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April  2019, 13(2): 285-307. doi: 10.3934/ipi.2019015

Incorporating structural prior information and sparsity into EIT using parallel level sets

Ville Kolehmainen 1,, , Matthias J. Ehrhardt 2, and  Simon R. Arridge 3,

1. 

Department of Applied Physics, University of Eastern Finland, POB 1627, FI-70211 Kuopio, Finland
2. 

Institute for Mathematical Innovation, University of Bath, Bath BA2 7AY, UK
3. 

Centre for Medical Image Computing, University College London, Gower Street, London, WC1E 6BT, UK

* Corresponding author: Ville Kolehmainen

Received  September 2017 Revised  June 2018 Published  January 2019

EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not provide complete information we consider in addition a sparsity regularization for the edges appearing in the EIT image. The combination of these approaches is conveniently described through the parallel level sets approach. We present an overview of previous methods for structural regularisation and then provide a variational setting for our approach and explain the numerical implementation. We present results on simulations and experimental data for different cases with accurate and inaccurate prior information. The results demonstrate that the structural prior information improves the reconstruction accuracy, even in cases when there is reasonable uncertainty in the prior about the location of the edges or only partial edge information is available.

Keywords: Electrical impedance tomography, structural prior, computational inverse problem, regularization, finite element method.
Mathematics Subject Classification: Primary: 65N21, 65F22; Secondary: 65N30.
Citation: Ville Kolehmainen, Matthias J. Ehrhardt, Simon R. Arridge. Incorporating structural prior information and sparsity into EIT using parallel level sets. Inverse Problems and Imaging, 2019, 13 (2) : 285-307. doi: 10.3934/ipi.2019015
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

M. Alsaker and J. Mueller, A d-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM Journal on Imaging Sciences, 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.

[3]

S. R. Arridge, V. Kolehmainen and M. J. Schweiger, Reconstruction and Regularisation in Optical Tomography, in Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation, 2007.

[4]

B. Bai, Q. Li and R. M. Leahy, Magnetic Resonance-guided Positron Emission Tomography Image Reconstruction, Seminars in Nuclear Medicine, 43 (2013), 30-44.  doi: 10.1053/j.semnuclmed.2012.08.006.

[5]

C. Ballester, V. Caselles, L. Igual, J. Verdera and B. Rougé, A Variational Model for P+XS Image Fusion, International Journal of Computer Vision, 69 (2006), 43-58.  doi: 10.1007/s11263-006-6852-x.

[6]

C. Bathke, T. Kluth, C. Brandt and P. Maa, Improved image reconstruction in magnetic particle imaging using structural a priori information, International Journal on Magnetic Particle Imaging, 3, URL https://journal.iwmpi.org/index.php/iwmpi/article/view/64.

[7]

I. Bayram and M. E. Kamasak, A Directional Total Variation, IEEE Signal Processing Letters, 19 (2012), 781-784.  doi: 10.1109/LSP.2012.2220349.

[8]

A. Beck and M. Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.

[9]

A. Björck, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971484.

[10]

J. E. Bowsher, H. Yuan, L. W. Hedlund, T. G. Turkington, G. Akabani, A. Badea, W. C. Kurylo, C. T. Wheeler, G. P. Cofer, M. W. Dewhirst and G. A. Johnson, Utilizing MRI Information to Estimate F18-FDG Distributions in Rat Flank Tumors, in IEEE Nuclear Science Symposium and Medical Imaging Conference, 2004, 2488–2492, http://ieeexplore.ieee.org/xpl/login.jsp?arnumber=1462760 & http://ieeexplore.ieee.org/xpls/abs{_}all.jsp?arnumber=1462760. doi: 10.1109/NSSMIC.2004.1462760.

[11]

K. Bredies, Y. Dong and M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration, International Journal of Computer Mathematics, 90 (2013), 109-123.  doi: 10.1080/00207160.2012.700400.

[12]

L. Bungert, D. A. Coomes, M. J. Ehrhardt, J. Rasch, R. Reisenhofer and C.-B. Schönlieb, Blind image fusion for hyperspectral imaging with the directional total variation, Inverse Problems, 34 (2008), 044003, 23 pp, URL http://arXiv.org/abs/1710.05705. doi: 10.1088/1361-6420/aaaf63.

[13]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[14]

K.-S. Cheng, D. Isaacson, J. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions Biomed. Eng., 36 (1989), 918–924.

[15]

S. R. Cherry, Multimodality in vivo imaging systems: twice the power or double the trouble?, Annual Review of Biomedical Engineering, 8 (2006), 35-62.  doi: 10.1146/annurev.bioeng.8.061505.095728.

[16]

T. Dowrick, C. Blochet and D. Holder, In vivo bioimpedance measurement of healthy and ischaemic rat brain: implications for stroke imaging using electrical impedance tomography, Physiological Measurement, 36 (2015), 1273, http://stacks.iop.org/0967-3334/36/i=6/a=1273.

[17]

M. J. Ehrhardt and M. M. Betcke, Multi-contrast MRI reconstruction with structure-guided total variation, SIAM Journal on Imaging Sciences, 9 (2016), 1084-1106.  doi: 10.1137/15M1047325.

[18]

M. J. Ehrhardt, P. Markiewicz, M. Liljeroth, A. Barnes, V. Kolehmainen, J. Duncan, L. Pizarro, D. Atkinson, B. F. Hutton, S. Ourselin, K. Thielemans and S. R. Arridge, PET reconstruction with an anatomical MRI Prior using parallel level sets, IEEE Transactions on Medical Imaging, 35 (2016), 2189-2199, URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7452643. doi: 10.1109/TMI.2016.2549601.

[19]

V. Estellers, S. Soatto and X. Bresson, Adaptive Regularization With the Structure Tensor, IEEE Transactions on Image Processing, 24 (2015), 1777-1790.  doi: 10.1109/TIP.2015.2409562.

[20]

F. Fang, F. Li, C. Shen and G. Zhang, A Variational Approach for Pan-Sharpening, IEEE Transactions on Image Processing, 22 (2013), 2822-2834.  doi: 10.1109/TIP.2013.2258355.

[21]

I. Frerichs, M. B. P. Amato, A. H. van Kaam, D. G. Tingay, Z. Zhao, B. Grychtol, M. Bodenstein, H. Gagnon, S. H. Bohm, E. Teschner, O. Stenqvist, T. Mauri, V. Torsani, L. Camporota, A. Schibler, G. K. Wolf, D. Gommers, S. Leonhardt and A. Adler, Chest electrical impedance tomography examination, data analysis, terminology, clinical use and recommendations: consensus statement of the translational eit development study group, Thorax, 72 (2017), 83-93, URL http://thorax.bmj.com/content/72/1/83. doi: 10.1136/thoraxjnl-2016-208357.

[22]

M. Grasmair, Locally adaptive total variation regularization, in SSVM, LNCS, 5567 (2009), 331–342. doi: 10.1007/978-3-642-02256-2_28.

[23]

M. Grasmair and F. Lenzen, Anisotropic total variation filtering, Optimization, 62 (2010), 323-339.  doi: 10.1007/s00245-010-9105-x.

[24]

L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: Ii. laboratory experiments, Meas. Sci. Technol., 13 (2002), 1855-1861.  doi: 10.1088/0957-0233/13/12/308.

[25]

B. F. Hutton, B. A. Thomas, K. Erlandsson, A. Bousse, A. Reilhac-Laborde, D. Kazantsev, S. Pedemonte, K. Vunckx, S. R. Arridge and S. Ourselin, What Approach to Brain Partial Volume Correction is best for PET/MRI?, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 702 (2013), 29-33.  doi: 10.1016/j.nima.2012.07.059.

[26]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inv Probl, 16 (2000), 1487-1522.  doi: 10.1088/0266-5611/16/5/321.

[27]

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.

[28]

D. Kazantsev, W. R. B. Lionheart, P. J. Withers and P. D. Lee, Multimodal Image Reconstruction using Supplementary Structural Information in Total Variation Regularization, Sensing and Imaging, 15 (2014), 97. doi: 10.1007/s11220-014-0097-5.

[29]

F. Knoll, Y. Dong, C. Langskammer, M. Hintermüller and R. Stollberger, Total variation denoising with spatially dependent regularization, in Proc. Intl. Soc. Mag. Reson. Med., 18 (2010), 5088.

[30]

V. Kolehmainen, M. Lassas and P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE Transactions on Medical Imaging, 27 (2008), 1404-1414.  doi: 10.1109/ISBI.2006.1625120.

[31]

J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen and L. M. Heikkinen, Suitability of a pxi platform for an electrical impedance tomography system, Meas. Sci. Technol., 20 (2009), 015503. doi: 10.1088/0957-0233/20/1/015503.

[32]

R. M. Leahy and X. Yan, Incorporation of Anatomical MR Data for Improved Functional Imaging with PET, in Information Processing in Medical Imaging, Springer, 1991,105–120. doi: 10.1007/BFb0033746.

[33]

F. Lenzen and J. Berger, Solution-driven adaptive total variation regularization, in SSVM, 9087 (2015), 203-215, URL http://adsabs.harvard.edu/abs/2009LNCS.5567.....T＄\delimiter"026E30F＄n http://link.springer.com/10.1007/978-3-642-24785-9. doi: 10.1007/978-3-319-18461-6_17.

[34]

E. Malone, M. Jehl, S. Arridge, T. Betcke and D. Holder, Stroke type differentiation using spectrally constrained multifrequency eit: Evaluation of feasibility in a realistic head model, Physiological Measurement, 35 (2014), 1051. doi: 10.1088/0967-3334/35/6/1051.

[35]

N. Mandache, Exponential instability in an inverse problem for the schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[36]

M. Möller, T. Wittman, A. L. Bertozzi and M. Burger, A variational approach for sharpening high dimensional images, SIAM Journal on Imaging Sciences, 5 (2012), 150-178.  doi: 10.1137/100810356.

[37]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, New York, NY, USA, 2006.

[38]

J. Rasch, V. Kolehmainen, R. Nivajrvi, M. Kettunen, O. Grhn, M. Burger and E.-M. Brinkmann, Dynamic mri reconstruction from undersampled data with an anatomical prescan, Inverse Problems, 34 (2018), 074001, 30pp, URL http://stacks.iop.org/0266-5611/34/i=/a=074001. doi: 10.1088/1361-6420/aac3af.

[39]

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.

[40]

D. M. Strong, P. Blomgren and T. F. Chan, Spatially adaptive local feature-driven total variation minimizing image restoration, in Statistical and Stochastic Methods in Image Processing II, vol. 3167 of Proc. SPIE, 1997,222–233.

[41]

M. Vauhkonen, D. Vadasz, J. P. Kaipio, E. Somersalo and P. Karjalainen, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-293.  doi: 10.1109/42.700740.

[42]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. doi: 10.1137/1.9780898717570.

[43]

K. Vunckx, A. Atre, K. Baete, A. Reilhac, C. M. Deroose, K. Van Laere and J. Nuyts, Evaluation of three MRI-based anatomical priors for quantitative PET brain imaging, IEEE Transactions on Medical Imaging, 31 (2012), 599-612.  doi: 10.1109/TMI.2011.2173766.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

M. Alsaker and J. Mueller, A d-bar algorithm with a priori information for 2-dimensional electrical impedance tomography, SIAM Journal on Imaging Sciences, 9 (2016), 1619-1654.  doi: 10.1137/15M1020137.

[3]

S. R. Arridge, V. Kolehmainen and M. J. Schweiger, Reconstruction and Regularisation in Optical Tomography, in Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation, 2007.

[4]

B. Bai, Q. Li and R. M. Leahy, Magnetic Resonance-guided Positron Emission Tomography Image Reconstruction, Seminars in Nuclear Medicine, 43 (2013), 30-44.  doi: 10.1053/j.semnuclmed.2012.08.006.

[5]

C. Ballester, V. Caselles, L. Igual, J. Verdera and B. Rougé, A Variational Model for P+XS Image Fusion, International Journal of Computer Vision, 69 (2006), 43-58.  doi: 10.1007/s11263-006-6852-x.

[6]

C. Bathke, T. Kluth, C. Brandt and P. Maa, Improved image reconstruction in magnetic particle imaging using structural a priori information, International Journal on Magnetic Particle Imaging, 3, URL https://journal.iwmpi.org/index.php/iwmpi/article/view/64.

[7]

I. Bayram and M. E. Kamasak, A Directional Total Variation, IEEE Signal Processing Letters, 19 (2012), 781-784.  doi: 10.1109/LSP.2012.2220349.

[8]

A. Beck and M. Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.

[9]

A. Björck, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971484.

[10]

J. E. Bowsher, H. Yuan, L. W. Hedlund, T. G. Turkington, G. Akabani, A. Badea, W. C. Kurylo, C. T. Wheeler, G. P. Cofer, M. W. Dewhirst and G. A. Johnson, Utilizing MRI Information to Estimate F18-FDG Distributions in Rat Flank Tumors, in IEEE Nuclear Science Symposium and Medical Imaging Conference, 2004, 2488–2492, http://ieeexplore.ieee.org/xpl/login.jsp?arnumber=1462760 & http://ieeexplore.ieee.org/xpls/abs{_}all.jsp?arnumber=1462760. doi: 10.1109/NSSMIC.2004.1462760.

[11]

K. Bredies, Y. Dong and M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration, International Journal of Computer Mathematics, 90 (2013), 109-123.  doi: 10.1080/00207160.2012.700400.

[12]

L. Bungert, D. A. Coomes, M. J. Ehrhardt, J. Rasch, R. Reisenhofer and C.-B. Schönlieb, Blind image fusion for hyperspectral imaging with the directional total variation, Inverse Problems, 34 (2008), 044003, 23 pp, URL http://arXiv.org/abs/1710.05705. doi: 10.1088/1361-6420/aaaf63.

[13]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[14]

K.-S. Cheng, D. Isaacson, J. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions Biomed. Eng., 36 (1989), 918–924.

[15]

S. R. Cherry, Multimodality in vivo imaging systems: twice the power or double the trouble?, Annual Review of Biomedical Engineering, 8 (2006), 35-62.  doi: 10.1146/annurev.bioeng.8.061505.095728.

[16]

T. Dowrick, C. Blochet and D. Holder, In vivo bioimpedance measurement of healthy and ischaemic rat brain: implications for stroke imaging using electrical impedance tomography, Physiological Measurement, 36 (2015), 1273, http://stacks.iop.org/0967-3334/36/i=6/a=1273.

[17]

M. J. Ehrhardt and M. M. Betcke, Multi-contrast MRI reconstruction with structure-guided total variation, SIAM Journal on Imaging Sciences, 9 (2016), 1084-1106.  doi: 10.1137/15M1047325.

[18]

M. J. Ehrhardt, P. Markiewicz, M. Liljeroth, A. Barnes, V. Kolehmainen, J. Duncan, L. Pizarro, D. Atkinson, B. F. Hutton, S. Ourselin, K. Thielemans and S. R. Arridge, PET reconstruction with an anatomical MRI Prior using parallel level sets, IEEE Transactions on Medical Imaging, 35 (2016), 2189-2199, URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7452643. doi: 10.1109/TMI.2016.2549601.

[19]

V. Estellers, S. Soatto and X. Bresson, Adaptive Regularization With the Structure Tensor, IEEE Transactions on Image Processing, 24 (2015), 1777-1790.  doi: 10.1109/TIP.2015.2409562.

[20]

F. Fang, F. Li, C. Shen and G. Zhang, A Variational Approach for Pan-Sharpening, IEEE Transactions on Image Processing, 22 (2013), 2822-2834.  doi: 10.1109/TIP.2013.2258355.

[21]

I. Frerichs, M. B. P. Amato, A. H. van Kaam, D. G. Tingay, Z. Zhao, B. Grychtol, M. Bodenstein, H. Gagnon, S. H. Bohm, E. Teschner, O. Stenqvist, T. Mauri, V. Torsani, L. Camporota, A. Schibler, G. K. Wolf, D. Gommers, S. Leonhardt and A. Adler, Chest electrical impedance tomography examination, data analysis, terminology, clinical use and recommendations: consensus statement of the translational eit development study group, Thorax, 72 (2017), 83-93, URL http://thorax.bmj.com/content/72/1/83. doi: 10.1136/thoraxjnl-2016-208357.

[22]

M. Grasmair, Locally adaptive total variation regularization, in SSVM, LNCS, 5567 (2009), 331–342. doi: 10.1007/978-3-642-02256-2_28.

[23]

M. Grasmair and F. Lenzen, Anisotropic total variation filtering, Optimization, 62 (2010), 323-339.  doi: 10.1007/s00245-010-9105-x.

[24]

L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: Ii. laboratory experiments, Meas. Sci. Technol., 13 (2002), 1855-1861.  doi: 10.1088/0957-0233/13/12/308.

[25]

B. F. Hutton, B. A. Thomas, K. Erlandsson, A. Bousse, A. Reilhac-Laborde, D. Kazantsev, S. Pedemonte, K. Vunckx, S. R. Arridge and S. Ourselin, What Approach to Brain Partial Volume Correction is best for PET/MRI?, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 702 (2013), 29-33.  doi: 10.1016/j.nima.2012.07.059.

[26]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inv Probl, 16 (2000), 1487-1522.  doi: 10.1088/0266-5611/16/5/321.

[27]

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.

[28]

D. Kazantsev, W. R. B. Lionheart, P. J. Withers and P. D. Lee, Multimodal Image Reconstruction using Supplementary Structural Information in Total Variation Regularization, Sensing and Imaging, 15 (2014), 97. doi: 10.1007/s11220-014-0097-5.

[29]

F. Knoll, Y. Dong, C. Langskammer, M. Hintermüller and R. Stollberger, Total variation denoising with spatially dependent regularization, in Proc. Intl. Soc. Mag. Reson. Med., 18 (2010), 5088.

[30]

V. Kolehmainen, M. Lassas and P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE Transactions on Medical Imaging, 27 (2008), 1404-1414.  doi: 10.1109/ISBI.2006.1625120.

[31]

J. Kourunen, T. Savolainen, A. Lehikoinen, M. Vauhkonen and L. M. Heikkinen, Suitability of a pxi platform for an electrical impedance tomography system, Meas. Sci. Technol., 20 (2009), 015503. doi: 10.1088/0957-0233/20/1/015503.

[32]

R. M. Leahy and X. Yan, Incorporation of Anatomical MR Data for Improved Functional Imaging with PET, in Information Processing in Medical Imaging, Springer, 1991,105–120. doi: 10.1007/BFb0033746.

[33]

F. Lenzen and J. Berger, Solution-driven adaptive total variation regularization, in SSVM, 9087 (2015), 203-215, URL http://adsabs.harvard.edu/abs/2009LNCS.5567.....T＄\delimiter"026E30F＄n http://link.springer.com/10.1007/978-3-642-24785-9. doi: 10.1007/978-3-319-18461-6_17.

[34]

E. Malone, M. Jehl, S. Arridge, T. Betcke and D. Holder, Stroke type differentiation using spectrally constrained multifrequency eit: Evaluation of feasibility in a realistic head model, Physiological Measurement, 35 (2014), 1051. doi: 10.1088/0967-3334/35/6/1051.

[35]

N. Mandache, Exponential instability in an inverse problem for the schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.

[36]

M. Möller, T. Wittman, A. L. Bertozzi and M. Burger, A variational approach for sharpening high dimensional images, SIAM Journal on Imaging Sciences, 5 (2012), 150-178.  doi: 10.1137/100810356.

[37]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, New York, NY, USA, 2006.

[38]

J. Rasch, V. Kolehmainen, R. Nivajrvi, M. Kettunen, O. Grhn, M. Burger and E.-M. Brinkmann, Dynamic mri reconstruction from undersampled data with an anatomical prescan, Inverse Problems, 34 (2018), 074001, 30pp, URL http://stacks.iop.org/0266-5611/34/i=/a=074001. doi: 10.1088/1361-6420/aac3af.

[39]

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.

[40]

D. M. Strong, P. Blomgren and T. F. Chan, Spatially adaptive local feature-driven total variation minimizing image restoration, in Statistical and Stochastic Methods in Image Processing II, vol. 3167 of Proc. SPIE, 1997,222–233.

[41]

M. Vauhkonen, D. Vadasz, J. P. Kaipio, E. Somersalo and P. Karjalainen, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-293.  doi: 10.1109/42.700740.

[42]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. doi: 10.1137/1.9780898717570.

[43]

K. Vunckx, A. Atre, K. Baete, A. Reilhac, C. M. Deroose, K. Van Laere and J. Nuyts, Evaluation of three MRI-based anatomical priors for quantitative PET brain imaging, IEEE Transactions on Medical Imaging, 31 (2012), 599-612.  doi: 10.1109/TMI.2011.2173766.

Figure 1.  Numerical experiment (Case 1). Rows top to bottom: $ \sigma_{{\rm true}} $ and reference images $ p(r) $ (top row), weighting function $ \gamma(r) $ (second row), reconstructions using $ {\rm SH}_1 $ regularisation (third row) and STV regularisation (fourth row)
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Figure 2.  Plot of standard deviation with respect to bias of the reconstructed conductivity with different values of the regularisation parameter $ \alpha $ for the simulation in figure 1. The left image shows the curves for the mean conductivity in the area of the true inclusion on the top in $ {\sigma}_{\rm true} $, and the image on the right for the inclusion on the bottom right. The triangle denotes the point corresponding to the smallest value of $ \alpha $ in the curves
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Figure 3.  Numerical experiment (Case 2). Rows top to bottom: $ \sigma_{{\rm true}} $ and reference images $ p(r) $ (top row), weighting function $ \gamma(r) $ (second row), reconstructions using $ {\rm SH}_1 $ regularisation (third row) and STV regularisation (fourth row)
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Figure 4.  Numerical experiment (Case 3). Rows top to bottom: $ \sigma_{{\rm true}} $ and reference images $ p(r) $ (top row), weighting function $ \gamma(r) $ (second row), reconstructions using $ {\rm SH}_1 $ regularisation (third row) and STV regularisation (fourth row)
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Figure 5.  Numerical experiment (Case 3): Reconstructions with increasing uncertainty about the edge location in the partial edge information. Rows top to bottom: $ \sigma_{{\rm true}} $ and reference images $ p(r) $ (top row), weighting function $ \gamma(r) $ (second row), reconstructions using $ {\rm SH}_1 $ regularisation (third row) and STV regularisation (fourth row)
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Figure 6.  Numerical experiment (Case 4). Rows top to bottom: $ \sigma_{{\rm true}} $ and reference images $ p(r) $ (top row), weighting function $ \gamma(r) $ (second row), reconstructions using $ {\rm SH}_1 $ regularisation (third row) and STV regularisation (fourth row)
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Figure 7.  Physical experiment. Top section: Photograph of the target and the reference images $ p(r) $. The second row shows the weighting functions $ \gamma(r) $. Bottom section: Reconstructions. $ {\rm SH}_1 $ regularisation (third row), STV regularisation (fourth row). (Color scales of the reconstructions are arbitrary in the sense that they are reconstructed 2D values from 3D data)
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Table 1.  Examples of $ \psi $ for different regularisation schemes in variational form. $ \psi $ is the mapping which defines the penalty for the gradient magnitude in (4) and $ \kappa $ is the corresponding local diffusivity function in (6).
$ \psi(t) $ $ \kappa(t) $
$ 1^{\rm st} $-order Tikhonov $ \frac{ t ^2}2 $ $ 1 $
TV $ t $ $ \frac{1}{ t } $
Smoothed TV $ T\left( t ^2 + T^2\right)^{1/2} -T^2 $ $ T \left( t ^2 + T^2\right)^{-1/2} $
Perona-Malik (1) $ \frac{T^2}2 \log\left(1 + \frac{ t ^2}{T^2}\right) $ $ T^2 \left( t ^2 + T^2\right)^{-1} $
Perona-Malik (2) $ \frac{T^2}2 \left[1 - \exp\left(-\frac{ t ^2}{T^2}\right)\right] $ $ \exp\left(- \frac{ t ^2}{T^2}\right) $
Huber $\left\{ \begin{array}{l} Tt - \frac{{{T^2}}}{2}\\ \frac{{{t^2}}}{2} \end{array} \right.$ $ \frac{T}{ t } $
1		 if $ t > T $
else
Tukey $\left\{ \begin{array}{l} \frac{{{T^2}}}{6}\\ \frac{{{T^2}}}{6}\left[ {1 - {{\left( {1 - \frac{{{t^2}}}{{{T^2}}}} \right)}^3}} \right] \end{array} \right.$ $ 0 $
$ \left(1 - \frac{ t ^2}{T^2}\right)^2 $ 		if $ t> T $
else
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$ \psi(t) $ $ \kappa(t) $
$ 1^{\rm st} $-order Tikhonov $ \frac{ t ^2}2 $ $ 1 $
TV $ t $ $ \frac{1}{ t } $
Smoothed TV $ T\left( t ^2 + T^2\right)^{1/2} -T^2 $ $ T \left( t ^2 + T^2\right)^{-1/2} $
Perona-Malik (1) $ \frac{T^2}2 \log\left(1 + \frac{ t ^2}{T^2}\right) $ $ T^2 \left( t ^2 + T^2\right)^{-1} $
Perona-Malik (2) $ \frac{T^2}2 \left[1 - \exp\left(-\frac{ t ^2}{T^2}\right)\right] $ $ \exp\left(- \frac{ t ^2}{T^2}\right) $
Huber $\left\{ \begin{array}{l} Tt - \frac{{{T^2}}}{2}\\ \frac{{{t^2}}}{2} \end{array} \right.$ $ \frac{T}{ t } $
1		 if $ t > T $
else
Tukey $\left\{ \begin{array}{l} \frac{{{T^2}}}{6}\\ \frac{{{T^2}}}{6}\left[ {1 - {{\left( {1 - \frac{{{t^2}}}{{{T^2}}}} \right)}^3}} \right] \end{array} \right.$ $ 0 $
$ \left(1 - \frac{ t ^2}{T^2}\right)^2 $ 		if $ t> T $
else
Table 2.  Reconstruction errors (28) for the simulated test cases for varying regularizations ($ {\rm SH}_1 $, STV) and reference images (no structure, correct, partial). Cases 1-4 refer to reconstructions in the figures 1, 3, 4 and 6 respectively. Errors are given in percentages
SH1 STV
case no structure correct partial no structure correct partial
1 12.3 3.6 8.6 8.8 3.5 4.9
2 15.6 5.6 10.9 13.8 3.3 10.1
3 10.6 3.5 6.5 8.0 2.4 4.9
4 15.3 11.5 12.9 14.3 11.1 12.7
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SH1 STV
case no structure correct partial no structure correct partial
1 12.3 3.6 8.6 8.8 3.5 4.9
2 15.6 5.6 10.9 13.8 3.3 10.1
3 10.6 3.5 6.5 8.0 2.4 4.9
4 15.3 11.5 12.9 14.3 11.1 12.7
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