February  2014, 8(1): 127-148. doi: 10.3934/ipi.2014.8.127

Adaptive meshing approach to identification of cracks with electrical impedance tomography

1. 

Department of Applied Physics, University of Eastern Finland, 70211 Kuopio, Finland, Finland

2. 

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142, New Zealand

Received  March 2013 Revised  September 2013 Published  March 2014

Electrical impedance tomography (EIT) is a non-invasive imaging modality in which the internal conductivity distribution is reconstructed based on boundary voltage measurements. In this work, we consider the application of EIT to non-destructive testing (NDT) of materials and, especially, crack detection. The main goal is to estimate the location, depth and orientation of a crack in three dimensions. We formulate the crack detection task as a shape estimation problem for boundaries imposed with Neumann zero boundary conditions. We propose an adaptive meshing algorithm that iteratively seeks the maximum a posteriori estimate for the shape of the crack. The approach is tested both numerically and experimentally. In all test cases, the EIT measurements are collected using a set of electrodes attached on only a single planar surface of the target -- this is often the only realizable configuration in NDT of large building structures, such as concrete walls. The results show that with the proposed computational method, it is possible to recover the position and size of the crack, even in cases where the background conductivity is inhomogeneous.
Citation: Kimmo Karhunen, Aku Seppänen, Jari P. Kaipio. Adaptive meshing approach to identification of cracks with electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (1) : 127-148. doi: 10.3934/ipi.2014.8.127
References:
[1]

G. Alessandrini and E. Di Benedetto, Determining 2-dimensional cracks in 3-dimensional bodies: Uniqueness and stability,, Indiana Univ. Math. J., 46 (1997), 1.   Google Scholar

[2]

G. Alessandrini and L. Rondi, Stable determination of a crack in a planar inhomogeneous conductor,, SIAM Journal on Mathematical Analysis, 30 (1998), 326.  doi: 10.1137/S0036141097325502.  Google Scholar

[3]

D. Álvarez, O. Dorn, N. Irishina and M. Moscoso, Crack reconstruction using a level-set strategy,, Journal of Computational Physics, 228 (2009), 5710.  doi: 10.1016/j.jcp.2009.04.038.  Google Scholar

[4]

K. E. Andersen, S. P. Brooks and M. B. Hansen., A Bayesian approach to crack detection in electrically conducting media,, Inverse Problems, 17 (2001), 121.  doi: 10.1088/0266-5611/17/1/310.  Google Scholar

[5]

S. Andrieux, A. B. Abda and H. D. Bui., Reciprocity principle and crack identification,, Inverse Problems, 15 (1999), 59.  doi: 10.1088/0266-5611/15/1/010.  Google Scholar

[6]

T. Bannour, A. B. Abda and M. Jaoua, A semi-explicit algorithm for the reconstruction of 3D planar cracks,, Inverse Problems, 13 (1997), 899.  doi: 10.1088/0266-5611/13/4/002.  Google Scholar

[7]

Z. B. Bazant and J. Planas, Fracture and Size Effect in Concrete and Other Quasibrittle Materials (New Directions in Civil Engineering),, CRC Press, (1997).   Google Scholar

[8]

L. R. Bentley and M. Gharibi, Two-and three-dimensional electrical resistivity imaging at a heterogeneous remediation site,, Geophysics, 69 (2004), 674.  doi: 10.1190/1.1759453.  Google Scholar

[9]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, Mathematical Modelling and Numerical Analysis, 35 (2001), 595.  doi: 10.1051/m2an:2001128.  Google Scholar

[10]

K. Bryan and M. Vogelius, Reconstruction of multiple cracks from experimental electrostatic boundary measurements,, Inverse Problems and Optimal Design in Industry, 7 (1993), 147.   Google Scholar

[11]

K. Bryan and M. S. Vogelius, A review of selected works on crack identification,, Geometric Methods in Inverse Problems and PDE Control, 137 (2004), 25.  doi: 10.1007/978-1-4684-9375-7_3.  Google Scholar

[12]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85.  doi: 10.1137/S0036144598333613.  Google Scholar

[13]

K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Transactions on Biomedical Engineering, 36 (1989), 918.   Google Scholar

[14]

P. Church, J. E. McFee, S. Gagnon and P. Wort, Electrical impedance tomographic imaging of buried landmines,, IEEE Transactions on Geoscience and Remote Sensing, 44 (2006), 2407.  doi: 10.1109/TGRS.2006.873208.  Google Scholar

[15]

W. Daily, A. Ramirez, A. Binley and D. LeBrecque, Electrical resistance tomography,, The Leading Edge, 23 (2004).  doi: 10.1190/1.1729225.  Google Scholar

[16]

J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography,, SIAM Journal on Imaging Sciences, 6 (2013), 176.  doi: 10.1137/120877301.  Google Scholar

[17]

A. R. Elcrat and C. Hu, Determination of surface and interior cracks from electrostatic measurements using Schwarz-Christoffel transformations,, International Journal of Engineering Science, 34 (1996), 1165.  doi: 10.1016/0020-7225(96)00011-0.  Google Scholar

[18]

I. Frerichs, G. Hahn and G. Hellige, Thoracic electrical impedance tomographic measurements during volume controlled ventilation-effects of tidal volume and positive end-expiratory pressure,, IEEE Trans. Med. Imaging, 18 (1999), 764.  doi: 10.1109/42.802754.  Google Scholar

[19]

A. Friedman and M. Vogelius, Determining cracks by boundary measurements,, Indiana Univ. Math. J., 38 (1989), 527.  doi: 10.1512/iumj.1989.38.38025.  Google Scholar

[20]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement,, Inverse Problems, 14 (1998), 67.  doi: 10.1088/0266-5611/14/1/008.  Google Scholar

[21]

T. C. Hou and J. P. Lynch, Electrical Impedance Tomographic Methods for Sensing Strain Fields and Crack Damage in Cementitious Structures,, Journal of Intelligent Material Systems and Structures, (2008).  doi: 10.1177/1045389X08096052.  Google Scholar

[22]

N. Hyvönen, K. Karhunen and A. Seppänen, Fréchet derivative with respect to the shape of an internal electrode in Electrical Impedance Tomography,, SIAM Journal on Applied Mathematics, 70 (2010).  doi: 10.1137/09075929X.  Google Scholar

[23]

S. Järvenpää, Finite Element Model for the Inverse Conductivity Problem,, Phil. Lic. thesis, (1996).   Google Scholar

[24]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in EIT,, Inverse Problems, 16 (2000), 1487.  doi: 10.1088/0266-5611/16/5/321.  Google Scholar

[25]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, 15 (1999), 713.  doi: 10.1088/0266-5611/15/3/306.  Google Scholar

[26]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Springer Science+ Business Media, (2005).   Google Scholar

[27]

K. Karhunen, A. Seppänen, A. Lehikoinen, J. Blunt, J. P. Kaipio and P. J. M. Monteiro, Electrical resistance tomography for assessment of cracks in concrete., ACI Materials Journal, 107 (2010).   Google Scholar

[28]

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.  doi: 10.1016/j.cemconres.2009.08.023.  Google Scholar

[29]

H. Kim and J. K. Seo, Unique determination of a collection of a finite number of cracks from two boundary measurements,, SIAM Journal on Mathematical Analysis, 27 (1996), 1336.  doi: 10.1137/S0036141094275488.  Google Scholar

[30]

V. Kolehmainen, S. R. Arridge, W. R. B. Lionheart, M. Vauhkonen and J. P. Kaipio, Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,, Inverse Problems, 15 (1999), 1375.  doi: 10.1088/0266-5611/15/5/318.  Google Scholar

[31]

J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen and L. M. Heikkinen, Suitability of a PXI platform for an electrical impedance tomography system,, Measurement Science and Technology, 20 (2009).  doi: 10.1088/0957-0233/20/1/015503.  Google Scholar

[32]

P. W. A. Kunst, A. V. Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. De Vries, Ventilation and perfusion imaging by electrical impedance tomography: A comparison with radionuclide scanning,, Physiological Measurement, 19 (1998), 481.  doi: 10.1088/0967-3334/19/4/003.  Google Scholar

[33]

J. F. Lataste, C. Sirieix, D. Breysse and M. Frappa, Electrical resistivity measurement applied to cracking assessment on reinforced concrete structures in civil engineering,, NDT and E International, 36 (2003), 383.  doi: 10.1016/S0963-8695(03)00013-6.  Google Scholar

[34]

C. Lieberman, K. Willcox and O. Ghattas, Parameter and state model reduction for large-scale statistical inverse problems,, SIAM Journal on Scientific Computing, 32 (2010), 2523.  doi: 10.1137/090775622.  Google Scholar

[35]

V. Liepa, F. Santosa and M. Vogelius, Crack determination from boundary measurements-reconstruction using experimental data,, Journal of Nondestructive Evaluation, 12 (1993), 163.  doi: 10.1007/BF00567084.  Google Scholar

[36]

K. J. Loh, T-C Hou, J. P. Lynch and N. A Kotov, Carbon nanotube sensing skins for spatial strain and impact damage identification,, Journal of Nondestructuctive Evaluation, 28 (2009), 9.  doi: 10.1007/s10921-009-0043-y.  Google Scholar

[37]

P. R. McGillivray and D. W. Oldenburg, Methods for calculating Fréchet derivatives and sensitivities for the non-linear inverse problem: A comparative study,, Geophysical Prospecting, 38 (1990), 499.   Google Scholar

[38]

K. S. Osterman, T. E. Kerner, D. B. Williams, A. Hartov, S. P. Poplack and K. D. Paulsen, Multifrequency electrical impedance imaging: Preliminary in vivo experience in breast,, Physiological Measurement, 21 (2000), 99.   Google Scholar

[39]

R. B. Polder, Test methods for on site measurement of resistivity of concrete-a RILEM TC-154 technical recommendation,, Construction and building materials, 15 (2001), 125.  doi: 10.1016/S0950-0618(00)00061-1.  Google Scholar

[40]

F. Santosa and M. Vogelius, A computational algorithm to determine cracks from electrostatic boundary measurements,, International Journal of Engineering Science, 29 (1991), 917.  doi: 10.1016/0020-7225(91)90166-Z.  Google Scholar

[41]

J. Schöberl, NETGEN An advancing front 2D/3D-mesh generator based on abstract rules,, Computing and Visualization in Science, 1 (1997), 41.   Google Scholar

[42]

E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 1023.  doi: 10.1137/0152060.  Google Scholar

[43]

O.-P. Tossavainen, V. Kolehmainen and M. Vauhkonen, Free-surface and admittivity estimation in electrical impedance tomography,, International Journal for Numerical Methods in Engineering, 66 (2006), 1991.  doi: 10.1002/nme.1603.  Google Scholar

[44]

O.-P. Tossavainen, M. Vauhkonen, L. M. Heikkinen and T. Savolainen, Estimating shapes and free surfaces with EIT,, Measurement Science and Technology, 15 (2004), 1402.   Google Scholar

[45]

P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on thecomplete electrode model,, IEEE Transactions on Biomedical Engineering, 46 (1999), 1150.   Google Scholar

[46]

T. Vilhunen, J. P. Kaipio, P. J. Vauhkonen, T. Savolainen and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory,, Measurement Science and Technology, 13 (2002), 1848.  doi: 10.1088/0957-0233/13/12/307.  Google Scholar

[47]

R. A. Williams and M. S. Beck, Process Tomography: Principles, Techniques, and Applications,, Butterworth-Heinemann, (1995).   Google Scholar

show all references

References:
[1]

G. Alessandrini and E. Di Benedetto, Determining 2-dimensional cracks in 3-dimensional bodies: Uniqueness and stability,, Indiana Univ. Math. J., 46 (1997), 1.   Google Scholar

[2]

G. Alessandrini and L. Rondi, Stable determination of a crack in a planar inhomogeneous conductor,, SIAM Journal on Mathematical Analysis, 30 (1998), 326.  doi: 10.1137/S0036141097325502.  Google Scholar

[3]

D. Álvarez, O. Dorn, N. Irishina and M. Moscoso, Crack reconstruction using a level-set strategy,, Journal of Computational Physics, 228 (2009), 5710.  doi: 10.1016/j.jcp.2009.04.038.  Google Scholar

[4]

K. E. Andersen, S. P. Brooks and M. B. Hansen., A Bayesian approach to crack detection in electrically conducting media,, Inverse Problems, 17 (2001), 121.  doi: 10.1088/0266-5611/17/1/310.  Google Scholar

[5]

S. Andrieux, A. B. Abda and H. D. Bui., Reciprocity principle and crack identification,, Inverse Problems, 15 (1999), 59.  doi: 10.1088/0266-5611/15/1/010.  Google Scholar

[6]

T. Bannour, A. B. Abda and M. Jaoua, A semi-explicit algorithm for the reconstruction of 3D planar cracks,, Inverse Problems, 13 (1997), 899.  doi: 10.1088/0266-5611/13/4/002.  Google Scholar

[7]

Z. B. Bazant and J. Planas, Fracture and Size Effect in Concrete and Other Quasibrittle Materials (New Directions in Civil Engineering),, CRC Press, (1997).   Google Scholar

[8]

L. R. Bentley and M. Gharibi, Two-and three-dimensional electrical resistivity imaging at a heterogeneous remediation site,, Geophysics, 69 (2004), 674.  doi: 10.1190/1.1759453.  Google Scholar

[9]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements,, Mathematical Modelling and Numerical Analysis, 35 (2001), 595.  doi: 10.1051/m2an:2001128.  Google Scholar

[10]

K. Bryan and M. Vogelius, Reconstruction of multiple cracks from experimental electrostatic boundary measurements,, Inverse Problems and Optimal Design in Industry, 7 (1993), 147.   Google Scholar

[11]

K. Bryan and M. S. Vogelius, A review of selected works on crack identification,, Geometric Methods in Inverse Problems and PDE Control, 137 (2004), 25.  doi: 10.1007/978-1-4684-9375-7_3.  Google Scholar

[12]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85.  doi: 10.1137/S0036144598333613.  Google Scholar

[13]

K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Transactions on Biomedical Engineering, 36 (1989), 918.   Google Scholar

[14]

P. Church, J. E. McFee, S. Gagnon and P. Wort, Electrical impedance tomographic imaging of buried landmines,, IEEE Transactions on Geoscience and Remote Sensing, 44 (2006), 2407.  doi: 10.1109/TGRS.2006.873208.  Google Scholar

[15]

W. Daily, A. Ramirez, A. Binley and D. LeBrecque, Electrical resistance tomography,, The Leading Edge, 23 (2004).  doi: 10.1190/1.1729225.  Google Scholar

[16]

J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography,, SIAM Journal on Imaging Sciences, 6 (2013), 176.  doi: 10.1137/120877301.  Google Scholar

[17]

A. R. Elcrat and C. Hu, Determination of surface and interior cracks from electrostatic measurements using Schwarz-Christoffel transformations,, International Journal of Engineering Science, 34 (1996), 1165.  doi: 10.1016/0020-7225(96)00011-0.  Google Scholar

[18]

I. Frerichs, G. Hahn and G. Hellige, Thoracic electrical impedance tomographic measurements during volume controlled ventilation-effects of tidal volume and positive end-expiratory pressure,, IEEE Trans. Med. Imaging, 18 (1999), 764.  doi: 10.1109/42.802754.  Google Scholar

[19]

A. Friedman and M. Vogelius, Determining cracks by boundary measurements,, Indiana Univ. Math. J., 38 (1989), 527.  doi: 10.1512/iumj.1989.38.38025.  Google Scholar

[20]

F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement,, Inverse Problems, 14 (1998), 67.  doi: 10.1088/0266-5611/14/1/008.  Google Scholar

[21]

T. C. Hou and J. P. Lynch, Electrical Impedance Tomographic Methods for Sensing Strain Fields and Crack Damage in Cementitious Structures,, Journal of Intelligent Material Systems and Structures, (2008).  doi: 10.1177/1045389X08096052.  Google Scholar

[22]

N. Hyvönen, K. Karhunen and A. Seppänen, Fréchet derivative with respect to the shape of an internal electrode in Electrical Impedance Tomography,, SIAM Journal on Applied Mathematics, 70 (2010).  doi: 10.1137/09075929X.  Google Scholar

[23]

S. Järvenpää, Finite Element Model for the Inverse Conductivity Problem,, Phil. Lic. thesis, (1996).   Google Scholar

[24]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in EIT,, Inverse Problems, 16 (2000), 1487.  doi: 10.1088/0266-5611/16/5/321.  Google Scholar

[25]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, 15 (1999), 713.  doi: 10.1088/0266-5611/15/3/306.  Google Scholar

[26]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Springer Science+ Business Media, (2005).   Google Scholar

[27]

K. Karhunen, A. Seppänen, A. Lehikoinen, J. Blunt, J. P. Kaipio and P. J. M. Monteiro, Electrical resistance tomography for assessment of cracks in concrete., ACI Materials Journal, 107 (2010).   Google Scholar

[28]

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.  doi: 10.1016/j.cemconres.2009.08.023.  Google Scholar

[29]

H. Kim and J. K. Seo, Unique determination of a collection of a finite number of cracks from two boundary measurements,, SIAM Journal on Mathematical Analysis, 27 (1996), 1336.  doi: 10.1137/S0036141094275488.  Google Scholar

[30]

V. Kolehmainen, S. R. Arridge, W. R. B. Lionheart, M. Vauhkonen and J. P. Kaipio, Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,, Inverse Problems, 15 (1999), 1375.  doi: 10.1088/0266-5611/15/5/318.  Google Scholar

[31]

J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen and L. M. Heikkinen, Suitability of a PXI platform for an electrical impedance tomography system,, Measurement Science and Technology, 20 (2009).  doi: 10.1088/0957-0233/20/1/015503.  Google Scholar

[32]

P. W. A. Kunst, A. V. Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. De Vries, Ventilation and perfusion imaging by electrical impedance tomography: A comparison with radionuclide scanning,, Physiological Measurement, 19 (1998), 481.  doi: 10.1088/0967-3334/19/4/003.  Google Scholar

[33]

J. F. Lataste, C. Sirieix, D. Breysse and M. Frappa, Electrical resistivity measurement applied to cracking assessment on reinforced concrete structures in civil engineering,, NDT and E International, 36 (2003), 383.  doi: 10.1016/S0963-8695(03)00013-6.  Google Scholar

[34]

C. Lieberman, K. Willcox and O. Ghattas, Parameter and state model reduction for large-scale statistical inverse problems,, SIAM Journal on Scientific Computing, 32 (2010), 2523.  doi: 10.1137/090775622.  Google Scholar

[35]

V. Liepa, F. Santosa and M. Vogelius, Crack determination from boundary measurements-reconstruction using experimental data,, Journal of Nondestructive Evaluation, 12 (1993), 163.  doi: 10.1007/BF00567084.  Google Scholar

[36]

K. J. Loh, T-C Hou, J. P. Lynch and N. A Kotov, Carbon nanotube sensing skins for spatial strain and impact damage identification,, Journal of Nondestructuctive Evaluation, 28 (2009), 9.  doi: 10.1007/s10921-009-0043-y.  Google Scholar

[37]

P. R. McGillivray and D. W. Oldenburg, Methods for calculating Fréchet derivatives and sensitivities for the non-linear inverse problem: A comparative study,, Geophysical Prospecting, 38 (1990), 499.   Google Scholar

[38]

K. S. Osterman, T. E. Kerner, D. B. Williams, A. Hartov, S. P. Poplack and K. D. Paulsen, Multifrequency electrical impedance imaging: Preliminary in vivo experience in breast,, Physiological Measurement, 21 (2000), 99.   Google Scholar

[39]

R. B. Polder, Test methods for on site measurement of resistivity of concrete-a RILEM TC-154 technical recommendation,, Construction and building materials, 15 (2001), 125.  doi: 10.1016/S0950-0618(00)00061-1.  Google Scholar

[40]

F. Santosa and M. Vogelius, A computational algorithm to determine cracks from electrostatic boundary measurements,, International Journal of Engineering Science, 29 (1991), 917.  doi: 10.1016/0020-7225(91)90166-Z.  Google Scholar

[41]

J. Schöberl, NETGEN An advancing front 2D/3D-mesh generator based on abstract rules,, Computing and Visualization in Science, 1 (1997), 41.   Google Scholar

[42]

E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 1023.  doi: 10.1137/0152060.  Google Scholar

[43]

O.-P. Tossavainen, V. Kolehmainen and M. Vauhkonen, Free-surface and admittivity estimation in electrical impedance tomography,, International Journal for Numerical Methods in Engineering, 66 (2006), 1991.  doi: 10.1002/nme.1603.  Google Scholar

[44]

O.-P. Tossavainen, M. Vauhkonen, L. M. Heikkinen and T. Savolainen, Estimating shapes and free surfaces with EIT,, Measurement Science and Technology, 15 (2004), 1402.   Google Scholar

[45]

P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on thecomplete electrode model,, IEEE Transactions on Biomedical Engineering, 46 (1999), 1150.   Google Scholar

[46]

T. Vilhunen, J. P. Kaipio, P. J. Vauhkonen, T. Savolainen and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory,, Measurement Science and Technology, 13 (2002), 1848.  doi: 10.1088/0957-0233/13/12/307.  Google Scholar

[47]

R. A. Williams and M. S. Beck, Process Tomography: Principles, Techniques, and Applications,, Butterworth-Heinemann, (1995).   Google Scholar

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