Incorporating a Spatial Prior into Nonlinear D-Bar EIT imaging for Complex Admittivities
Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filtering of nonlinear (nonphysical) Fourier transform data specific to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass filter to include higher frequency components in the reconstruction, in particular, features that are known with high confidence. This information is additionally included in the system of D-bar equations with an independent regularization parameter from that of the extended scattering transform. In this paper, this approach is used in the 2-D D-bar method for admittivity (conductivity as well as permittivity) EIT imaging. Noise-robust reconstructions are presented for simulated EIT data on chest-shaped phantoms with a simulated pneumothorax and pleural effusion. No assumption of the pathology is used in the construction of the prior, yet the method still produces significant enhancements of the underlying pathology (pneumothorax or pleural effusion) even in the presence of strong noise.
Electrical Impedance Tomography (EIT) is a non-invasive radiation-free imaging modality in which low amplitude current is applied through electrodes placed on the surface of a body and the resulting voltages are measured. From these surface measurements, images of the interior conductivity and permittivity can be obtained. The severe ill-posedness of the inverse conductivity/permittivity problem limits the spatial resolution of the reconstructed images, which hinders their clinical applicability. The use of spatial a priori information in the solution of the inverse problem provides a means of including anatomical information that is present with high confidence, while still allowing unknown features such as lung pathologies to emerge in the reconstructed image without any assumption of their presence. In patients with serious respiratory illness, it is often the case that a CT scan is performed to obtain a diagnosis or for a regular exam in the case of a chronic illness, and the condition is monitored with one or more follow-up scans. The initial scan can provide basic a priori information for the reconstruction algorithm such as chest shape, and approximate lung and heart sizes, and relative positions in the plane of the electrodes.
A priori information has been used successfully in iterative reconstruction algorithms to enhance image quality [1, 2, 3, 4, 5, 6, 7, 8, 9], and more recently in  in the direct 2-D D-bar method for (real-valued) conductivity reconstruction. In this paper, the method of  is extended to the 2-D D-bar algorithm for the reconstruction of complex admittivities [11, 12, 13]. The reconstruction algorithm for complex admittivities differs from the D-bar algorithm for real-valued conductivities in the construction of the complex geometrical optics (CGO) solutions. While the well-developed real-valued case [14, 15, 16, 17, 18, 19, 20, 21] utilizes the familiar transformation of the generalized Laplace equation governing the physical EIT problem to a Schrödinger equation, the complex admittivity algorithm requires transforming the problem to a first order elliptic system and constructing two sets of CGO solutions. The algorithm is described briefly in Section 2, and the reader is referred to [11, 12, 13] for further detail.
The method incorporates spatial a priori information about the admittivity distribution in the scattering transform, as well as in the system of D-bar equations, and includes regularization parameters in each place that can be adjusted to control the amount of influence the prior has on the reconstruction. The effectiveness of the method is tested here on simulated data with 0.1% and 1.0% added Gaussian relative noise for a 2-D phantom chest with a simulated pleural effusion and with a simulated pneumothorax. No a priori information about the presence of the effusion or the pneumothorax is used in the reconstruction, only a priori spatial information about the heart and lung boundaries. Nevertheless, both the effusion and pneumothorax become considerably sharper than in images computed without the a priori organ boundary information.
The “heart and lungs prior” is depicted in Figure 1. While the initial prior is piecewise constant, after conductivity and permittivity values have been assigned, the prior is mollified to obtain a smooth function since the method of computing the scattering transform for the prior requires that it be differentiated. Assigning the initial admittivity values to the prior can be done in a number of ways, and the a priori reconstruction algorithm presented here is valid for any assignment method. In our tests, we computed an initial reconstruction with no prior from the noisy data (which we will refer to as a standard D-bar reconstruction), then computed the average conductivity/permittivity in each region of the piecewise constant “heart and lungs prior” prior and assigned those values to each region of the piecewise constant prior. Further implementation details are found in Section 3.2.
The paper is organized as follows. The a priori method is presented in Section 2, which first provides a brief description of the forward model in Subsection 2.1 used to simulate the EIT data, followed by a summary of the D-bar method for complex admittivity imaging in Subsection 2.2, with the modifications for the a priori method described in Subsection 2.3. The D-bar method for admittivity reconstruction is admittedly mathematically complicated, and the reader is referred to the papers [13, 12, 22] for further details. The numerical implementation of the method is outlined in Section 3.2, the test problems described in Section 3.1, and the discussion and conclusions presented in Section 4.
2.1. The forward model
The electric potential inside the 2-D region is modeled by the admittivity equation, a generalized Laplace equation,
where denotes the complex valued admittivity, the electrical conductivity (bounded away from zero ), the electrical permittivity (assumed to be non-negative), and the angular frequency of the applied current. The boundary data for the inverse problem is the Dirichlet-to-Neumann (DN) map which maps a voltage at the boundary to the corresponding current density, i.e.,
where denotes the outward unit normal vector to the boundary . In practice, to dampen rather than amplify the noise in the measured data, currents are applied and the resulting voltages are measured. This corresponds to knowledge of the Neumann-to-Dirichlet ND map
Ensuring conservation of charge and specifying a ground, the ND map can be inverted to obtain the DN map .
For the simulation of the data, a finite element implementation of the complete electrode model (CEM) was used. The CEM  takes into account both the shunting effect of the electrodes and the contact impedances between the electrodes and tissue. The complete electrode model consists of the admittivity equation (1) and the following boundary conditions on electrodes:
where is the effective contact impedance between the electrode and the medium, is the applied current, and is the measured voltage. In addition, Kirchhoff’s Law and the choice of ground must be imposed to ensure existence and uniqueness of the result:
The uniqueness and existence of a solution to the CEM has been proven in .
2.2. The D-bar method for complex admittivities
D-bar methods are named for the partial derivatives with respect to the complex conjugates that arise in the equations in the methods. The operator with respect to the complex variable and the related operator are defined by
Throughout the paper, is associated with via .
The method described below is based on the uniqueness proof for the inverse admittivity problem , which was completed as a constructive proof in [12, 11]. With the introduction of a non-physical complex parameter , the admittivity equation (1) admits solutions with special exponentially growing behavior known as CGO solutions. In particular, it was shown in  that there exist separate solutions and to (1) such that and .
Defining an operator vector , the change of variables
transform the admittivity equation into the first order elliptic system 
Equation (5) has a unique solution for for some .
D-bar methods follow the basic computational outline:
DN map Admittivity.
The scattering data is a matrix function , not physically measurable from the data, with zero entries on the diagonal and off-diagonal entries given by
where and supp .
The DN map uniquely determines the scattering data , and the scattering data uniquely determines the admittivity . However, the relationship between the scattering data and the DN map relies on the intermediate computation of the CGO solutions and on the boundary of as well as functions and . This is described in Step 1 below.
Step 1: From Boundary Measurements to Scattering Data:
For each , solve the following two boundary integral equations
and where denotes the DN map corresponding to a constant admittivity .
Next, compute the traces of the CGO solutions and from the second set of boundary integral equations
where denotes the principal value of the integral.
Then, compute the scattering transforms and :
All of these computations are performed in practice with to stabilize the reconstruction in the presence of noise. The scattering data is set to zero for . This approach has been proved to be a nonlinear regularization strategy in the case of real-valued conductivities . Parallel computing can be used to solve equations (7)-(9) since each of these equations is solved for each independently. Further implementation details are found in Section 3.2.
Step 2: Computation of CGO Solutions:
Let be a domain slightly larger than . This will be needed to numerically compute the and derivatives of the CGO solutions required to form the matrix potential in Step 3. For each , solve the equation
using the fundamental solution for the operator, by solving the decoupled systems
The convolutions take place in over the disc of radius .
Step 3: From CGO Solutions to the Admittivity:
Using the CGO solutions corresponding to , compute the potentials (only one is actually needed)
and from these, compute the admittivity using either
where the convolution in takes place over since has compact support.
2.3. Inclusion of a priori admittivity information
The low pass-filtering (setting for ) in the non-physical scattering domain has an effect similar to that of traditional low-pass filtering in the standard Fourier domain. As , the scattering data and , and thus, for large the scattering data are essentially Fourier transforms of the potential . Hence, it is reasonable to expect a loss of sharp edges in reconstructions of from the low-pass filtered scattering data.
In practice, the scattering data computed via the boundary integral equations (9) “blows up” to as increases, sometimes as early as in the presence of noise. Therefore, a natural question arises. Is it possible to obtain the scattering data for a larger radius ? While methods based on post-processing D-bar conductivity images have been proposed [29, 30], the work of Alsaker and Mueller  is the first D-bar method which directly includes spatial a priori information into the nonlinear reconstruction method. This information is used in the the scattering transform and in the D-bar equation with parameters that can be adjusted to control the amount of influence the prior has on the reconstruction.
The scattering data is augmented by the scattering data that corresponds to the prior outside the feasible region of computation of the true scattering data. Denoting the scattering data from the admittivity prior by , and the feasible region of computation by , we form the new extended scattering data via the formula
where is computed from current and voltage measurements using (9) for . The truncation radius controls the amount of influence the inclusion of has on the reconstruction. The larger , the greater the influence. When , there is no inclusion of . Note that since as , the influence of does not grow without bound as increases.
The second place that a priori information is included in the reconstruction method is in the integral forms of the D-bar equations, systems (11) and (12). The and terms in (11), (12) arise from terms of the form
whose limits are for and and for and . Analogously to , to include a priori information encoded in the CGO solutions, the terms in (16) are replaced by a weighted integral, which we will denote by
Note, when and the method reduces to the original D-bar method of Subsection 2.2 without a priori information.
We summarize the steps of the a priori method. The final approximation to the admittivity is denoted by .
Step 0: Setup:
Compute the DN map from the voltage and current measurements and determine an admittivity prior .
Step I: Computation of Scattering Data :
Compute the extendend scattering via (15). This involves using Step 1 of Subsection 2.2 to compute the traditional scattering data for . To obtain computationally, the smoothed admittivity prior is first used to compute the potential via (4). Then, for , the system (5) is solved, and the resulting matrix of CGO solutions is denoted by . The scattering data is then computed via (6) using and in these equations.
3. Simulation and Implementation
3.1. Simulation of Voltage Data
The FEM was used to simulate voltages for each of the test problems using the Complete Electrode Model (CEM) on the chest-shaped domain in Figure 1 of perimeter 1016 mm, with electrodes of length mm and height mm. The contact impedance was set to on all electrodes, and trigonometric current patterns with amplitude mA were used. The trigonometric current patterns are given by
where corresponds to the angle of the center point of the -th electrode . The quantity therefore represents the current applied on corresponding to the -th current pattern. Note that linearly independent current patterns were applied since electrodes were used in the simulations.
Zero mean Gaussian relative noise was added to each complex-valued vector of simulated voltages in the same manner as  as follows. Let denote the the desired noise level and a vector of Gaussian zero mean noise that is unique for each current pattern (and each test scenario). Then, the real and imaginary parts of the noisy voltage data were computed as
The discrete approximation to the D-N map was computed as in [16, 17], which we summarize briefly here. Denoting by the -th entry of the matrix of applied currents with each column normalized with respect to the -vector norm, , let denote the entries of the -th voltage vector normalized so that and . Let denote the area of the -th electrode. Then where the -th entry of is given by
Figure 2 shows the two simulations: (a) a pneumothorax, (b) a pleural effusion. For both sets of simulated data, the admittivity of the heart was S/m, the lungs S/m, and the background S/m. The pneumothorax was set to S/m and the pleural effusion to S/m.
3.2. Implementation of the a priori method
In this paper, the admittivity prior was computed using a standard D-bar reconstruction recovered using Steps 1-3 of Section 2.2 with the measured data . However, in practice, any initial prior can be used, making the method easily adaptable to other approaches.
Step 0: The matrix approximation to the DN map was formed using the noisy voltages computed from the CEM. The admittivity prior was formed as follows. First the standard D-bar reconstruction was computed using Steps 1-3 of Section 2.2 (see  for details regarding the computation of ). Next, using the spatial heart and lungs prior (see the red dots of Figure 1), the average value of the pixels in each region (heart, left lung, right lung, and background) were computed and the corresponding average assigned to each region to form the admittivity prior . Note that the spatial prior does not assume any pathology is present. The prior was then mollified to a smooth version and computed using finite differences for the and derivatives of .
Step I: The extended scattering data was computed via (15). Using the DN maps and , the traditional scattering data for was determined via Step 1 of Section 2.2. The reader is referred to  for the computational details of computing and and subsequently and . Briefly, the Fredholm integral equations for and (7) are solved by a Galerkin method, and the integrals for evaluating and and scattering data , in (9) are computed using a Simpson’s rule. The scattering prior is determined as follows. First, the admittivity prior is smoothed to compute the potential via (4). Then, for , the system (5) is solved for using Fourier transforms on the following two decoupled systems:
where the convolutions take place in over . Using a uniform -grid of size with stepsize , convolutions such as can be implemented as
Step II: Choose a regularization weight . Using the combined scattering data , the CGO solutions and were recovered using Fourier transforms to solve the modified equations (22)
where the convolutions take place in over and is computed from (17) using a Simpson’s rule. An analogous system is solved to recover and .
In this work, two noise levels were considered: added relative noise and relative noise. For each example, we present results with three values of the truncation radius in the prior, and three regularization weights for the D-bar equation: . Recall that corresponds to the strongest weight and to no weight given (see (17)). Due to the ill-posedness of the inverse problem, the radii of admissible scattering data is problem specific, and the scattering transform will blow up in the presence of noise at a rate that is more rapid in some directions in the -plane than others. The value chosen for each example was chosen empirically to be as large as possible without exhibiting blow up in the initial reconstruction without a priori information. The blow-up was more rapid in the case of 1% noise, and so in those examples a non-uniform truncation of the scattering transform was used. In such cases a threshold of the scattering data was enforced by setting if or , where the value was chosen empirically to be the largest permissible value of the magnitude. Determining such a threshold is intuitive from a plot of the scattering data since the blowup rate is exponential.
The admittivity prior consisted of approximate knowledge of the organ boundaries (see Figure 1) with no assumption of pathology in the lungs. These average values for the prior are given in Tables 1 and 2.
|Admittivities||Background||Left Lung||Right Lung||Heart||Pneumothorax|
|Prior 0.1% Noise||0.79 + 0.40i||0.66 + 0.28i||0.64 + 0.29i||0.89 + 0.48i||N/A|
|Prior 1.0% Noise||0.79+0.39i||0.66 + 0.25i||0.64 + 0.28i||0.84 + 0.47i||N/A|
|Admittivities||Background||Left Lung||Right Lung||Heart||Pleural Effusion|
|Prior 0.1% Noise||0.80+0.40i||0.77 + 0.39i||0.62 + 0.29i||0.92 + 0.47i||N/A|
|Prior 1.0% Noise||0.79 + 0.40i||0.74 + 0.39i||0.59 + 0.31i||0.91 + 0.52i||N/A|
3.3.1. Example 1: Simulated Pneumothorax:
This test problem corresponds to phantom (a) in Figure 2. The preliminary reconstruction with no prior was computed for the 0.1% added noise case using a radius of , and for the 1% added noise case using a nonuniform truncation with a maximum radius of . Table 1 contains the values of the true admittivity in each region as well as the values assigned to the heart and lung prior for 0.1% and 1% noise. We emphasize that we assume only approximate knowledge of the boundaries of the heart and lungs (see the red dots in Figure 2 (a)), and no knowledge of the presence of a pneumothorax. Reconstructions for the 0.1% added noise case with truncation radii for the prior and weights are found in Figure 3. Reconstructions for the added relative noise case with truncation radii for the prior and weights are shown in Figure 4.
3.3.2. Example 2: Simulated Pleural Effusion
This test problem corresponds to phantom (b) in Figure 2. The preliminary reconstruction with no prior was computed for the 0.1% added noise case using a radius of and for the 1% added noise case using a nonuniform truncation with a maximum radius of . Table 2 presents the average values used in the prior for each noise level. Reconstructions for the 0.1% added noise case with truncation radii for the prior and weight are found in Figure 5. Reconstructions for the added relative noise case with truncation radii for the prior and weight are shown in Figure 6.
4. Discussion and Conclusions
The reader is advised to view the images on a computer screen if possible, since details in the color map are likely masked in printed versions.
In Figures 3, 4, 5, and 6, the upper left figure is the same as the preliminary reconstruction (ie, no prior), and it is evident that the spatial resolution of the organ boundaries improves with the introduction of the prior and as the influence of the prior increases. In the case of the pneumothorax, no pathology is evident in the preliminary reconstruction, but as the influence of the prior increases, even though the prior includes no assumption of pathology, the pneumothorax is clearly visible in the reconstructions. However, in both the conductivity and permittivity images, a lower conductivity and permittivity region becomes evident in the dorsal right lung as well, which is an artifact of the reconstruction, and it becomes stronger as the weighting of increases ( and .) This artifact is less pronounced in the permittivity images, and is arguably not present in the added noise case in the permittivity images.
The presence of the simulated pleural effusion, on the other hand, is clearly evident in the preliminary reconstructions for both conductivity and permittivity and for both noise levels. The presence of the prior improves the spatial resolution of the organs and the actual conductivity and permittivity values in the region of the effusion, but since the regularization results in reconstructed conductivity and permittivity functions that are smooth, there is a smooth transition from the healthy ventral portion of the left lung to the effusion, and so the boundary is far from as sharp as in the piecewise constant phantom. In practice, image segmentation is often used on reconstructed EIT images, which would likely improve the appearance of the reconstructed images. Alternatively, once a pathology is visible, an iterative method could then be invoked as in  which segments the prior in the region of a possible pathology potentially sharpening the pathology even more. Post-processing approaches are left for future work.
Figures 7 and 9 include side-by-side images of the (a) truth, (b) standard D-bar reconstruction with no prior, and (c) the reconstructed conductivity and permittivity images with the strongest weights on the prior considered here, all displayed on the same scale for ease of comparison. The true boundaries of the organs and pathologies are superimposed with black outlines. Figures 8 and 10 show the new reconstructions alone for and for the 0.1% pneumothorax example, and for the 0.1% pleural effusion example, to demonstrate the spatial improvement in the reconstructions.
It is clear from all of these images that this method is highly effective when organ boundaries are known with some confidence for improving the reconstructions without any bias of prior knowledge of the pathology. The influence of various qualities of prior knowledge of the boundary and organ boundaries is left for future work, as are results from experimental data. In practice, this high quality knowledge of organ boundaries corresponds to electrodes placed in the same plane as a CT scan slice. This can be accomplished with careful use of fiducial markers, and averaging of several slices to account for the fact that EIT electrodes are typically much higher than a CT scan slice, resulting in an image that corresponds to a much thicker slice.
S. J. Hamilton was supported by the 2015 Summer Faculty Fellowship from Marquette University. The project described was additionally supported by Award Number 1R21EB016869-01A1 from the National Institute Of Biomedical Imaging And Bioengineering. The content is solely the responsibility of the authors and does not necessarily represent the official view of the National Institute Of Biomedical Imaging And Bioengineering or the National Institutes of Health.
-  N. Avis and D. Barber, “Incorporating a priori information into the Sheffield filtered backprojection algorithm,” Physiological Measurement, vol. 16, no. 3A, pp. A111–A122, 1995.
-  U. Baysal and B. Eyüboglu, “Use of a priori information in estimating tissue resistivities - a simulation study,” Physics in Medicine and Biology, vol. 43, no. 12, pp. 3589–3606, 1998.
-  E. Camargo, “Development of an absolute electrical impedance imaging algorithm for clinical use,” Ph.D. dissertation, University of São Paulo, 2013.
-  H. Dehghani, D. Barber, and I. Basarab-Horwath, “Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography,” Physiological Measurement, vol. 20, no. 1, pp. 87–102, 1999.
-  D. C. Dobson and F. Santosa, “An image enhancement technique for electrical impedance tomography,” Inverse Problems, vol. 10, pp. 317–334, 1994.
-  D. Ferrario, B. Grychtol, A. Adler, J. Sola, S. Bohm, and M. Bodenstein, “Toward morphological thoracic EIT: Major signal sources correspond to respective organ locations in CT,” Biomedical Engineering, IEEE Transactions on, vol. 59, no. 11, pp. 3000–3008, 2012.
-  D. Flores-Tapia and S. Pistorius, “Electrical impedance tomography reconstruction using a monotonicity approach based on a priori knowledge,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, Aug 2010, pp. 4996–4999.
-  M. Soleimani, “Electrical impedance tomography imaging using a priori ultrasound data,” BioMedical Engineering OnLine, vol. 5, no. 8, 2006.
-  M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Transactions on Medical Imaging, vol. 17, pp. 285–293, 1998.
-  M. Alsaker and J. L. Mueller, “A D-bar algorithm with a priori information for 2-D Electrical Impedance Tomography,” arXiv: 1505.01196v1, 2015.
-  S. J. Hamilton, “A direct D-bar reconstruction algorithm for complex admittivies in for the 2-D EIT problem,” Ph.D. Thesis, Colorado State University, Fort Collins, CO, Summer 2012.
-  S. Hamilton, C. Herrera, J. L. Mueller, and A. VonHerrmann, “A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D,” Inverse Problems, vol. 28:(095005), 2012.
-  S. J. Hamilton and J. L. Mueller, “Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D.” IEEE transactions on medical imaging, vol. 32, no. 4, pp. 757–769, 2013.
-  M. DeAngelo and J. L. Mueller, “2d D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform,” Physiological Measurement, vol. 31, pp. 221–232, 2010.
-  M. Dodd and J. L. Mueller, “A real-time d-bar algorithm for 2-d electrical impedance tomography data,” Inverse Problems and Imaging, vol. 8, no. 4, pp. 1013–1031, 2014. [Online]. Available: http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10528
-  D. Isaacson, J. L. Mueller, J. C. Newell, and S. Siltanen, “Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography,” IEEE Transactions on Medical Imaging, vol. 23, pp. 821–828, 2004.
-  D. Isaacson, J. Mueller, J. Newell, and S. Siltanen, “Imaging cardiac activity by the D-bar method for electrical impedance tomography,” Physiological Measurement, vol. 27, pp. S43–S50, 2006.
-  J. Mueller and S. Siltanen, “Direct reconstructions of conductivities from boundary measurements,” SIAM Journal on Scientific Computing, vol. 24, no. 4, pp. 1232–1266, 2003.
-  K. Knudsen, M. Lassas, J. Mueller, and S. Siltanen, “D-bar method for electrical impedance tomography with discontinuous conductivities,” SIAM Journal on Applied Mathematics, vol. 67, no. 3, p. 893, 2007.
-  E. K. Murphy and J. L. Mueller, “Effect of domain-shape modeling and measurement errors on the 2-d D-bar method for electrical impedance tomography,” IEEE Transactions on Medical Imaging, vol. 28, no. 10, pp. 1576–1584, 2009.
-  S. Siltanen, J. Mueller, and D. Isaacson, “An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem,” Inverse Problems, vol. 16, pp. 681–699, 2000.
-  C. Herrera, M. Vallejo, J. Mueller, and R. Lima, “Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest,” Medical Imaging, IEEE Transactions on, vol. 34, no. 1, pp. 267–274, Jan 2015.
-  K. Cheng, D. Isaacson, J. Newell, and D. Gisser, “Electrode models for electric current computed tomography,” IEEE Transactions on Biomedical Engineering, vol. 36, no. 9, pp. 918–924, sep 1989.
-  E. Somersalo, M. Cheney, and D. Isaacson, “Existence and uniqueness for electrode models for electric current computed tomography,” SIAM Journal on Applied Mathematics, vol. 52, no. 4, pp. 1023–1040, 1992.
-  E. Francini, “Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map,” Inverse Problems, vol. 16, pp. 107–119, 2000.
-  L. D. Faddeev, “Increasing solutions of the Schrödinger equation,” Soviet Physics Doklady, vol. 10, pp. 1033–1035, 1966.
-  J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications. SIAM, 2012.
-  K. Knudsen, M. Lassas, J. Mueller, and S. Siltanen, “Regularized D-bar method for the inverse conductivity problem,” Inverse Problems and Imaging, vol. 3, no. 4, pp. 599–624, 2009.
-  S. J. Hamilton, A. Hauptmann, and S. Siltanen, “A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography,” Inverse Problems and Imaging, vol. 8, no. 4, pp. 1053–1072, 2014.
-  S. J. Hamiton, J. M. Reyes, S. Siltanen, and X. Zhang, “A hybrid segmentation and d-bar method for Electrical Impedance Tomography,” SIAM J. on Imaging Sciences (accepted), 2016.