June  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

2. 

Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53233, USA

3. 

Department of Computer Science, University College London, WC1E 6BT London, UK

* 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 and 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.  doi: 10.1137/15M1020137.  Google Scholar

[2]

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D. CalvettiP. J. HadwinJ. M. HuttunenD. IsaacsonJ. P. KaipioD. McGivneyE. Somersalo and 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.  doi: 10.3934/ipi.2015.9.749.  Google Scholar

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D. CalvettiP. J. HadwinJ. M. HuttunenJ. P. Kaipio and 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.  doi: 10.3934/ipi.2015.9.767.  Google Scholar

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E. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use, PhD thesis, University of São Paulo, 2013. Google Scholar

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F. J. Chung, Partial data for the neumann-to-dirichlet map, Journal of Fourier Analysis and Applications, 21 (2015), 628-665.  doi: 10.1007/s00041-014-9379-5.  Google Scholar

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M. Dodd and J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031.  doi: 10.3934/ipi.2014.8.1013.  Google Scholar

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D. FerrarioB. GrychtolA. AdlerJ. SolaS. Bohm and 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.  doi: 10.1109/TBME.2012.2209116.  Google Scholar

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M. HallajiA. Seppänen and M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001.  doi: 10.1088/0964-1726/23/8/085001.  Google Scholar

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[20]

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[21]

B. Harrach and M. Ullrich, Local uniqueness for an inverse boundary value problem with partial data, Proc. Amer. Math. Soc., 145 (2017), 1087-1095.  doi: 10.1090/proc/12991.  Google Scholar

[22]

A. HauptmannM. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009.  doi: 10.1088/1361-6420/33/2/025009.  Google Scholar

[23]

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[24]

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T. Hou and 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.  doi: 10.1177/1045389X08096052.  Google Scholar

[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.  doi: 10.1142/S0218202509003759.  Google Scholar

[28]

N. HyvönenP. Piiroinen and 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.  doi: 10.1137/120872164.  Google Scholar

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[31]

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C. KaragiannidisA. D. WaldmannC. Ferrando OrtoláM. Muñoz MartinezA. VidalA. SantosP. L. RókaM. Perez MárquezS. H. Bohm and F. Suarez-Spimann, Position-dependent distribution of ventilation measured with electrical impedance tomography, European Respiratory Journal, 46 (2015), PA2144.  doi: 10.1183/13993003.congress-2015.PA2144.  Google Scholar

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K. KarhunenA. SeppänenA. LehikoinenP. J. M. Monteiro and J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145.  doi: 10.1016/j.cemconres.2009.08.023.  Google Scholar

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[35]

K. KnudsenM. LassasJ. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar

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D. LiuV. KolehmainenS. Siltanen and 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.  doi: 10.1088/0266-5611/31/3/035012.  Google Scholar

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show all references

References:
[1]

M. Alsaker and 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.  doi: 10.1137/15M1020137.  Google Scholar

[2]

N. Avis and D. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiological Measurement, 16 (1995), A111-A122.  doi: 10.1088/0967-3334/16/3A/011.  Google Scholar

[3]

U. Baysal and B. Eyüboglu, Use of a priori information in estimating tissue resistivities -a simulation study, Physics in Medicine and Biology, 43 (1998), 3589-3606.  doi: 10.1088/0967-3334/25/3/013.  Google Scholar

[4]

D. CalvettiP. J. HadwinJ. M. HuttunenD. IsaacsonJ. P. KaipioD. McGivneyE. Somersalo and 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.  doi: 10.3934/ipi.2015.9.749.  Google Scholar

[5]

D. CalvettiP. J. HadwinJ. M. HuttunenJ. P. Kaipio and 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.  doi: 10.3934/ipi.2015.9.767.  Google Scholar

[6]

E. Camargo, Development of an Absolute Electrical Impedance Imaging Algorithm for Clinical Use, PhD thesis, University of São Paulo, 2013. Google Scholar

[7]

F. J. Chung, Partial data for the neumann-to-dirichlet map, Journal of Fourier Analysis and Applications, 21 (2015), 628-665.  doi: 10.1007/s00041-014-9379-5.  Google Scholar

[8]

G. CinnellaS. GrassoP. RaimondoD. D'AntiniL. MirabellaM. Rauseo and 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.  doi: 10.1097/ALN.0000000000000862.  Google Scholar

[9]

E. CostaC. ChavesS. GomesM. BeraldoM. VolpeM. TucciI. SchettinoS. BohmC. CarvalhoH. TanakaR. G. Lima and M. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Critical Care Medicine, 36 (2008), 1230-1238.  doi: 10.1097/CCM.0b013e31816a0380.  Google Scholar

[10]

W. Daily and A. Ramirez, Electrical imaging of engineered hydraulic barriers, Symposium on the Application of Geophysics to Engineering and Environmental Problems, (1999), 683-691.  doi: 10.4133/1.2922667.  Google Scholar

[11]

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, 31 (2010), 221-232.  doi: 10.1088/0967-3334/31/2/008.  Google Scholar

[12]

H. DehghaniD. Barber and I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiological Measurement, 20 (1999), 87-102.  doi: 10.1088/0967-3334/20/1/007.  Google Scholar

[13]

M. Dodd and J. L. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Problems and Imaging, 8 (2014), 1013-1031.  doi: 10.3934/ipi.2014.8.1013.  Google Scholar

[14]

D. FerrarioB. GrychtolA. AdlerJ. SolaS. Bohm and 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.  doi: 10.1109/TBME.2012.2209116.  Google Scholar

[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. doi: 10.1109/IEMBS.2010.5627204.  Google Scholar

[16]

C. GrantT. PhamJ. HoughT. RiedelC. Stocker and A. Schibler, Measurement of ventilation and cardiac related impedance changes with electrical impedance tomography, Critical Care, 15 (2011), R37.  doi: 10.1186/cc9985.  Google Scholar

[17]

G. HahnA. JustT. DudykevychI. FrerichsJ. HinzM. Quintel and G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT, Physiological Measurement, 27 (2006), S187-S198.  doi: 10.1088/0967-3334/27/5/S16.  Google Scholar

[18]

M. HallajiA. Seppänen and M. Pour-Ghaz, Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete, Smart Materials and Structures, 23 (2014), 085001.  doi: 10.1088/0964-1726/23/8/085001.  Google Scholar

[19]

S. J. HamiltonJ. L. Mueller and M. Alsaker, Incorporating a spatial prior into nonlinear d-bar EIT imaging for complex admittivities, IEEE Trans. Med. Imaging, 36 (2017), 457-466.  doi: 10.1109/TMI.2016.2613511.  Google Scholar

[20]

S. J. Hamilton and S. Siltanen, Nonlinear inversion from partial data EIT: Computational experiments, Contemporary Mathematics: Inverse Problems and Applications, 615 (2014), 105-129.  doi: 10.1090/conm/615/12267.  Google Scholar

[21]

B. Harrach and M. Ullrich, Local uniqueness for an inverse boundary value problem with partial data, Proc. Amer. Math. Soc., 145 (2017), 1087-1095.  doi: 10.1090/proc/12991.  Google Scholar

[22]

A. HauptmannM. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Problems, 33 (2017), 025009.  doi: 10.1088/1361-6420/33/2/025009.  Google Scholar

[23]

L. M. HeikkinenM. VauhkonenT. SavolainenK. Leinonen and J. P. Kaipio, Electrical process tomography with known internal structures and resistivities, Inverse Probl. Eng., 9 (2001), 431-454.  doi: 10.1080/174159701088027775.  Google Scholar

[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. doi: 10.3997/2214-4609.20144397.  Google Scholar

[25]

J. Hola and 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.  doi: 10.1016/S1644-9665(12)60133-2.  Google Scholar

[26]

T. Hou and 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.  doi: 10.1177/1045389X08096052.  Google Scholar

[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.  doi: 10.1142/S0218202509003759.  Google Scholar

[28]

N. HyvönenP. Piiroinen and 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.  doi: 10.1137/120872164.  Google Scholar

[29]

O. ImanuvilovG. Uhlmann and M. Yamamoto, The neumann-to-dirichlet map in two dimensions, Advances in Mathematics, 281 (2015), 578-593.  doi: 10.1016/j.aim.2015.03.026.  Google Scholar

[30]

D. IsaacsonJ. L. MuellerJ. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), 821-828.  doi: 10.1109/TMI.2004.827482.  Google Scholar

[31]

J. KaipioV. KolehmainenM. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729.  doi: 10.1088/0266-5611/15/3/306.  Google Scholar

[32]

C. KaragiannidisA. D. WaldmannC. Ferrando OrtoláM. Muñoz MartinezA. VidalA. SantosP. L. RókaM. Perez MárquezS. H. Bohm and F. Suarez-Spimann, Position-dependent distribution of ventilation measured with electrical impedance tomography, European Respiratory Journal, 46 (2015), PA2144.  doi: 10.1183/13993003.congress-2015.PA2144.  Google Scholar

[33]

K. KarhunenA. SeppänenA. LehikoinenP. J. M. Monteiro and J. P. Kaipio, Electrical resistance tomography imaging of concrete, Cement and Concrete Research, 40 (2010), 137-145.  doi: 10.1016/j.cemconres.2009.08.023.  Google Scholar

[34]

P. Kaup and F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, Journal of Nondestructive Evaluation, 14 (1995), 127-136.  doi: 10.1007/BF01183118.  Google Scholar

[35]

K. KnudsenM. LassasJ. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.  doi: 10.3934/ipi.2009.3.599.  Google Scholar

[36]

D. LiuV. KolehmainenS. SiltanenA.-m. Laukkanen and A. Seppänen, Estimation of conductivity changes in a region of interest with electrical impedance tomography, Inverse Problems and Imaging, 9 (2015), 211-229.  doi: 10.3934/ipi.2015.9.211.  Google Scholar

[37]

D. LiuV. KolehmainenS. Siltanen and 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.  doi: 10.1088/0266-5611/31/3/035012.  Google Scholar

[38]

J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science and Engineering, SIAM, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[39]

J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266.  doi: 10.1137/S1064827501394568.  Google Scholar

[40]

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, 28 (2009), 1576-1584.  doi: 10.1109/TMI.2009.2021611.  Google Scholar

[41]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.  doi: 10.2307/2118653.  Google Scholar

[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.  doi: 10.1007/BF01077418.  Google Scholar

[43]

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Figure 1.  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)
Figure 2.  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
Figure 3.  The A Priori D-bar Method with Partial Data
Figure 4.  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
Figure 5.  Blind priors used for the thoracic (top) and industrial (bottom) imaging examples. Take particular note that the priors do not assume any pathology/defect
Figure 6.  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 ${\mu ^{{\rm{int}}}}\sim 1$
Figure 7.  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)
Figure 8.  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 $R=4$, $\alpha=0$
Figure 9.  Left: Original prior. Right: Updated Pneumothorax prior. The left lung in the updated prior was segmented into two regions
Figure 10.  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.
Figure 11.  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 $R=4$, $\alpha=0$
Figure 12.  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
Figure 13.  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 $R=4$, $\alpha=0$
Figure 14.  Left: Original prior. Right: Updated Pleural Effusion prior with the left lung segmented into two regions
Figure 15.  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
Figure 16.  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 $R=4$, $\alpha=0$
Figure 17.  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
Figure 18.  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 $\alpha=0$
Figure 19.  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 $\alpha=0$
Figure 20.  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
Figure 21.  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 $R_2$
Table 1.  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
Table 2.  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
Table 3.  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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