# American Institue of Mathematical Sciences

2017, 11(3): 427-454. doi: 10.3934/ipi.2017020

## A direct D-bar method for partial boundary data electrical impedance tomography with a priori information

 1 Gonzaga University, Mathematics Department, 502 E. Boone Ave. MSC 2615, Spokane, WA 99258-0072, USA, United States 2 Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53233, USA, United States 3 Department of Computer Science, University College London, WC1E 6BT London, UK, United Kingdom

* Corresponding author

Received  September 2016 Revised  October 2016 Published  April 2017

Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios.
Citation: Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020
##### References:
 [1] M. Alsaker, J. L. Mueller, A D-bar algorithm with a priori information for 2-D Electrical Impedance Tomography, SIAM J. on Imaging Sciences, 9 (2016), 1619-1654. [2] N. Avis, D. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiological Measurement, 16 (1995), A111-A122. [3] U. Baysal, B. Eyüboglu, Use of a priori information in estimating tissue resistivities -a simulation study, Physics in Medicine and Biology, 43 (1998), 3589-3606. [4] D. Calvetti, P. J. Hadwin, J. M. Huttunen, D. Isaacson, J. P. Kaipio, D. McGivney, E. Somersalo, J. Volzer, Artificial boundary conditions and domain truncation in electrical impedance tomography. part Ⅰ: Theory and preliminary results, Inverse Problems & Imaging, 9 (2015), 749-766. [5] D. Calvetti, P. J. Hadwin, J. M. Huttunen, J. P. Kaipio, E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. part Ⅱ: Stochastic extension of the boundary map., Inverse Problems & Imaging, 9 (2015), 767-789. [6] E. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use PhD thesis, University of São Paulo, 2013. [7] F. J. Chung, Partial data for the neumann-to-dirichlet map, Journal of Fourier Analysis and Applications, 21 (2015), 628-665. [8] G. Cinnella, S. Grasso, P. Raimondo, D. D'Antini, L. Mirabella, M. Rauseo, M. Dambrosio, Physiological effects of the open lung approach in patients with early, mild, diffuse acute respiratory distress syndromean electrical impedance tomography study, The Journal of the American Society of Anesthesiologists, 123 (2015), 1113-1121. [9] E. Costa, C. Chaves, S. Gomes, M. Beraldo, M. Volpe, M. Tucci, I. Schettino, S. Bohm, C. Carvalho, H. Tanaka, R. G. Lima, M. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Critical Care Medicine, 36 (2008), 1230-1238. [10] W. Daily, A. Ramirez, Electrical imaging of engineered hydraulic barriers, Symposium on the Application of Geophysics to Engineering and Environmental Problems, (1999), 683-691. [11] M. DeAngelo, J. L. Mueller, 2d D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232. [12] H. Dehghani, D. Barber, I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiological Measurement, 20 (1999), 87-102. [13] M. Dodd, J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031. [14] D. Ferrario, B. Grychtol, A. Adler, J. Sola, S. Bohm, M. Bodenstein, Toward morphological thoracic EIT: Major signal sources correspond to respective organ locations in CT, Biomedical Engineering, IEEE Transactions on, 59 (2012), 3000-3008. [15] 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, 2010,4996–4999. [16] C. Grant, T. Pham, J. Hough, T. Riedel, C. Stocker, A. Schibler, Measurement of ventilation and cardiac related impedance changes with electrical impedance tomography, Critical Care, 15 (2011), R37. [17] G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. Quintel, G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT, Physiological Measurement, 27 (2006), S187-S198. [18] M. Hallaji, A. Seppänen, M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001. [19] S. J. Hamilton, J. L. Mueller, M. Alsaker, Incorporating a spatial prior into nonlinear d-bar EIT imaging for complex admittivities, IEEE Trans. Med. Imaging, 36 (2017), 457-466. [20] S. J. Hamilton, S. Siltanen, Nonlinear inversion from partial data EIT: Computational experiments, Contemporary Mathematics: Inverse Problems and Applications, 615 (2014), 105-129. [21] B. Harrach, M. Ullrich, Local uniqueness for an inverse boundary value problem with partial data, Proc. Amer. Math. Soc., 145 (2017), 1087-1095. [22] A. Hauptmann, M. Santacesaria, S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009. [23] L. M. Heikkinen, M. Vauhkonen, T. Savolainen, K. Leinonen, J. P. Kaipio, Electrical process tomography with known internal structures and resistivities, Inverse Probl. Eng., 9 (2001), 431-454. [24] T. Hermans, D. Caterina, R. Martin, A. Kemna, T. Robert and F. Nguyen, How to incorporate prior information in geophysical inverse problems-deterministic and geostatistical approaches in Near Surface 2011-17th EAGE European Meeting of Environmental and Engineering Geophysics, 2011. [25] J. Hola, K. Schabowicz, State-of-the-art non-destructive methods for diagnostic testing of building structures — anticipated development trends, Archives of Civil and Mechanical Engineering, 10 (2010), 5-18. [26] T. Hou, J. Lynch, Electrical impedance tomographic methods for sensing strain fields and crack damage in cementitious structures, Journal of Intelligent Material Systems and Structures, 20 (2009), 1363-1379. [27] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Mathematical Models and Methods in Applied Sciences, 19 (2009), 1185-1202. [28] N. Hyvönen, P. Piiroinen, O. Seiskari, Point measurements for a neumann-to-dirichlet map and the calderón problem in the plane, SIAM Journal on Mathematical Analysis, 44 (2012), 3526-3536. [29] O. Imanuvilov, G. Uhlmann, M. Yamamoto, The neumann-to-dirichlet map in two dimensions, Advances in Mathematics, 281 (2015), 578-593. [30] D. Isaacson, J. L. Mueller, J. C. Newell, S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), 821-828. [31] J. Kaipio, V. Kolehmainen, M. Vauhkonen, E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. [32] C. Karagiannidis, A. D. Waldmann, C. Ferrando Ortolá, M. Muñoz Martinez, A. Vidal, A. Santos, P. L. Róka, M. Perez Márquez, S. H. Bohm, F. Suarez-Spimann, Position-dependent distribution of ventilation measured with electrical impedance tomography, European Respiratory Journal, 46 (2015), PA2144. [33] K. Karhunen, A. Seppänen, A. Lehikoinen, P. J. M. Monteiro, J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145. [34] P. Kaup, F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, Journal of Nondestructive Evaluation, 14 (1995), 127-136. [35] K. Knudsen, M. Lassas, J. Mueller, S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. [36] D. Liu, V. Kolehmainen, S. Siltanen, A.-m. Laukkanen, A. Seppänen, Estimation of conductivity changes in a region of interest with electrical impedance tomography, Inverse Problems and Imaging, 9 (2015), 211-229. [37] D. Liu, V. Kolehmainen, S. Siltanen, A. Seppänen, A nonlinear approach to difference imaging in EIT; assessment of the robustness in the presence of modelling errors, Inverse Problems, 31 (2015), 035012, 25pp. [38] J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications vol. 10 of Computational Science and Engineering, SIAM, 2012. [39] J. Mueller, S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266. [40] E. K. Murphy, 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, 28 (2009), 1576-1584. [41] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. [42] R. Novikov, A multidimensional inverse spectral problem for the equation $-δ ψ+(v(x)-eu(x))ψ = 0$, Functional Analysis and Its Applications, 22 (1988), 263-272. [43] A. Pesenti, G. Musch, D. Lichtenstein, F. Mojoli, M. B. P. Amato, G. Cinnella, L. Gattinoni, M. Quintel, Imaging in acute respiratory distress syndrome, Intensive Care Medicine, 42 (2016), 686-698. [44] H. Reinius, J. B. Borges, F. Fredén, L. Jideus, E. D. L. B. Camargo, M. B. P. Amato, G. L. A. Hedenstierna, F. Lennmyr, Real-time ventilation and perfusion distributions by electrical impedance tomography during one-lung ventilation with capnothorax, Acta Anaesthesiologica Scandinavica, 59 (2015), 354-368. [45] A. Schlibler, T. Pham, A. Moray, C. Stocker, Ventilation and cardiac related impedance changes in children undergoing corrective open heart surgery, Physiological Measurement, 34 (2013), 1319-1327. [46] A. Seppänen, K. Karhunen, A. Lehikoinen, J. Kaipio and P. Monteiro, Electrical resistance tomography imaging of concrete, in Concrete Repair, Rehabilitation and Retrofitting Ⅱ: 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting, 2009,571–577. [47] S. Siltanen, J. Mueller, D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699. [48] M. Soleimani, Electrical impedance tomography imaging using a priori ultrasound data BioMedical Engineering OnLine 5. [49] M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-293.

show all references

##### References:
 [1] M. Alsaker, J. L. Mueller, A D-bar algorithm with a priori information for 2-D Electrical Impedance Tomography, SIAM J. on Imaging Sciences, 9 (2016), 1619-1654. [2] N. Avis, D. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiological Measurement, 16 (1995), A111-A122. [3] U. Baysal, B. Eyüboglu, Use of a priori information in estimating tissue resistivities -a simulation study, Physics in Medicine and Biology, 43 (1998), 3589-3606. [4] D. Calvetti, P. J. Hadwin, J. M. Huttunen, D. Isaacson, J. P. Kaipio, D. McGivney, E. Somersalo, J. Volzer, Artificial boundary conditions and domain truncation in electrical impedance tomography. part Ⅰ: Theory and preliminary results, Inverse Problems & Imaging, 9 (2015), 749-766. [5] D. Calvetti, P. J. Hadwin, J. M. Huttunen, J. P. Kaipio, E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. part Ⅱ: Stochastic extension of the boundary map., Inverse Problems & Imaging, 9 (2015), 767-789. [6] E. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use PhD thesis, University of São Paulo, 2013. [7] F. J. Chung, Partial data for the neumann-to-dirichlet map, Journal of Fourier Analysis and Applications, 21 (2015), 628-665. [8] G. Cinnella, S. Grasso, P. Raimondo, D. D'Antini, L. Mirabella, M. Rauseo, M. Dambrosio, Physiological effects of the open lung approach in patients with early, mild, diffuse acute respiratory distress syndromean electrical impedance tomography study, The Journal of the American Society of Anesthesiologists, 123 (2015), 1113-1121. [9] E. Costa, C. Chaves, S. Gomes, M. Beraldo, M. Volpe, M. Tucci, I. Schettino, S. Bohm, C. Carvalho, H. Tanaka, R. G. Lima, M. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Critical Care Medicine, 36 (2008), 1230-1238. [10] W. Daily, A. Ramirez, Electrical imaging of engineered hydraulic barriers, Symposium on the Application of Geophysics to Engineering and Environmental Problems, (1999), 683-691. [11] M. DeAngelo, J. L. Mueller, 2d D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiological Measurement, 31 (2010), 221-232. [12] H. Dehghani, D. Barber, I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiological Measurement, 20 (1999), 87-102. [13] M. Dodd, J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031. [14] D. Ferrario, B. Grychtol, A. Adler, J. Sola, S. Bohm, M. Bodenstein, Toward morphological thoracic EIT: Major signal sources correspond to respective organ locations in CT, Biomedical Engineering, IEEE Transactions on, 59 (2012), 3000-3008. [15] 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, 2010,4996–4999. [16] C. Grant, T. Pham, J. Hough, T. Riedel, C. Stocker, A. Schibler, Measurement of ventilation and cardiac related impedance changes with electrical impedance tomography, Critical Care, 15 (2011), R37. [17] G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. Quintel, G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT, Physiological Measurement, 27 (2006), S187-S198. [18] M. Hallaji, A. Seppänen, M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001. [19] S. J. Hamilton, J. L. Mueller, M. Alsaker, Incorporating a spatial prior into nonlinear d-bar EIT imaging for complex admittivities, IEEE Trans. Med. Imaging, 36 (2017), 457-466. [20] S. J. Hamilton, S. Siltanen, Nonlinear inversion from partial data EIT: Computational experiments, Contemporary Mathematics: Inverse Problems and Applications, 615 (2014), 105-129. [21] B. Harrach, M. Ullrich, Local uniqueness for an inverse boundary value problem with partial data, Proc. Amer. Math. Soc., 145 (2017), 1087-1095. [22] A. Hauptmann, M. Santacesaria, S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009. [23] L. M. Heikkinen, M. Vauhkonen, T. Savolainen, K. Leinonen, J. P. Kaipio, Electrical process tomography with known internal structures and resistivities, Inverse Probl. Eng., 9 (2001), 431-454. [24] T. Hermans, D. Caterina, R. Martin, A. Kemna, T. Robert and F. Nguyen, How to incorporate prior information in geophysical inverse problems-deterministic and geostatistical approaches in Near Surface 2011-17th EAGE European Meeting of Environmental and Engineering Geophysics, 2011. [25] J. Hola, K. Schabowicz, State-of-the-art non-destructive methods for diagnostic testing of building structures — anticipated development trends, Archives of Civil and Mechanical Engineering, 10 (2010), 5-18. [26] T. Hou, J. Lynch, Electrical impedance tomographic methods for sensing strain fields and crack damage in cementitious structures, Journal of Intelligent Material Systems and Structures, 20 (2009), 1363-1379. [27] N. Hyvönen, Approximating idealized boundary data of electric impedance tomography by electrode measurements, Mathematical Models and Methods in Applied Sciences, 19 (2009), 1185-1202. [28] N. Hyvönen, P. Piiroinen, O. Seiskari, Point measurements for a neumann-to-dirichlet map and the calderón problem in the plane, SIAM Journal on Mathematical Analysis, 44 (2012), 3526-3536. [29] O. Imanuvilov, G. Uhlmann, M. Yamamoto, The neumann-to-dirichlet map in two dimensions, Advances in Mathematics, 281 (2015), 578-593. [30] D. Isaacson, J. L. Mueller, J. C. Newell, S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), 821-828. [31] J. Kaipio, V. Kolehmainen, M. Vauhkonen, E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. [32] C. Karagiannidis, A. D. Waldmann, C. Ferrando Ortolá, M. Muñoz Martinez, A. Vidal, A. Santos, P. L. Róka, M. Perez Márquez, S. H. Bohm, F. Suarez-Spimann, Position-dependent distribution of ventilation measured with electrical impedance tomography, European Respiratory Journal, 46 (2015), PA2144. [33] K. Karhunen, A. Seppänen, A. Lehikoinen, P. J. M. Monteiro, J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145. [34] P. Kaup, F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, Journal of Nondestructive Evaluation, 14 (1995), 127-136. [35] K. Knudsen, M. Lassas, J. Mueller, S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624. [36] D. Liu, V. Kolehmainen, S. Siltanen, A.-m. Laukkanen, A. Seppänen, Estimation of conductivity changes in a region of interest with electrical impedance tomography, Inverse Problems and Imaging, 9 (2015), 211-229. [37] D. Liu, V. Kolehmainen, S. Siltanen, A. Seppänen, A nonlinear approach to difference imaging in EIT; assessment of the robustness in the presence of modelling errors, Inverse Problems, 31 (2015), 035012, 25pp. [38] J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications vol. 10 of Computational Science and Engineering, SIAM, 2012. [39] J. Mueller, S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266. [40] E. K. Murphy, 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, 28 (2009), 1576-1584. [41] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. [42] R. Novikov, A multidimensional inverse spectral problem for the equation $-δ ψ+(v(x)-eu(x))ψ = 0$, Functional Analysis and Its Applications, 22 (1988), 263-272. [43] A. Pesenti, G. Musch, D. Lichtenstein, F. Mojoli, M. B. P. Amato, G. Cinnella, L. Gattinoni, M. Quintel, Imaging in acute respiratory distress syndrome, Intensive Care Medicine, 42 (2016), 686-698. [44] H. Reinius, J. B. Borges, F. Fredén, L. Jideus, E. D. L. B. Camargo, M. B. P. Amato, G. L. A. Hedenstierna, F. Lennmyr, Real-time ventilation and perfusion distributions by electrical impedance tomography during one-lung ventilation with capnothorax, Acta Anaesthesiologica Scandinavica, 59 (2015), 354-368. [45] A. Schlibler, T. Pham, A. Moray, C. Stocker, Ventilation and cardiac related impedance changes in children undergoing corrective open heart surgery, Physiological Measurement, 34 (2013), 1319-1327. [46] A. Seppänen, K. Karhunen, A. Lehikoinen, J. Kaipio and P. Monteiro, Electrical resistance tomography imaging of concrete, in Concrete Repair, Rehabilitation and Retrofitting Ⅱ: 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting, 2009,571–577. [47] S. Siltanen, J. Mueller, D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699. [48] M. Soleimani, Electrical impedance tomography imaging using a priori ultrasound data BioMedical Engineering OnLine 5. [49] M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-293.
Example simulating a patient with a pneumothorax in the left lung. The simulated noisy measurement is collected from 75% ventral data. The first image displays the true conductivity with the position of electrodes indicated. Using a partial data D-bar approach alone results in a reconstruction with low spatial resolution, where the pathology can be hardly seen (second). Incorporating a priori data corresponding to a healthy patient directly into the reconstruction method significantly improves the spatial resolution (third). Refining the prior improves the reconstruction further, allowing even sharper visualization of the pathology (fourth)
Illustration of mappings involved in the measurement modeling. Top row: Neumann data with the basis function $\varphi(\theta)=\cos(\theta)/\sqrt{\pi}$ on the left and the nonorthogonal projection $J\varphi$ on the right. Bottom row: Dirichlet data where $g=u|_{\partial\Omega}$ on the left is the solution of the partial differential equation (1) and on the right the orthogonal projection to the extended electrodes
The A Priori D-bar Method with Partial Data
Phantoms used in numerical examples with the corresponding boundaries of the priors outlined by white dots. Note that for each example, the prior does not assume a pathology/defect. Left: A simulated pneumothorax occurring near the heart in the left lung. Middle: A simulated pleural effusion occurring away from the heart in the left lung. Right: An enclosed diamond with an ovular defect
Blind priors used for the thoracic (top) and industrial (bottom) imaging examples. Take particular note that the priors do not assume any pathology/defect
The real part of the ${\mu ^{{\rm{int}}}}$ data (shown in the $z$ plane for $z\in\mathcal{D}$) corresponding to the blind thoracic prior given in Figure 5(top) computed from extended radii $R_2=4.0$, 6.5, and 9.0 in the $k$ plane. Note that as the radius increases, the integral term approaches its asymptotic behavior of ${\mu ^{{\rm{int}}}}\sim 1$
Scattering data corresponding to the pneumothorax example using the blind prior given in Figure 5(top). The original radius is $R=4$ and extended radius $R_2=9$. All scattering data is plotted on the same scale (real and imaginary, respectively)
Pneumothorax example for 62.5% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$, using the blind thoracic prior. The maximum value is 2.25, occurring of $R=4$, $\alpha=0$
Left: Original prior. Right: Updated Pneumothorax prior. The left lung in the updated prior was segmented into two regions
Pneumothorax example with 75% Ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 8. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4,6.5$ and various $\alpha$ using the SEG AVG or SEG MIN segmented thoracic priors. The maximum value is 2.70 and occurs in the $R_2=4$, $\alpha=0$ recon using the SEG MIN prior.
Pleural effusion example for 75% ventral data. TOP: The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$. BOTTOM: The recovered conductivity ${\sigma _{{R_2},\alpha }}$ using the blind thoracic prior. The maximum value is 2.90, occurring of $R=4$, $\alpha=0$
Pneumothorax Example. Results for $R_2=9.0$ and $\alpha=0.67$. The maximum is 2.71 and occurs in the 100% boundary data, BLIND prior reconstruction
Pleural effusion example for 62.5% ventral data. The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown at the top. Below, the recovered conductivity ${\sigma _{{R_2},\alpha }}$ is shown using the blind thoracic prior. The maximum value is 2.74, occurring of $R=4$, $\alpha=0$
Left: Original prior. Right: Updated Pleural Effusion prior with the left lung segmented into two regions
Pleural effusion example for 75% ventral data and segmented prior. The corresponding partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown in Figure 13. Here we display the recovered conductivity ${\sigma _{{R_2},\alpha }}$ for $R_2=4$, 6.5 and various $\alpha$ using the SEG AVG or SEG MAX segmented thoracic prior. The maximum value is 2.83 and occurs in the $R_2=4$, $\alpha=0$ reconstruction using the SEG MAX prior
Pleural effusion example for 62.5% ventral data. The partial data ND D-bar reconstruction ${\sigma ^{{\rm{ND}}}}$ is shown at the top. Below, the recovered conductivity ${\sigma _{{R_2},\alpha }}$ is shown using the blind thoracic prior. The maximum value is 2.74, occurring of $R=4$, $\alpha=0$
Pleural Effusion Example. Results for $R_2=6.5$ and $\alpha=0.67$. The maximum is 2.65 and occurs in the 100% boundary data, BLIND prior reconstruction
Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with scattering radius $R_2=4$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.12) occurs for the 50% data reconstruction with strongest weight $\alpha=0$
Industrial Example: From top to bottom, conductivity reconstructions ${\sigma _{{R_2},\alpha }}$ for 100%, 75%, 62.5%, and 50% boundary data are presented with extended scattering radius $R_2=6.5$ and various weights $\alpha$. The first column displays the ${\sigma ^{{\rm{ND}}}}$ reconstructions that do not include any a priori information. The maximum value (3.13) occurs for the 50% data reconstruction with strongest weight $\alpha=0$
Relative $\ell_2$-error of reconstructions from 75% ventral data of the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$. Recall that $\alpha=0$ corresponds to the heaviest weighting of the ${\mu ^{{\rm{int}}}}$ term, while $\alpha=1$ to the weakest expression of the prior. Errors from ${\sigma ^{{\rm{ND}}}}$ are compared to the new reconstructions ${\sigma _{{R_2},\alpha }}$ for the blind and segmented priors
Relative $\ell_2$-error in the lung region within the boundary of the pathology, for 75% ventral data for the pneumothorax example. The horizontal axis represents $\alpha$-values for increasing regularization radii $R_2$
Conductivity values of thoracic phantoms and assigned blind prior in S/m
 Heart Lungs Pathology Aorta Spine Background Pneumothorax 2.0 0.5 0.15 2.0 0.25 1 Pleural Effusion 2.0 0.5 1.8 2.0 0.25 1 Prior 2.05 0.45 - 2.05 0.23 1
 Heart Lungs Pathology Aorta Spine Background Pneumothorax 2.0 0.5 0.15 2.0 0.25 1 Pleural Effusion 2.0 0.5 1.8 2.0 0.25 1 Prior 2.05 0.45 - 2.05 0.23 1
Conductivity values of industrial phantom and assigned blind prior in S/m
 Diamond Inclusion Background Industrial 2.0 1.4 1 Prior 2.05 - 1
 Diamond Inclusion Background Industrial 2.0 1.4 1 Prior 2.05 - 1
Relative $\ell_2$-errors (%) for the conductivity reconstructions from §4, for the extended regularization radii $R_2=4$ and $6.5$
 D-BAR $\mathbf{R_2=4}$ $\mathbf{R_2=6.5}$ RECON $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$ $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$ PNEUMOTHORAX Blind Prior: 75% 35.13 29.65 26.74 24.86 24.44 26.82 25.36 24.20 23.39 Seg Avg Prior: 75% 35.13 29.14 26.22 24.65 24.95 25.75 24.16 22.92 22.11 Seg Min Prior: 75% 35.13 28.84 25.96 24.75 25.74 25.07 23.44 22.23 21.55 Blind Prior: 62.5% 38.95 32.71 30.02 28.06 27.12 30.12 28.74 27.56 26.63 Seg Avg Prior: 62.5% 38.95 32.33 29.62 27.83 27.30 29.43 27.99 26.78 25.84 Seg Min Prior: 62.5% 38.95 31.99 29.27 27.67 27.61 28.66 27.13 25.88 24.97 Pleural Effusion Blind Prior: 75% 27.40 24.82 24.43 25.94 29.23 25.44 25.54 26.13 27.20 Seg Avg Prior: 75% 27.40 24.24 22.27 21.88 23.33 22.40 21.47 20.96 20.90 Seg Max Prior: 75% 27.40 24.14 21.95 21.39 22.81 21.98 20.94 20.34 20.22 Blind Prior: 62.5% 32.56 29.80 29.22 30.00 32.21 29.34 29.10 29.20 29.67 Seg Avg Prior: 62.5% 32.56 29.18 27.66 27.22 28.08 27.53 26.80 26.35 26.21 Seg Max Prior: 62.5% 32.56 28.87 27.01 26.24 26.80 26.77 25.85 25.20 24.87 Industrial phantom Blind Prior: 100% 18.43 18.43 16.07 14.17 12.99 15.31 14.17 13.28 12.68 Blind Prior: 75% 18.46 17.91 16.10 14.99 14.80 15.23 14.42 13.93 13.80 Blind Prior: 62.5% 20.72 19.96 18.55 17.90 18.15 18.03 17.49 17.27 17.37 Blind Prior: 50% 22.14 21.24 20.25 20.04 20.70 19.68 19.34 19.31 19.60
 D-BAR $\mathbf{R_2=4}$ $\mathbf{R_2=6.5}$ RECON $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$ $\alpha=1$ $\alpha=\frac{2}{3}$ $\alpha=\frac{1}{3}$ $\alpha=0$ PNEUMOTHORAX Blind Prior: 75% 35.13 29.65 26.74 24.86 24.44 26.82 25.36 24.20 23.39 Seg Avg Prior: 75% 35.13 29.14 26.22 24.65 24.95 25.75 24.16 22.92 22.11 Seg Min Prior: 75% 35.13 28.84 25.96 24.75 25.74 25.07 23.44 22.23 21.55 Blind Prior: 62.5% 38.95 32.71 30.02 28.06 27.12 30.12 28.74 27.56 26.63 Seg Avg Prior: 62.5% 38.95 32.33 29.62 27.83 27.30 29.43 27.99 26.78 25.84 Seg Min Prior: 62.5% 38.95 31.99 29.27 27.67 27.61 28.66 27.13 25.88 24.97 Pleural Effusion Blind Prior: 75% 27.40 24.82 24.43 25.94 29.23 25.44 25.54 26.13 27.20 Seg Avg Prior: 75% 27.40 24.24 22.27 21.88 23.33 22.40 21.47 20.96 20.90 Seg Max Prior: 75% 27.40 24.14 21.95 21.39 22.81 21.98 20.94 20.34 20.22 Blind Prior: 62.5% 32.56 29.80 29.22 30.00 32.21 29.34 29.10 29.20 29.67 Seg Avg Prior: 62.5% 32.56 29.18 27.66 27.22 28.08 27.53 26.80 26.35 26.21 Seg Max Prior: 62.5% 32.56 28.87 27.01 26.24 26.80 26.77 25.85 25.20 24.87 Industrial phantom Blind Prior: 100% 18.43 18.43 16.07 14.17 12.99 15.31 14.17 13.28 12.68 Blind Prior: 75% 18.46 17.91 16.10 14.99 14.80 15.23 14.42 13.93 13.80 Blind Prior: 62.5% 20.72 19.96 18.55 17.90 18.15 18.03 17.49 17.27 17.37 Blind Prior: 50% 22.14 21.24 20.25 20.04 20.70 19.68 19.34 19.31 19.60
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