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Nonconvex regularization for blurred images with Cauchy noise
Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography
1. | Department of Mathematics, Gonzaga University, Spokane, WA 99258, USA |
2. | Boeing Co., 6001 S Air Depot Blvd, Oklahoma City, OK 73135, USA |
3. | Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824, USA |
4. | College of Medicine, University of Arizona, Phoenix, AZ 85004, USA |
For patients undergoing mechanical ventilation due to respiratory failure, 2D electrical impedance tomography (EIT) is emerging as a means to provide functional monitoring of pulmonary processes. In EIT, electrical current is applied to the body, and the internal conductivity distribution is reconstructed based on subsequent voltage measurements. However, EIT images are known to often suffer from large systematic artifacts arising from various limitations and exacerbated by the ill-posedness of the inverse problem. The direct D-bar reconstruction method admits a nonlinear Fourier analysis of the EIT problem, providing the ability to process and filter reconstructions in the nonphysical frequency regime. In this work, a technique is introduced for automated Fourier-domain filtering of known systematic artifacts in 2D D-bar reconstructions. The new method is validated using three numerically simulated static thoracic datasets with induced artifacts, plus two experimental dynamic human ventilation datasets containing systematic artifacts. Application of the method is shown to significantly reduce the appearance of artifacts and improve the shape of the lung regions in all datasets.
References:
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Impedance imaging of lung ventilation: Do we need to account for chest expansion?, IEEE Tran. Biomedical Engineering, 43 (1994), 414-420.
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A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiological Measurement, 30 (2009), 35pp. |
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A D-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM J. Imaging Sci, 9 (2016), 1619-1654.
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M. Alsaker, D. A. C. Cárdenas, S. S. Furuie and J. L. Mueller, Complementary use of priors for pulmonary imaging with electrical impedance and ultrasound computed tomography, J. Compu. Appl. Math., 395 (2021), 15pp.
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M. Alsaker and J. L. Mueller,
Use of an optimized spatial prior in D-bar reconstructions of EIT tank data, Inverse Probl. Imaging, 12 (2018), 883-901.
doi: 10.3934/ipi.2018037. |
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M. Alsaker, J. L. Mueller and R. Murthy,
Dynamic optimized priors for D-bar reconstructions of human ventilation using electrical impedance tomography, J. Comput. Appl. Math., 362 (2019), 276-294.
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Bioimpedance tomography (electrical impedance tomography), Annu. Rev. Biomed. Eng., 8 (2006), 63-91.
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R. Blue, D. Isaacson and J. C. Newell, Real-time three-dimensional electrical impedance imaging, Physiological Measurement, 21 (2000), 15pp.
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Effect of skin impedance on image quality and variability in electrical impedance tomography: A model study, Medical and Biological Engineering and Computing, 34 (1996), 351-354.
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|
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A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Probl. Imaging, 8 (2014), 1013-1031.
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Deep d-bar: Real-time electrical impedance tomography imaging with deep neural networks, IEEE Trans. Med. Imaging., 37 (2018), 2367-2377.
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Toward improved surveillance: The impact of ventilator-associated complications on length of stay and antibiotic use in patients in intensive care units, Clinical Infectious Diseases, 56 (2013), 471-477.
|
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P. Hua, E. J. Woo, J. G. Webster and W. J. Tompkins,
Iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography, IEEE Trans. Med. Imaging, 10 (1991), 621-628.
doi: 10.1109/42.108598. |
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J. Kobylianskii, A. Murray, D. Brace, E. Goligher and E. Fan,
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V. Kolehmainen, M. Vauhkonen, P. Karjalainen and J. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns, Physiological Measurement, 18 (1997).
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V. Kolehmainen, M. Lassas and P. Ola,
Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE Trans. Med. Imaging, 27 (2006), 1404-1414.
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[29] |
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A fidelity-embedded regularization method for robust electrical impedance tomography, IEEE Trans. Med. Imaging, 37 (2018), 1970-1977.
doi: 10.1109/TMI.2017.2762741. |
[30] |
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Electrical impedance tomography: The holy grail of ventilation and perfusion monitoring?, Intensive Care Medicine, 38 (2012), 1917-1929.
doi: 10.1007/s00134-012-2684-z. |
[31] |
D. Liu and J. Du,
A moving morphable components based shape reconstruction framework for electrical impedance tomography, IEEE Trans. Med. Imaging, 38 (2019), 2937-2948.
doi: 10.1109/TMI.2019.2918566. |
[32] |
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. Biomedical Engineering, 63 (2016), 1956-1965.
doi: 10.1109/TBME.2015.2509508. |
[33] |
D. Liu, D. Smyl, D. Gu and J. Du,
Shape-driven difference electrical impedance tomography, IEEE Trans. Med. Imaging, 39 (2020), 3801-3812.
doi: 10.1109/TMI.2020.3004806. |
[34] |
S. Liu, R. Cao, Y. Huang, T. Ouypornkochagorn and J. Jia,
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Image reconstruction in electrical impedance tomography based on structure-aware sparse bayesian learning, IEEE Trans. Med. Imaging, 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
[36] |
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. Imaging, 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
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|
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What's new in electrical impedance tomography, Intensive Care Medicine, 45 (2019), 674-677.
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M. M. Mellenthin et al.,
The ace1 electrical impedance tomography system for thoracic imaging, IEEE Tran. Instrumentation and Measurement, 68 (2019), 3137-3150.
|
[41] |
J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012.
doi: 10.1137/1.9781611972344. |
[42] |
P. A. Muller, J. L. Mueller and M. M. Mellenthin,
Real-time implementation of calderón's method on subject-specific domains, IEEE Tran. Medical Imaging, 36 (2017), 1868-1875.
|
[43] |
E. K. Murphy and J. L. Mueller,
Effect of domain shape modeling and measurement errors on the 2-d d-bar method for eit, IEEE Tran. Med. Imaging, 28 (2009), 1576-1584.
doi: 10.1109/TMI.2009.2021611. |
[44] |
A. I. Nachman,
Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96.
doi: 10.2307/2118653. |
[45] |
S. Oh, T. Tang and R. Sadleir, Quantitative analysis of shape change in electrical impedance tomography (EIT), In 13th International Conference on Electrical Bioimpedance and the 8th Conference on Electrical Impedance Tomography, (2007), 424–427.
doi: 10.1007/978-3-540-73841-1_110. |
[46] |
L. Pinhu, T. Whitehead, T. Evans and M. Griffiths,
Ventilator-associated lung injury, The Lancet, 361 (2003), 332-340.
doi: 10.1016/S0140-6736(03)12329-X. |
[47] |
F. Sadi, S. Y. Lee and M. K. Hasan,
Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter, Computers in Biology and Medicine, 40 (2010), 109-118.
doi: 10.1016/j.compbiomed.2009.11.007. |
[48] |
R. J. Sadleir and R. A. Fox,
Detection and quantification of intraperitoneal fluid using electrical impedance tomography, IEEE Tran. Biomedical Engineering, 48 (2001), 484-491.
doi: 10.1109/10.915715. |
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show all references
References:
[1] |
A. Adler, R. Guardo and Y. Berthiaume,
Impedance imaging of lung ventilation: Do we need to account for chest expansion?, IEEE Tran. Biomedical Engineering, 43 (1994), 414-420.
doi: 10.1109/IEMBS.1994.411917. |
[2] |
A. Adler et al., GREIT: A unified approach to 2D linear EIT reconstruction of lung images, Physiological Measurement, 30 (2009), 35pp. |
[3] |
G. Alessandrini,
Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[4] |
M. Alsaker and J. 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. |
[5] |
M. Alsaker, D. A. C. Cárdenas, S. S. Furuie and J. L. Mueller, Complementary use of priors for pulmonary imaging with electrical impedance and ultrasound computed tomography, J. Compu. Appl. Math., 395 (2021), 15pp.
doi: 10.1016/j.cam.2021.113591. |
[6] |
M. Alsaker and J. L. Mueller,
Use of an optimized spatial prior in D-bar reconstructions of EIT tank data, Inverse Probl. Imaging, 12 (2018), 883-901.
doi: 10.3934/ipi.2018037. |
[7] |
M. Alsaker, J. L. Mueller and R. Murthy,
Dynamic optimized priors for D-bar reconstructions of human ventilation using electrical impedance tomography, J. Comput. Appl. Math., 362 (2019), 276-294.
doi: 10.1016/j.cam.2018.07.039. |
[8] |
N. Avis and D. Barber, Incorporating a priori information into the Sheffield filtered backprojection algorithm, Physiological Measurement, 16 (1995).
doi: 10.1088/0967-3334/16/3A/011. |
[9] |
R. H. Bayford,
Bioimpedance tomography (electrical impedance tomography), Annu. Rev. Biomed. Eng., 8 (2006), 63-91.
doi: 10.1146/annurev.bioeng.8.061505.095716. |
[10] |
R. Blue, D. Isaacson and J. C. Newell, Real-time three-dimensional electrical impedance imaging, Physiological Measurement, 21 (2000), 15pp.
doi: 10.1088/0967-3334/21/1/303. |
[11] |
K. Boone and D. Holder,
Effect of skin impedance on image quality and variability in electrical impedance tomography: A model study, Medical and Biological Engineering and Computing, 34 (1996), 351-354.
doi: 10.1007/BF02520003. |
[12] |
A. Boyle and A. Adler, Electrode models under shape deformation in electrical impedance tomography, J. Phys.: Conf. Ser., 224 (2010).
doi: 10.1088/1742-6596/224/1/012051. |
[13] |
A. Boyle, A. Adler and W. R. Lionheart,
Shape deformation in two-dimensional electrical impedance tomography, IEEE Tran. Med. Imaging, 31 (2012), 2185-2193.
doi: 10.1109/TMI.2012.2204438. |
[14] |
C. Bozsak and E. Techner, Mini-Manual Electrical Impedance Tomography (EIT): Device Handling, Application Tips, and Examples, 2018. https://www.draeger.com/Library/Content/EIT-Mini-Manual.pdf |
[15] |
M. Cheney, D. Isaacson, J. C. Newell, S. Simske and J. Goble,
NOSER: An algorithm for solving the inverse conductivity problem, Inter. J. Imaging Systems and Technology, 2 (1990), 66-75.
|
[16] |
T. de Castro Martins, et al., A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images, Annual Reviews in Control. |
[17] |
H. Dehghani, D. Barber and I. Basarab-Horwath, Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography, Physiological Measurement, 20 (1999), 87pp. |
[18] |
M. Dodd and J. Mueller,
A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Probl. Imaging, 8 (2014), 1013-1031.
doi: 10.3934/ipi.2014.8.1013. |
[19] |
S. Guan, A. A. Khan, S. Sikdar and P. V. Chitnis,
Fully dense UNet for 2-D sparse photoacoustic tomography artifact removal, IEEE J. Biomedical and Health Informatics, 24 (2019), 568-576.
doi: 10.1109/JBHI.2019.2912935. |
[20] |
R. Guardo, C. Boulay, B. Murray and M. Bertrand,
An experimental study in electrical impedance tomography using backprojection reconstruction, IEEE Tran., 38 (1991), 617-627.
doi: 10.1109/10.83560. |
[21] |
S. Hamilton and A. Hauptmann,
Deep d-bar: Real-time electrical impedance tomography imaging with deep neural networks, IEEE Trans. Med. Imaging., 37 (2018), 2367-2377.
doi: 10.1109/TMI.2018.2828303. |
[22] |
Y. Hayashi et al.,
Toward improved surveillance: The impact of ventilator-associated complications on length of stay and antibiotic use in patients in intensive care units, Clinical Infectious Diseases, 56 (2013), 471-477.
|
[23] |
P. Hua, E. J. Woo, J. G. Webster and W. J. Tompkins,
Iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography, IEEE Trans. Med. Imaging, 10 (1991), 621-628.
doi: 10.1109/42.108598. |
[24] |
J. P. 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. |
[25] |
K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen,
Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging., 3 (2009), 599-624.
doi: 10.3934/ipi.2009.3.599. |
[26] |
J. Kobylianskii, A. Murray, D. Brace, E. Goligher and E. Fan,
Electrical impedance tomography in adult patients undergoing mechanical ventilation: A systematic review, J. Critical Care, 35 (2016), 33-50.
doi: 10.1016/j.jcrc.2016.04.028. |
[27] |
V. Kolehmainen, M. Vauhkonen, P. Karjalainen and J. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns, Physiological Measurement, 18 (1997).
doi: 10.1088/0967-3334/18/4/003. |
[28] |
V. Kolehmainen, M. Lassas and P. Ola,
Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE Trans. Med. Imaging, 27 (2006), 1404-1414.
doi: 10.1109/ISBI.2006.1625120. |
[29] |
K. Lee, E. J. Woo and J. K. Seo,
A fidelity-embedded regularization method for robust electrical impedance tomography, IEEE Trans. Med. Imaging, 37 (2018), 1970-1977.
doi: 10.1109/TMI.2017.2762741. |
[30] |
S. Leonhardt and B. Lachmann,
Electrical impedance tomography: The holy grail of ventilation and perfusion monitoring?, Intensive Care Medicine, 38 (2012), 1917-1929.
doi: 10.1007/s00134-012-2684-z. |
[31] |
D. Liu and J. Du,
A moving morphable components based shape reconstruction framework for electrical impedance tomography, IEEE Trans. Med. Imaging, 38 (2019), 2937-2948.
doi: 10.1109/TMI.2019.2918566. |
[32] |
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. Biomedical Engineering, 63 (2016), 1956-1965.
doi: 10.1109/TBME.2015.2509508. |
[33] |
D. Liu, D. Smyl, D. Gu and J. Du,
Shape-driven difference electrical impedance tomography, IEEE Trans. Med. Imaging, 39 (2020), 3801-3812.
doi: 10.1109/TMI.2020.3004806. |
[34] |
S. Liu, R. Cao, Y. Huang, T. Ouypornkochagorn and J. Jia,
Time sequence learning for electrical impedance tomography using bayesian spatiotemporal priors, IEEE Trans. Instrumentation and Measurement, 69 (2020), 6045-6057.
doi: 10.1109/TIM.2020.2972172. |
[35] |
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. Imaging, 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
[36] |
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. Imaging, 37 (2018), 2090-2102.
doi: 10.1109/TMI.2018.2816739. |
[37] |
A. Lyckegaard, G. Johnson and P. Tafforeau,
Correction of ring artifacts in X-ray tomographic images, Int. J. Tomo. Stat, 18 (2011), 1-9.
|
[38] |
S. Martin and C. T. M. Choi, A post-processing method for three-dimensional electrical impedance tomography, Scientific Reports, 7 (2017).
doi: 10.1038/s41598-017-07727-2. |
[39] |
T. Mauri, A. Mercat and G. Grasselli,
What's new in electrical impedance tomography, Intensive Care Medicine, 45 (2019), 674-677.
doi: 10.1007/s00134-018-5398-z. |
[40] |
M. M. Mellenthin et al.,
The ace1 electrical impedance tomography system for thoracic imaging, IEEE Tran. Instrumentation and Measurement, 68 (2019), 3137-3150.
|
[41] |
J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Computational Science & Engineering, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012.
doi: 10.1137/1.9781611972344. |
[42] |
P. A. Muller, J. L. Mueller and M. M. Mellenthin,
Real-time implementation of calderón's method on subject-specific domains, IEEE Tran. Medical Imaging, 36 (2017), 1868-1875.
|
[43] |
E. K. Murphy and J. L. Mueller,
Effect of domain shape modeling and measurement errors on the 2-d d-bar method for eit, IEEE Tran. Med. Imaging, 28 (2009), 1576-1584.
doi: 10.1109/TMI.2009.2021611. |
[44] |
A. I. Nachman,
Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., 143 (1996), 71-96.
doi: 10.2307/2118653. |
[45] |
S. Oh, T. Tang and R. Sadleir, Quantitative analysis of shape change in electrical impedance tomography (EIT), In 13th International Conference on Electrical Bioimpedance and the 8th Conference on Electrical Impedance Tomography, (2007), 424–427.
doi: 10.1007/978-3-540-73841-1_110. |
[46] |
L. Pinhu, T. Whitehead, T. Evans and M. Griffiths,
Ventilator-associated lung injury, The Lancet, 361 (2003), 332-340.
doi: 10.1016/S0140-6736(03)12329-X. |
[47] |
F. Sadi, S. Y. Lee and M. K. Hasan,
Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter, Computers in Biology and Medicine, 40 (2010), 109-118.
doi: 10.1016/j.compbiomed.2009.11.007. |
[48] |
R. J. Sadleir and R. A. Fox,
Detection and quantification of intraperitoneal fluid using electrical impedance tomography, IEEE Tran. Biomedical Engineering, 48 (2001), 484-491.
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