
-
Previous Article
Nonlinear fractional diffusion model for deblurring images with textures
- IPI Home
- This Issue
- Next Article
Electrical impedance tomography with multiplicative regularization
1. | State Key Laboratory on Microwave and Digital Communications, Beijing National Research Center for Information Science and Technology (BNRist), Department of Electronic Engineering, Tsinghua University, Beijing 100084, China |
2. | Schlumberger, Houston, TX 77056, USA |
It is known that EIT inversion is an ill-posed problem, meaning that the solution is unstable if noise exists in the measured data. Generally, a regularization scheme is needed to alleviate the ill-posedness. In this work, a multiplicative regularization scheme is applied to EIT inversion. In this regularization scheme, a cost functional is constructed in which the data misfit functional is multiplied by a regularization factor, and no regularization parameter is needed. The regularization factor is based on the weighted $ L2 $-norm favoring 'blocky' profiles in the reconstructed images. Gauss–Newton method is used to minimize the cost functional iteratively. In the implementation of the multiplicative regularization scheme, the spatial gradient and divergence need to be computed on triangular meshes. For this purpose, the discrete exterior calculus (DEC) theory is applied to formulate the related discrete operators. Numerical and experimental results show good anti-noise performance of the multiplicative regularization scheme in EIT inverse problem.
References:
[1] |
A. Abbasi, B. V. Vahdat and G. E. Fakhim,
An inverse solution for 2D electrical impedance tomography based on electrical properties of material blocks, Journal of Applied Sciences, 9 (2009), 1962-1967.
doi: 10.3923/jas.2009.1962.1967. |
[2] |
A. Abbasi and B. V. Vahdat,
A non-iterative linear inverse solution for the block approach in EIT, Journal of Computational Science, 1 (2010), 190-196.
doi: 10.1016/j.jocs.2010.09.001. |
[3] |
A. Adler and R. Guardo,
Electrical impedance tomography: Regularized imaging and contrast detection, IEEE Trans. Med. Imag., 15 (1996), 170-179.
doi: 10.1109/42.491418. |
[4] |
A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiol. Meas., 30 (2009), S35. |
[5] |
M. Alsaker and J. L. Mueller,
A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 1619-1654.
doi: 10.1137/15M1020137. |
[6] |
M. Alsaker, S. J. Hamilton and A. Hauptmann,
A direct D-bar method for partial boundary data electrical impedance tomography with a priori information, Inverse Probl. Imag., 11 (2017), 427-454.
doi: 10.3934/ipi.2017020. |
[7] |
A. Borsic, B. M. Graham, A. Adler and W. R. B. Lionheart,
In vivo impedance imaging with total variation regularization, IEEE Trans. Med. Imag., 29 (2010), 44-54.
doi: 10.1109/TMI.2009.2022540. |
[8] |
X. Chen, Ill-posed problems and regularization, Computational Methods for Electromagnetic Inverse Scattering, John Wiley & Sons, (2018), 281–289.
doi: 10.1002/9781119311997.app1. |
[9] |
D. Isaacson, M. Cheney, J. C. Newell, S. Simske and J. Goble,
NOSER: An algorithm for solving the inverse conductivity problem, Int. J. Imag. Syst. Technol., 2 (1990), 66-75.
|
[10] |
M. Cheney, D. Isaacson and J. C. Newell,
Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.
doi: 10.1137/S0036144598333613. |
[11] |
K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser,
Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924.
|
[12] |
E. T. Chung, T. F. Chan and X. C. Tai,
Electrical impedance tomography using level set representation and total variational regularization, J. Comput. Phy., 205 (2005), 357-372.
doi: 10.1016/j.jcp.2004.11.022. |
[13] |
C. Cohen-Bacrie, Y. Goussard and R. Guardo,
Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint, IEEE Trans. Med. Imag., 16 (1997), 562-571.
|
[14] |
E. L. V. Costa, J. B. Borges, A. Melo, F. Suarez-Sipmann, C. Toufen, S. H. Bohm and M. B. P. Amato, Bedside estimation of recruitable alveolar collapse and hyperdistension by electrical impedance tomography, Intens. Care Med., 2012,165–170.
doi: 10.1007/978-3-642-28270-6_34. |
[15] |
K. Crane, Discrete Differential Geometry: An Applied Introduction, Carnegie Mellon University, Pittsburgh, USA, 2018. |
[16] |
D. C. Dobson and F. Santosa,
An image-enhancement technique for electrical impedance tomography, Inverse Probl., 10 (1994), 317-334.
doi: 10.1088/0266-5611/10/2/008. |
[17] |
G. Franchineau, N. Brechot, G. Lebreton, G. Hekimian, A. Nieszkowska, J. Trouillet, P. Leprince, J. Chastre, C. Luyt, A. Combes and M. Schmidt,
Bedside contribution of electrical impedance tomography to setting positive end-expiratory pressure for extracorporeal membrane oxygenation-treated patients with severe acute respiratory distress syndrome, Am. J. Respir. Crit. Care Med., 196 (2017), 447-457.
doi: 10.1164/rccm.201605-1055OC. |
[18] |
I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, M. Quintel and G. Hellige,
Regional lung perfusion as determined by electrical impedance tomography in comparison with electron beam CT imaging, IEEE Trans. Med. Imag., 21 (2002), 646-652.
doi: 10.1109/TMI.2002.800585. |
[19] |
I. Frerichs, Z. Zhao, T. Becher, P. Zabel, N. Weiler and B. Vogt,
Regional lung function determined by electrical impedance tomography during bronchodilator reversibility testing in patients with asthma, Physiol. Meas., 37 (2016), 698-712.
doi: 10.1088/0967-3334/37/6/698. |
[20] |
N. P. Galatsanos and A. K. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[21] |
B. Gebauer and N. Hyvönen,
Factorization method and irregular inclusions in electrical impedance tomography, Inverse Probl., 23 (2007), 2159-2170.
doi: 10.1088/0266-5611/23/5/020. |
[22] |
M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. P. Kaipio and P. Maass,
Sparsity reconstruction in electrical impedance tomography: An experimental evaluation, J. Comput. Appl. Math., 236 (2012), 2126-2136.
doi: 10.1016/j.cam.2011.09.035. |
[23] |
G. H. Golub, M. Heath and G. Wahba,
Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.1080/00401706.1979.10489751. |
[24] |
G. González, J. M. J. Huttunen, V. Kolehmainen, A. Seppänen and M. Vauhkonen,
Experimental evaluation of 3D electrical impedance tomography with total variation prior, Inverse Probl. Sci. En., 24 (2016), 1411-1431.
|
[25] |
G. González, V. Kolehmainen and A. Seppänen,
Isotropic and anisotropic total variation regularization in electrical impedance tomography, Comput. Math. Appl., 74 (2017), 564-576.
doi: 10.1016/j.camwa.2017.05.004. |
[26] |
T. M. Habashy and A. Abubakar,
A general framework for constraint minimization for the inversion of electromagnetic measurements, Prog. Electromagn. Res., 46 (2004), 265-312.
doi: 10.2528/PIER03100702. |
[27] |
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, Physiol. Meas., 27 (2006), S187–S198.
doi: 10.1088/0967-3334/27/5/S16. |
[28] |
S. J. Hamilton, J. M. Reyes, S. Siltanen and X. Zhang,
A hybrid segmentation and D-bar method for electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 770-793.
doi: 10.1137/15M1025992. |
[29] |
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. Imag., 36 (2017), 457-466.
doi: 10.1109/TMI.2016.2613511. |
[30] |
M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Special section on imaging, Inverse probl., 19 (2003), S65–S90.
doi: 10.1088/0266-5611/19/6/055. |
[31] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.
doi: 10.1137/1034115. |
[32] |
B. Harrach, Recent progress on the factorization method for electrical impedance tomography, Comput. Math. Methods Med., 2013 (2013), Art. ID 425184, 8 pp.
doi: 10.1155/2013/425184. |
[33] |
A. Hauptmann, M. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Probl., 33 (2017), 025009, 26pp.
doi: 10.1088/1361-6420/33/2/025009. |
[34] |
A. Hauptmann, V. Kolehmainen, N. M. Mach, T. Savolainen, A. Seppänen and S. Siltanen, Open 2D electrical impedance tomography data archive, preprint, arXiv: 1704.01178. |
[35] |
A. N. Hirani, Discrete Exterior Calculus, Ph.D thesis, California Institute of Technology, 2003. |
[36] |
B. Jin, T. Khan and P. Maass,
A reconstruction algorithm for electrical impedance tomography based on sparsity regularization, Int. J. Numer. Meth. Eng., 89 (2012), 337-353.
doi: 10.1002/nme.3247. |
[37] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen,
Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imag., 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[38] |
P. W. A. Kunst, A. V. Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. M. J. M. De Vries,
Ventilation and perfusion imaging by electrical impedance tomography: A comparison with radionuclide scanning, Physiol. Meas., 19 (1998), 481-490.
doi: 10.1088/0967-3334/19/4/003. |
[39] |
K. Lee, E. J. Woo and J. K. Seo,
A fidelity-embedded regularization method for robust electrical impedance tomography, IEEE Trans. Med. Imag., 37 (2018), 1970-1977.
doi: 10.1109/TMI.2017.2762741. |
[40] |
S. Lehmann, S. Leonhardt, C. Ngo, L. Bergmann, S. Schrading, K. Heimann, N. Wagner and K. Tenbrock,
Electrical impedance tomography as possible guidance for individual positioning of patients with multiple lung injury, Clin. Respir. J., 12 (2018), 68-75.
doi: 10.1111/crj.12481. |
[41] |
D. Liu, V. Kolehmainen, S. Siltanen, A. M. Laukkanen and A. Seppänen,
Nonlinear difference imaging approach to three-dimensional electrical impedance tomography in the presence of geometric modeling errors, IEEE Trans. Biomed. Eng., 63 (2016), 1956-1965.
doi: 10.1109/TBME.2015.2509508. |
[42] |
D. Liu, A. K. Khambampati and J. Du,
A parametric level set method for electrical impedance tomography, IEEE Trans. Med. Imag., 37 (2018), 451-460.
doi: 10.1109/TMI.2017.2756078. |
[43] |
D. Liu, D. Smyl and J. Du,
A parametric level set-based approach to difference imaging in electrical impedance tomography, IEEE Trans. Med. Imag., 38 (2019), 145-155.
doi: 10.1109/TMI.2018.2857839. |
[44] |
S. Liu, J. Jia, Y. D. Zhang and Y. Yang,
Image reconstruction in electrical impedance tomography based on structure-aware sparse Bayesian learning, IEEE Trans. Med. Imag., 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
[45] |
M. B. Mazzoni, A. Perri, A. M. Plebani, S. Ferrari, G. Amelio, A. Rocchi, D. Consonni, G. P. Milani and E. F. Fossali,
Electrical impedance tomography in children with community acquired pneumonia: Preliminary data, Resp. Med., 130 (2017), 9-12.
doi: 10.1016/j.rmed.2017.07.001. |
[46] |
M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, Math. Vis., Springer, Berlin, (2003), 35–57. |
[47] |
J. L. Mueller, S. Siltanen and D. Isaacson,
A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Imag., 21 (2002), 555-559.
doi: 10.1109/TMI.2002.800574. |
[48] |
J. L. Mueller, P. Muller, M. Mellenthin, R. Murthy, M. Capps, M. Alsaker, R. Deterding, S. D. Sagel and E. DeBoer, Estimating regions of air trapping from electrical impedance tomography data, Physiol. Meas., 39 (2018), 05NT01.
doi: 10.1088/1361-6579/aac295. |
[49] |
P. A. Muller, J. L. Mueller, M. Mellenthin, R. Murthy, M. Capps, B. D. Wagner, M. Alsaker, R. Deterding, S. D. Sagel and J. Hoppe, Evaluation of surrogate measures of pulmonary function derived from electrical impedance tomography data in children with cystic fibrosis, Physiol. Meas., 39 (2018), 045008.
doi: 10.1088/1361-6579/aab8c4. |
[50] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Experimental mathematics: Computational issues in nonlinear science (Los Alamos, NM, 1991), Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[51] |
J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, Applied Computational Geometry Towards Geometric Engineering, (eds. M. C. Lin and D. Manocha), Springer, Berlin, Heidelberg, (1996), 203–222. |
[52] |
S. Siltanen, J. Mueller and D. Isaacson,
An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem, Inverse probl., 16 (2000), 681-699.
doi: 10.1088/0266-5611/16/3/310. |
[53] |
M. Soleimani, C. E. Powell and N. Polydorides,
Improving the forward solver for the complete electrode model in EIT using algebraic multigrid, IEEE Trans. Med. Imag., 24 (2005), 577-583.
doi: 10.1109/TMI.2005.843741. |
[54] |
E. Somersalo, M. Cheney and D. Isaacson,
Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[55] |
J. N. Tehrani, A. McEwan, C. Jin and A. Van Schaik,
L1 regularization method in electrical impedance tomography by using the L1-curve (Pareto frontier curve), Appl. Math. Model., 36 (2012), 1095-1105.
doi: 10.1016/j.apm.2011.07.055. |
[56] |
C. J. Trepte, C. R. Phillips, J. Solà, A. Adler, S. A. Haas, M. Rapin, S. H. Böhm and D. A. Reuter, Electrical impedance tomography (EIT) for quantification of pulmonary edema in acute lung injury, Crit. Care, 20 (2016), Article 18.
doi: 10.1186/s13054-015-1173-5. |
[57] |
P. M. van den Berg, A. L. Van Broekhoven and A. Abubakar,
Extended contrast source inversion, Inverse Probl., 15 (1999), 1325-1344.
doi: 10.1088/0266-5611/15/5/315. |
[58] |
P. M. van den Berg, A. Abubakar and J. T. Fokkema, Multiplicative regularization for contrast profile inversion, Radio Sci., 38 (2003), 23-1–23-10. |
[59] |
M. Vauhkonen, D. Vadasz, P. A. Karjalainen, E. Somersalo and J. P. Kaipio,
Tikhonov regularization and prior information in electrical impedance tomography, IEEE Trans. Med. Imag., 17 (1998), 285-293.
doi: 10.1109/42.700740. |
[60] |
P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio,
Three-dimensional electrical impedance tomography based on the complete electrode model, IEEE Trans. Biomed. Eng., 46 (1999), 1150-1160.
doi: 10.1109/10.784147. |
[61] |
J. A. Victorino, J. B. Borges, V. N. Okamoto, G. F. J. Matos, M. R. Tucci, M. P. R. Caramez, H. Tanaka, F. S. Sipmann, D. C. B. Santos, C. S. V. Barbas, C. R. R. Carvalho and M. B. P. Amato,
Imbalances in regional lung ventilation: A validation study on electrical impedance tomography, Am. J. Respir. Crit. Care Med., 169 (2004), 791-800.
|
[62] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli,
Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[63] |
E. J. Woo, P. Hua, J. G. Webster and W. J. Tompkins,
A robust image reconstruction algorithm and its parallel implementation in electrical impedance tomography, IEEE Trans. Med. Imag., 12 (1993), 137-146.
doi: 10.1109/42.232242. |
[64] |
Y. Yang, H. Wu and J. Jia,
Image reconstruction for electrical impedance tomography using enhanced adaptive group sparsity with total variation, IEEE Sens. J., 17 (2017), 5589-5598.
doi: 10.1109/JSEN.2017.2728179. |
[65] |
T. J. Yorkey, J. G. Webster and W. J. Tompkins,
Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomed. Eng., BME-34 (1987), 843-852.
doi: 10.1109/TBME.1987.326032. |
[66] |
A. Zakaria and J. LoVetri,
Application of multiplicative regularization to the finite-element contrast source inversion method, IEEE Trans. Antenn. Propag., 59 (2011), 3495-3498.
doi: 10.1109/TAP.2011.2161564. |
[67] |
A. Zakaria, I. Jefirey and J. LoVetri,
Full-vectorial parallel finite-element contrast source inversion method, Prog. Electromagn. Res., 142 (2013), 463-483.
doi: 10.2528/PIER13080706. |
[68] |
L. Zhou, B. Harrach and J. K. Seo, Monotonicity-based electrical impedance tomography for lung imaging, Inverse Probl., 34 (2018), 045005, 25pp.
doi: 10.1088/1361-6420/aaaf84. |
show all references
References:
[1] |
A. Abbasi, B. V. Vahdat and G. E. Fakhim,
An inverse solution for 2D electrical impedance tomography based on electrical properties of material blocks, Journal of Applied Sciences, 9 (2009), 1962-1967.
doi: 10.3923/jas.2009.1962.1967. |
[2] |
A. Abbasi and B. V. Vahdat,
A non-iterative linear inverse solution for the block approach in EIT, Journal of Computational Science, 1 (2010), 190-196.
doi: 10.1016/j.jocs.2010.09.001. |
[3] |
A. Adler and R. Guardo,
Electrical impedance tomography: Regularized imaging and contrast detection, IEEE Trans. Med. Imag., 15 (1996), 170-179.
doi: 10.1109/42.491418. |
[4] |
A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiol. Meas., 30 (2009), S35. |
[5] |
M. Alsaker and J. L. Mueller,
A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 1619-1654.
doi: 10.1137/15M1020137. |
[6] |
M. Alsaker, S. J. Hamilton and A. Hauptmann,
A direct D-bar method for partial boundary data electrical impedance tomography with a priori information, Inverse Probl. Imag., 11 (2017), 427-454.
doi: 10.3934/ipi.2017020. |
[7] |
A. Borsic, B. M. Graham, A. Adler and W. R. B. Lionheart,
In vivo impedance imaging with total variation regularization, IEEE Trans. Med. Imag., 29 (2010), 44-54.
doi: 10.1109/TMI.2009.2022540. |
[8] |
X. Chen, Ill-posed problems and regularization, Computational Methods for Electromagnetic Inverse Scattering, John Wiley & Sons, (2018), 281–289.
doi: 10.1002/9781119311997.app1. |
[9] |
D. Isaacson, M. Cheney, J. C. Newell, S. Simske and J. Goble,
NOSER: An algorithm for solving the inverse conductivity problem, Int. J. Imag. Syst. Technol., 2 (1990), 66-75.
|
[10] |
M. Cheney, D. Isaacson and J. C. Newell,
Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.
doi: 10.1137/S0036144598333613. |
[11] |
K.-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser,
Electrode models for electric current computed tomography, IEEE Trans. Biomed. Eng., 36 (1989), 918-924.
|
[12] |
E. T. Chung, T. F. Chan and X. C. Tai,
Electrical impedance tomography using level set representation and total variational regularization, J. Comput. Phy., 205 (2005), 357-372.
doi: 10.1016/j.jcp.2004.11.022. |
[13] |
C. Cohen-Bacrie, Y. Goussard and R. Guardo,
Regularized reconstruction in electrical impedance tomography using a variance uniformization constraint, IEEE Trans. Med. Imag., 16 (1997), 562-571.
|
[14] |
E. L. V. Costa, J. B. Borges, A. Melo, F. Suarez-Sipmann, C. Toufen, S. H. Bohm and M. B. P. Amato, Bedside estimation of recruitable alveolar collapse and hyperdistension by electrical impedance tomography, Intens. Care Med., 2012,165–170.
doi: 10.1007/978-3-642-28270-6_34. |
[15] |
K. Crane, Discrete Differential Geometry: An Applied Introduction, Carnegie Mellon University, Pittsburgh, USA, 2018. |
[16] |
D. C. Dobson and F. Santosa,
An image-enhancement technique for electrical impedance tomography, Inverse Probl., 10 (1994), 317-334.
doi: 10.1088/0266-5611/10/2/008. |
[17] |
G. Franchineau, N. Brechot, G. Lebreton, G. Hekimian, A. Nieszkowska, J. Trouillet, P. Leprince, J. Chastre, C. Luyt, A. Combes and M. Schmidt,
Bedside contribution of electrical impedance tomography to setting positive end-expiratory pressure for extracorporeal membrane oxygenation-treated patients with severe acute respiratory distress syndrome, Am. J. Respir. Crit. Care Med., 196 (2017), 447-457.
doi: 10.1164/rccm.201605-1055OC. |
[18] |
I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, M. Quintel and G. Hellige,
Regional lung perfusion as determined by electrical impedance tomography in comparison with electron beam CT imaging, IEEE Trans. Med. Imag., 21 (2002), 646-652.
doi: 10.1109/TMI.2002.800585. |
[19] |
I. Frerichs, Z. Zhao, T. Becher, P. Zabel, N. Weiler and B. Vogt,
Regional lung function determined by electrical impedance tomography during bronchodilator reversibility testing in patients with asthma, Physiol. Meas., 37 (2016), 698-712.
doi: 10.1088/0967-3334/37/6/698. |
[20] |
N. P. Galatsanos and A. K. Katsaggelos,
Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1 (1992), 322-336.
doi: 10.1109/83.148606. |
[21] |
B. Gebauer and N. Hyvönen,
Factorization method and irregular inclusions in electrical impedance tomography, Inverse Probl., 23 (2007), 2159-2170.
doi: 10.1088/0266-5611/23/5/020. |
[22] |
M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. P. Kaipio and P. Maass,
Sparsity reconstruction in electrical impedance tomography: An experimental evaluation, J. Comput. Appl. Math., 236 (2012), 2126-2136.
doi: 10.1016/j.cam.2011.09.035. |
[23] |
G. H. Golub, M. Heath and G. Wahba,
Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.1080/00401706.1979.10489751. |
[24] |
G. González, J. M. J. Huttunen, V. Kolehmainen, A. Seppänen and M. Vauhkonen,
Experimental evaluation of 3D electrical impedance tomography with total variation prior, Inverse Probl. Sci. En., 24 (2016), 1411-1431.
|
[25] |
G. González, V. Kolehmainen and A. Seppänen,
Isotropic and anisotropic total variation regularization in electrical impedance tomography, Comput. Math. Appl., 74 (2017), 564-576.
doi: 10.1016/j.camwa.2017.05.004. |
[26] |
T. M. Habashy and A. Abubakar,
A general framework for constraint minimization for the inversion of electromagnetic measurements, Prog. Electromagn. Res., 46 (2004), 265-312.
doi: 10.2528/PIER03100702. |
[27] |
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, Physiol. Meas., 27 (2006), S187–S198.
doi: 10.1088/0967-3334/27/5/S16. |
[28] |
S. J. Hamilton, J. M. Reyes, S. Siltanen and X. Zhang,
A hybrid segmentation and D-bar method for electrical impedance tomography, SIAM J. Imaging Sci., 9 (2016), 770-793.
doi: 10.1137/15M1025992. |
[29] |
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. Imag., 36 (2017), 457-466.
doi: 10.1109/TMI.2016.2613511. |
[30] |
M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Special section on imaging, Inverse probl., 19 (2003), S65–S90.
doi: 10.1088/0266-5611/19/6/055. |
[31] |
P. C. Hansen,
Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.
doi: 10.1137/1034115. |
[32] |
B. Harrach, Recent progress on the factorization method for electrical impedance tomography, Comput. Math. Methods Med., 2013 (2013), Art. ID 425184, 8 pp.
doi: 10.1155/2013/425184. |
[33] |
A. Hauptmann, M. Santacesaria and S. Siltanen, Direct inversion from partial-boundary data in electrical impedance tomography, Inverse Probl., 33 (2017), 025009, 26pp.
doi: 10.1088/1361-6420/33/2/025009. |
[34] |
A. Hauptmann, V. Kolehmainen, N. M. Mach, T. Savolainen, A. Seppänen and S. Siltanen, Open 2D electrical impedance tomography data archive, preprint, arXiv: 1704.01178. |
[35] |
A. N. Hirani, Discrete Exterior Calculus, Ph.D thesis, California Institute of Technology, 2003. |
[36] |
B. Jin, T. Khan and P. Maass,
A reconstruction algorithm for electrical impedance tomography based on sparsity regularization, Int. J. Numer. Meth. Eng., 89 (2012), 337-353.
doi: 10.1002/nme.3247. |
[37] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen,
Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imag., 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[38] |
P. W. A. Kunst, A. V. Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. M. J. M. De Vries,
Ventilation and perfusion imaging by electrical impedance tomography: A comparison with radionuclide scanning, Physiol. Meas., 19 (1998), 481-490.
doi: 10.1088/0967-3334/19/4/003. |
[39] |
K. Lee, E. J. Woo and J. K. Seo,
A fidelity-embedded regularization method for robust electrical impedance tomography, IEEE Trans. Med. Imag., 37 (2018), 1970-1977.
doi: 10.1109/TMI.2017.2762741. |
[40] |
S. Lehmann, S. Leonhardt, C. Ngo, L. Bergmann, S. Schrading, K. Heimann, N. Wagner and K. Tenbrock,
Electrical impedance tomography as possible guidance for individual positioning of patients with multiple lung injury, Clin. Respir. J., 12 (2018), 68-75.
doi: 10.1111/crj.12481. |
[41] |
D. Liu, V. Kolehmainen, S. Siltanen, A. M. Laukkanen and A. Seppänen,
Nonlinear difference imaging approach to three-dimensional electrical impedance tomography in the presence of geometric modeling errors, IEEE Trans. Biomed. Eng., 63 (2016), 1956-1965.
doi: 10.1109/TBME.2015.2509508. |
[42] |
D. Liu, A. K. Khambampati and J. Du,
A parametric level set method for electrical impedance tomography, IEEE Trans. Med. Imag., 37 (2018), 451-460.
doi: 10.1109/TMI.2017.2756078. |
[43] |
D. Liu, D. Smyl and J. Du,
A parametric level set-based approach to difference imaging in electrical impedance tomography, IEEE Trans. Med. Imag., 38 (2019), 145-155.
doi: 10.1109/TMI.2018.2857839. |
[44] |
S. Liu, J. Jia, Y. D. Zhang and Y. Yang,
Image reconstruction in electrical impedance tomography based on structure-aware sparse Bayesian learning, IEEE Trans. Med. Imag., 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
[45] |
M. B. Mazzoni, A. Perri, A. M. Plebani, S. Ferrari, G. Amelio, A. Rocchi, D. Consonni, G. P. Milani and E. F. Fossali,
Electrical impedance tomography in children with community acquired pneumonia: Preliminary data, Resp. Med., 130 (2017), 9-12.
doi: 10.1016/j.rmed.2017.07.001. |
[46] |
M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, Math. Vis., Springer, Berlin, (2003), 35–57. |
[47] |
J. L. Mueller, S. Siltanen and D. Isaacson,
A direct reconstruction algorithm for electrical impedance tomography, IEEE Trans. Med. Imag., 21 (2002), 555-559.
doi: 10.1109/TMI.2002.800574. |
[48] |
J. L. Mueller, P. Muller, M. Mellenthin, R. Murthy, M. Capps, M. Alsaker, R. Deterding, S. D. Sagel and E. DeBoer, Estimating regions of air trapping from electrical impedance tomography data, Physiol. Meas., 39 (2018), 05NT01.
doi: 10.1088/1361-6579/aac295. |
[49] |
P. A. Muller, J. L. Mueller, M. Mellenthin, R. Murthy, M. Capps, B. D. Wagner, M. Alsaker, R. Deterding, S. D. Sagel and J. Hoppe, Evaluation of surrogate measures of pulmonary function derived from electrical impedance tomography data in children with cystic fibrosis, Physiol. Meas., 39 (2018), 045008.
doi: 10.1088/1361-6579/aab8c4. |
[50] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms, Experimental mathematics: Computational issues in nonlinear science (Los Alamos, NM, 1991), Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[51] |
J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, Applied Computational Geometry Towards Geometric Engineering, (eds. M. C. Lin and D. Manocha), Springer, Berlin, Heidelberg, (1996), 203–222. |
[52] |
S. Siltanen, J. Mueller and D. Isaacson,
An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem, Inverse probl., 16 (2000), 681-699.
doi: 10.1088/0266-5611/16/3/310. |
[53] |
M. Soleimani, C. E. Powell and N. Polydorides,
Improving the forward solver for the complete electrode model in EIT using algebraic multigrid, IEEE Trans. Med. Imag., 24 (2005), 577-583.
doi: 10.1109/TMI.2005.843741. |
[54] |
E. Somersalo, M. Cheney and D. Isaacson,
Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023-1040.
doi: 10.1137/0152060. |
[55] |
J. N. Tehrani, A. McEwan, C. Jin and A. Van Schaik,
L1 regularization method in electrical impedance tomography by using the L1-curve (Pareto frontier curve), Appl. Math. Model., 36 (2012), 1095-1105.
doi: 10.1016/j.apm.2011.07.055. |
[56] |
C. J. Trepte, C. R. Phillips, J. Solà, A. Adler, S. A. Haas, M. Rapin, S. H. Böhm and D. A. Reuter, Electrical impedance tomography (EIT) for quantification of pulmonary edema in acute lung injury, Crit. Care, 20 (2016), Article 18.
doi: 10.1186/s13054-015-1173-5. |
[57] |
P. M. van den Berg, A. L. Van Broekhoven and A. Abubakar,
Extended contrast source inversion, Inverse Probl., 15 (1999), 1325-1344.
doi: 10.1088/0266-5611/15/5/315. |
[58] |
P. M. van den Berg, A. Abubakar and J. T. Fokkema, Multiplicative regularization for contrast profile inversion, Radio Sci., 38 (2003), 23-1–23-10. |
[59] |
M. Vauhkonen, D. Vadasz, P. A. Karjalainen, E. Somersalo and J. P. Kaipio,
Tikhonov regularization and prior information in electrical impedance tomography, IEEE Trans. Med. Imag., 17 (1998), 285-293.
doi: 10.1109/42.700740. |
[60] |
P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio,
Three-dimensional electrical impedance tomography based on the complete electrode model, IEEE Trans. Biomed. Eng., 46 (1999), 1150-1160.
doi: 10.1109/10.784147. |
[61] |
J. A. Victorino, J. B. Borges, V. N. Okamoto, G. F. J. Matos, M. R. Tucci, M. P. R. Caramez, H. Tanaka, F. S. Sipmann, D. C. B. Santos, C. S. V. Barbas, C. R. R. Carvalho and M. B. P. Amato,
Imbalances in regional lung ventilation: A validation study on electrical impedance tomography, Am. J. Respir. Crit. Care Med., 169 (2004), 791-800.
|
[62] |
Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli,
Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.
doi: 10.1109/TIP.2003.819861. |
[63] |
E. J. Woo, P. Hua, J. G. Webster and W. J. Tompkins,
A robust image reconstruction algorithm and its parallel implementation in electrical impedance tomography, IEEE Trans. Med. Imag., 12 (1993), 137-146.
doi: 10.1109/42.232242. |
[64] |
Y. Yang, H. Wu and J. Jia,
Image reconstruction for electrical impedance tomography using enhanced adaptive group sparsity with total variation, IEEE Sens. J., 17 (2017), 5589-5598.
doi: 10.1109/JSEN.2017.2728179. |
[65] |
T. J. Yorkey, J. G. Webster and W. J. Tompkins,
Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomed. Eng., BME-34 (1987), 843-852.
doi: 10.1109/TBME.1987.326032. |
[66] |
A. Zakaria and J. LoVetri,
Application of multiplicative regularization to the finite-element contrast source inversion method, IEEE Trans. Antenn. Propag., 59 (2011), 3495-3498.
doi: 10.1109/TAP.2011.2161564. |
[67] |
A. Zakaria, I. Jefirey and J. LoVetri,
Full-vectorial parallel finite-element contrast source inversion method, Prog. Electromagn. Res., 142 (2013), 463-483.
doi: 10.2528/PIER13080706. |
[68] |
L. Zhou, B. Harrach and J. K. Seo, Monotonicity-based electrical impedance tomography for lung imaging, Inverse Probl., 34 (2018), 045005, 25pp.
doi: 10.1088/1361-6420/aaaf84. |











2D numerical example 1 | 2D numerical example 2 | |||||||
Noise level | 0% | 1% | 3% | 5% | 0% | 1% | 2% | 3% |
MR-WL2 |
0.9532 | 0.9007 | 0.8845 | 0.8718 | 0.9571 | 0.9107 | 0.8894 | 0.8807 |
AR-TV |
0.9591 | 0.8954 | 0.8755 | 0.8530 | 0.9581 | 0.9005 | 0.8805 | 0.8636 |
* Multiplicative weighted L2-norm regularization † Additive total variation regularization |
2D numerical example 1 | 2D numerical example 2 | |||||||
Noise level | 0% | 1% | 3% | 5% | 0% | 1% | 2% | 3% |
MR-WL2 |
0.9532 | 0.9007 | 0.8845 | 0.8718 | 0.9571 | 0.9107 | 0.8894 | 0.8807 |
AR-TV |
0.9591 | 0.8954 | 0.8755 | 0.8530 | 0.9581 | 0.9005 | 0.8805 | 0.8636 |
* Multiplicative weighted L2-norm regularization † Additive total variation regularization |
Phantom 2.2 | Phantom 3.1 | Phantom 4.1 | Phantom 5.2 | |
Phantom 2.2 | Phantom 3.1 | Phantom 4.1 | Phantom 5.2 | |
[1] |
Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems and Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561 |
[2] |
Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems and Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353 |
[3] |
Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems and Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251 |
[4] |
Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066 |
[5] |
Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems and Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323 |
[6] |
Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55 |
[7] |
Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems and Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531 |
[8] |
Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems and Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217 |
[9] |
Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen, Patrick F. Bray. 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods. Inverse Problems and Imaging, 2021, 15 (5) : 1135-1169. doi: 10.3934/ipi.2021032 |
[10] |
Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems and Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211 |
[11] |
Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems and Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417 |
[12] |
Gen Nakamura, Päivi Ronkanen, Samuli Siltanen, Kazumi Tanuma. Recovering conductivity at the boundary in three-dimensional electrical impedance tomography. Inverse Problems and Imaging, 2011, 5 (2) : 485-510. doi: 10.3934/ipi.2011.5.485 |
[13] |
Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423 |
[14] |
Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen. Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps. Inverse Problems and Imaging, 2018, 12 (2) : 373-400. doi: 10.3934/ipi.2018017 |
[15] |
Erfang Ma. Integral formulation of the complete electrode model of electrical impedance tomography. Inverse Problems and Imaging, 2020, 14 (2) : 385-398. doi: 10.3934/ipi.2020017 |
[16] |
Kimmo Karhunen, Aku Seppänen, Jari P. Kaipio. Adaptive meshing approach to identification of cracks with electrical impedance tomography. Inverse Problems and Imaging, 2014, 8 (1) : 127-148. doi: 10.3934/ipi.2014.8.127 |
[17] |
Jérémi Dardé, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis. Fine-tuning electrode information in electrical impedance tomography. Inverse Problems and Imaging, 2012, 6 (3) : 399-421. doi: 10.3934/ipi.2012.6.399 |
[18] |
Haigang Li, Jenn-Nan Wang, Ling Wang. Refined stability estimates in electrical impedance tomography with multi-layer structure. Inverse Problems and Imaging, 2022, 16 (1) : 229-249. doi: 10.3934/ipi.2021048 |
[19] |
Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163 |
[20] |
Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021074 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]