy assessment plots in Figure 5. Therefore, the suitability of sincconvolution for accurate phantom reconstructions is acknowledged.
Neonate chest
Twodimensional conductivity images of the spontaneously breathing neonate chest are reconstructed using all three methods. The results are depicted in Figure 7. Note that, in these images anterior is at the top and right side of the neonate chest is reconstructed on the left side of the images. Images in the left, middle and right columns of Figure 7 correspond to 45th, 70th and 173th frames of data. These images illustrate the conductivity distribution of the neonate’s thoracic region in three endinspirations.
It is worth noting that tidal volumes in the neonate’s left lung were reported less than those in his right lung [3]. That is, the conductivity of right lung is expected to be less than that of left one in reconstructed images. Comparing reconstructed images depicted in Figure 7, it is clear that this fact is well recovered in sincconvolution results. In addition, the sincconvolution reconstructions seem physiologically most accurate, demonstrating conductivity contrast of heart and lungs and recovering the approximate position of organs with least degree of ringing and deformation. It is evident that the reconstructions of other two methods are relatively distorted. One can easily notice an excellent agreement between numerical results obtained via parametric assessments and the quality of reconstructed images in Figure 7. As a result, the high degree of blurring in MG images may be caused by its low resolution and amplitude response. Similarly, the high degree of deformation of lungs and considerable ringing around them in NOSER images are previously predicted by SD and RNG curves of this method in Figure 5.
Note that, since exact information about the conductivity distribution inside the neonate’s chest is not available, no parametric evaluation and comparison could be planned. However, the representative results of this experiment and their correspondence to parametric evaluations confirm the feasibility of precise clinical EIT reconstruction using sincconvolution.
Conclusions
The feasibility of accurate practical conductivity image reconstruction via use of sincconvolution algorithm in Dbar framework was investigated in this study. In the meantime, the performance of this algorithm was compared with two practical methods including, multigrid and NOSER. In this regard, a twofold scenario was employed. In the first step, the quality of sincconvolution reconstructions from noisy boundary data collected on specific synthetic models were evaluated against GREIT agreed accuracy parameters. Results show that the amplitude response and resolution of images are relatively better in sincconvolution reconstructions. In addition, the effect of the distortions like position error, ringing and shape deformation is considerably reduced in the images produced by sincconvolution method. Moreover, comparing the convergence rate of the sincconvolution with that of MG shows that the new sincconvolution method is computationally more efficient than its Dbar based competitor.
In the second step, conductivity images of an experimental phantom chest were reconstructed using all three methods. Excellent agreement between their qualities and parametric assessment results support the sincconvolution suitability for experimental EIT. As a complementary experiment, twodimensional conductivity images of the chest crosssection of a spontaneously breathing neonate were reconstructed using all three methods. A watchful comparison shows that the related physiological problem is best revealed in sincconvolution images. In addition, position, size and orientation of organs are well recovered in sincconvolution images.
These reasons, suggest the sincconvolution as an efficient algorithm for precise clinical EIT applications.
Appendix A: Computing γ _{ best }
The best constant conductivity approximation to the measured boundary data can be computed according to the following formula, which is found in [9, 21].
Let ρ denote the resistivity (the reciprocal of the conductivity), then for a medium of homogenous resistivity, the voltage on the l th electrode from the k th current pattern is proportional to the voltage arising from a constant distribution of one. That is
Let {U _{ l } ^{k}} denote the set of measured voltage data and V _{ l } ^{k}(ρ) the calculated voltage on the electrodes. To find the best constant resistivity fit to the data, one must solve the minimization problem
The solution ρ _{ best } to this minimization problem is given by
The best constant conductivity is then ${\gamma}_{\mathit{best}}=\frac{1}{{\rho}_{\mathit{best}}}$.
Abbreviations
 EIT:

Electrical impedance tomography
 PI:

Product integral
 MG:

Multigrid
 CR:

Convergence rate
 DT:

Degree of truth
 AR:

Amplitude response
 PE:

Position error
 RES:

Resolution
 RNG:

Ringing
 SD:

Shape deformation.
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Acknowledgements
Authors appreciate professor Jin Keun Seo for helpful discussions. Also, the authors thank the anonymous referees for their indepth reviews and constructive comments.
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MA designed and performed the experiments and numerical modeling; ARNN analyzed the experiments and numerical modeling. Both of authors read and approved the final manuscript.
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Abbasi, M., NaghshNilchi, A. Precise twodimensional Dbar reconstructions of human chest and phantom tank via sincconvolution algorithm. BioMed Eng OnLine 11, 34 (2012). https://doi.org/10.1186/1475925X1134
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Keywords
 EIT
 Dbar
 Sincconvolution
 Accuracy measures
 Chest phantom
 Human chest