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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 |
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.
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. |
[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. |
[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. |
[4] |
D. Calvetti, P. J. Hadwin, J. M. Huttunen, D. Isaacson, J. P. Kaipio, D. McGivney, E. 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. |
[5] |
D. Calvetti, P. J. Hadwin, J. M. Huttunen, J. 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. |
[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.
doi: 10.1007/s00041-014-9379-5. |
[8] |
G. Cinnella, S. Grasso, P. Raimondo, D. D'Antini, L. Mirabella, M. 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. |
[9] |
E. Costa, C. Chaves, S. Gomes, M. Beraldo, M. Volpe, M. Tucci, I. Schettino, S. Bohm, C. Carvalho, H. Tanaka, R. 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. |
[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. |
[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. |
[12] |
H. Dehghani, D. 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. |
[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. |
[14] |
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, 59 (2012), 3000-3008.
doi: 10.1109/TBME.2012.2209116. |
[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. |
[16] |
C. Grant, T. Pham, J. Hough, T. Riedel, C. 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. |
[17] |
G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. 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. |
[18] |
M. Hallaji, A. 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. |
[19] |
S. J. Hamilton, J. 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. |
[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. |
[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. |
[22] |
A. Hauptmann, M. 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. |
[23] |
L. M. Heikkinen, M. Vauhkonen, T. Savolainen, K. 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. |
[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. |
[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. |
[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. |
[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. |
[28] |
N. Hyvönen, P. 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. |
[29] |
O. Imanuvilov, G. 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. |
[30] |
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, 23 (2004), 821-828.
doi: 10.1109/TMI.2004.827482. |
[31] |
J. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo,
Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729.
doi: 10.1088/0266-5611/15/3/306. |
[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 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. |
[33] |
K. Karhunen, A. Seppänen, A. Lehikoinen, P. 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. |
[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. |
[35] |
K. Knudsen, M. Lassas, J. 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. |
[36] |
D. Liu, V. Kolehmainen, S. Siltanen, A.-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. |
[37] |
D. Liu, V. Kolehmainen, S. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
A. Pesenti, G. Musch, D. Lichtenstein, F. Mojoli, M. B. P. Amato, G. Cinnella, L. Gattinoni and M. Quintel,
Imaging in acute respiratory distress syndrome, Intensive Care Medicine, 42 (2016), 686-698.
doi: 10.1007/s00134-016-4328-1. |
[44] |
H. Reinius, J. B. Borges, F. Fredén, L. Jideus, E. D. L. B. Camargo, M. B. P. Amato, G. L. A. Hedenstierna and 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.
doi: 10.1111/aas.12455. |
[45] |
A. Schlibler, T. Pham, A. Moray and C. Stocker,
Ventilation and cardiac related impedance changes in children undergoing corrective open heart surgery, Physiological Measurement, 34 (2013), 1319-1327.
doi: 10.1088/0967-3334/34/10/1319. |
[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 and D. Isaacson,
An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.
doi: 10.1088/0266-5611/16/3/310. |
[48] |
M. Soleimani, Electrical impedance tomography imaging using a priori ultrasound data BioMedical Engineering OnLine, 5.
doi: 10.1186/1475-925X-5-8. |
[49] |
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, 17 (1998), 285-293.
doi: 10.1109/42.700740. |
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. |
[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. |
[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. |
[4] |
D. Calvetti, P. J. Hadwin, J. M. Huttunen, D. Isaacson, J. P. Kaipio, D. McGivney, E. 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. |
[5] |
D. Calvetti, P. J. Hadwin, J. M. Huttunen, J. 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. |
[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.
doi: 10.1007/s00041-014-9379-5. |
[8] |
G. Cinnella, S. Grasso, P. Raimondo, D. D'Antini, L. Mirabella, M. 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. |
[9] |
E. Costa, C. Chaves, S. Gomes, M. Beraldo, M. Volpe, M. Tucci, I. Schettino, S. Bohm, C. Carvalho, H. Tanaka, R. 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. |
[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. |
[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. |
[12] |
H. Dehghani, D. 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. |
[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. |
[14] |
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, 59 (2012), 3000-3008.
doi: 10.1109/TBME.2012.2209116. |
[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. |
[16] |
C. Grant, T. Pham, J. Hough, T. Riedel, C. 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. |
[17] |
G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. 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. |
[18] |
M. Hallaji, A. 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. |
[19] |
S. J. Hamilton, J. 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. |
[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. |
[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. |
[22] |
A. Hauptmann, M. 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. |
[23] |
L. M. Heikkinen, M. Vauhkonen, T. Savolainen, K. 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. |
[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. |
[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. |
[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. |
[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. |
[28] |
N. Hyvönen, P. 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. |
[29] |
O. Imanuvilov, G. 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. |
[30] |
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, 23 (2004), 821-828.
doi: 10.1109/TMI.2004.827482. |
[31] |
J. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo,
Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729.
doi: 10.1088/0266-5611/15/3/306. |
[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 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. |
[33] |
K. Karhunen, A. Seppänen, A. Lehikoinen, P. 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. |
[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. |
[35] |
K. Knudsen, M. Lassas, J. 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. |
[36] |
D. Liu, V. Kolehmainen, S. Siltanen, A.-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. |
[37] |
D. Liu, V. Kolehmainen, S. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
A. Pesenti, G. Musch, D. Lichtenstein, F. Mojoli, M. B. P. Amato, G. Cinnella, L. Gattinoni and M. Quintel,
Imaging in acute respiratory distress syndrome, Intensive Care Medicine, 42 (2016), 686-698.
doi: 10.1007/s00134-016-4328-1. |
[44] |
H. Reinius, J. B. Borges, F. Fredén, L. Jideus, E. D. L. B. Camargo, M. B. P. Amato, G. L. A. Hedenstierna and F. Lennmyr,
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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 |
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 |
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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