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On finding the surface admittance of an obstacle via the time domain enclosure method
Incorporating structural prior information and sparsity into EIT using parallel level sets
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |







TV | |||
Smoothed TV | |||
Perona-Malik (1) | |||
Perona-Malik (2) | |||
Huber | 1 |
if else |
|
Tukey | if else |
TV | |||
Smoothed TV | |||
Perona-Malik (1) | |||
Perona-Malik (2) | |||
Huber | 1 |
if else |
|
Tukey | if else |
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 |
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 |
[1] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
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Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 |
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Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 |
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Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
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Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
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Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074 |
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Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008 |
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Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
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Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 |
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Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021021 |
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Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
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Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
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Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021018 |
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Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021022 |
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Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021076 |
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Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021082 |
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Andreas Neubauer. On Tikhonov-type regularization with approximated penalty terms. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021027 |
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Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021046 |
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Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021046 |
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Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032 |
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