Discontinuous Galerkin Method for Electrical Impedance Tomography Based on Sparsity Regularization with Adam

Abstract

In this work, we develop a novel sparsity reconstruction algorithm for Electrical Impedance Tomography (EIT), focusing on problems where the region of interest contains relatively simple, localized inhomogeneities. The optimization framework employs $l^1$-norm regularization to promote sparsity, combined with the soft shrinkage algorithm. Our key ingredient is incorporating the Adam (Adaptive Moment Estimation) algorithm for adaptive step sizes and directions, enhancing both convergence speed and accuracy. The optimization problem is solved numerically using a high-order Discontinuous Galerkin (DG) method with quadratic polynomials. Numerical results demonstrate the efficiency and accuracy of the proposed sparsity reconstruction method with Adam, with comparisons to the smoothness regularization also presented.

Keywords

Electrical impedance tomography (EIT) / Sparsity / Adam / Discontinuous Galerkin (DG) method / Partial data / 35R30 / 65J20 / 65N21

References

[1] Adler A, Guardo R. A neural network image reconstruction technique for electrical impedance tomography. IEEE Trans. Med. Imaging, 1994, 13: 594-600

[2] Alessandrini G. Stable determination of conductivity by boundary measurements. Appl. Anal., 1988, 27: 153-172.

[3] Alessandrini G, Vessella S. Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math., 2005, 35: 207-241.

[4] Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 1982, 39: 742-760.

[5] Astala K, Päivärinta L. Calderón’s inverse conductivity problem in the plane. Ann. Math., 2006, 163: 265-299.

[6] Bao G, Ye X, Zang Y, Zhou H. Numerical solution of inverse problems by weak adversarial networks. Inverse Prob., 2020, 36: 1-31.

[7] Bar L, Sochen N. Strong solutions for PDE-based tomography by unsupervised learning. SIAM J. Imaging Sci., 2021, 14(1): 128-155.

[8] Barceló T, Faraco D, Ruiz A. Stability of Calderón’s inverse problem in the plane. Journal des Mathématiques Pures et Appliquées, 2007, 88: 522-556

[9] Barzilai J, Borwein JM. Two-point step size gradient methods. IMA J. Numer. Anal., 1988, 8(1): 141-148.

[10] Bonesky T, Bredies K, Lorenz DA, Maass P. A generalized conditional gradient method for nonlinear operator equations with sparsity constraints. Inverse Prob., 2007, 23(5): 2041.

[11] Borcea L. Electrical impedance tomography. Inverse Prob., 2002, 18: R99-R136.

[12] Borcea L, Gray G, Zhang Y. Variationally constrained numerical solution of electrical impedance tomography. Inverse Prob., 2003, 19: 1159-1184.

[13] Bredies K, Lorenz DA, Maass P. A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl., 2009, 42: 173-193.

[14] Calderón AP. On an inverse boundary value problem. Comput. Appl. Math., 2006, 25: 133-138.

[15] Cheney M, Isaacson D, Newell JC. Electrical impedance tomography. SIAM Rev., 1999, 41: 85-101.

[16] Cheney, M., Isaacson, D., Newell, J.C., Simske, S., Goble, J.: NOSER: an algorithm for solving the inverse conductivity problem. Int. J. Imaging Syst. Technol. 2, 66–75 (1990)

[17] Chow YT, Ito K, Zou J. A direct sampling method for electrical impedance tomography. Inverse Prob., 2014, 30: 1-25.

[18] Chung ET, Chan TF, Tai X-C. Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys., 2005, 205: 357-372.

[19] Cockburn B, Dong B. An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput., 2007, 32: 233-262.

[20] Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds) Discontinuous Galerkin Methods: Theory, Computation and Applications, vol. 11. Springer (2000)

[21] Cockburn B, Shu C-W. Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput., 2005, 22–23: 1-3

[22] Costa E, Chaves C, Gomes S, Beraldo M, Volpe M, Tucci M, Schettino I, Bohm S, Carvalho C, Tanaka H, Lima R, Amato M. Real-time detection of pneumothorax using electrical impedance tomography. Crit. Care Med., 2008, 36: 1230-1238

[23] Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

[24] Dawson, C.: Foreword for the special issue on discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 195(25/26/27/28), 3183 (2006)

[25] Dobson DC. Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math., 1992, 52(2): 442-458.

[26] Dobson DC, Santosa F. An image-enhancement technique for electrical impedance tomography. Inverse Prob., 1994, 10(2): 317.

[27] Dozat, T.: Incorporating Nesterov momentum into Adam. In: Proc, ICLR (2016)

[28] Fan Y, Ying L. Solving electrical impedance tomography with deep learning. J. Comput. Phys., 2020, 404: 1-19.

[29] Garde H, Knudsen K. Sparsity prior for electrical impedance tomography with partial data. Inverse Probl. Sci. Eng., 2016, 24(3): 524-541.

[30] Guo R, Jiang J. Construct deep neural networks based on direct sampling methods for solving electrical impedance tomography. SIAM J. Sci. Comput., 2021, 43(3): B678-B711.

[31] Hanke M. Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numer. Funct. Anal. Optim., 1997, 18: 971-993.

[32] Isaacson, D., Newell, J.C., Goble, J.C., Cheney, M.: Thoracic impedance images during ventilation. In: Proceedings of the Twelfth Annual International Conference of the IEEE Engineering in Medicine and Biology Society 12, 106–107 (1990)

[33] Jin B, Khan T, Maass P. A reconstruction algorithm for electrical impedance tomography based on sparsity regularization. Int. J. Numer. Methods Eng., 2012, 89: 337-353.

[34] Jin B, Maass P. An analysis of electrical impedance tomography with applications to Tikhonov regularization. ESAIM Control Optim. Calc. Var., 2012, 18: 1027-1048.

[35] Jin B, Maass P. Sparsity regularization for parameter identification problems. Inverse Prob., 2012, 28: 1-70.

[36] Jordana J, Gasulla M, Pallás-Areny R. Electrical resistance tomography to detect leaks from buried pipes. Meas. Sci. Technol., 2001, 12: 1061-1068

[37] Kaipio J, Kolehmainen V, Somersalo E, Vauhkonen M. Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Prob., 2000, 16: 1487-1522.

[38] Kaipio J, Somersalo EStatistical and Computational Inverse Problems, 2005, New York: Springer

[39] Kaipio JP, Kolehmainen V, Vauhkonen M, Somersalo E. Inverse problems with structural prior information. Inverse Prob.1999, 15(3): 713.

[40] Khan TA, Ling SH. Review on electrical impedance tomography: artificial intelligence methods and its applications. Algorithms, 2019, 12: 1-18

[41] Kingma, D., Ba, J.: Adam: a method for stochastic optimization. arXiv:1412.6980 (2014)

[42] Kirsch A, Grinberg NThe Factorization Method for Inverse Problems, 2008, Oxford: Oxford University Press

[43] Klibanov M, Li J, Zhang W. Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data. Inverse Prob.2019, 35: 1-33.

[44] Knudsen, K., Lassas, M., Mueller, J.L., Siltanen, S.: Regularized D-bar method for the inverse conductivity problem. Inverse Probl. Imaging 3, 599–624 (2009)

[45] Kohn R, Vogelius M. Relaxation of a variational method for impedance computed tomography. Commun. Pure Appl. Math., 1987, 40(6): 745-777.

[46] Lange, K.: MM Optimization Algorithms. SIAM, Philadelphia (2016)

[47] Lechleiter A, Rieder A. Newton regularizations for impedance tomography: a numerical study. Inverse Prob., 2006, 22: 1967-1987.

[48] Li X, Wang W. A high order discontinuous Galerkin method for the recovery of the conductivity in electrical impedance tomography. J. Comput. Appl. Math., 2023, 434115344

[49] Li X, Wang W. An analysis of discontinuous Galerkin method for electrical impedance tomography with partial data. J. Comput. Appl. Math., 2025, 459116376

[50] Lorenz DA. Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl., 2008, 16(5): 463-478.

[51] Lukaschewitsch M, Maass P, Pidcock M. Tikhonov regularization for electrical impedance tomography on unbounded domains. Inverse Prob., 2003, 19(3): 585.

[52] Mandache N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Prob., 2001, 17: 1435-1444.

[53] Nachman A. Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math., 1996, 143: 71-96.

[54] Ramirez A, Daily W, Binley B, LaBreque D, Roelant D. Detection of leaks in underground storage tanks using electrical resistance methods. J. Environ. Eng. Geophys., 1996, 1: 189-203

[55] Ramlau R, Teschke G. A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints. Numer. Math., 2006, 104: 177-203.

[56] Rondi L, Santosa F. Enhanced electrical impedance tomography via the Mumford-Shah functional. ESAIM Control Optim. Calc. Var., 2001, 6: 517-538.

[57] Siltanen S, Mueller J, Isaacson D. An implementation of the reconstruction algorithm of a Nachman for the 2D inverse conductivity problem. Inverse Prob., 2000, 16: 681-699.

[58] Sylvester J. A convergent layer stripping algorithm for the radially symmetric impedance tomography problem. Comm. Part. Differ. Equ., 1992, 17: 1955-1994.

[59] Sylvester J, Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann. Math., 1987, 125: 153-169.

[60] Tanyu DN, Ning J, Hauptmann A, Jin B, Maass P. Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches. Data-driven Models Inverse Probl., 2024, 31: 437

[61] Uhlmann G. Inverse problems: seeing the unseen. Bull. Math. Sci., 2014, 4: 209-279.

[62] Vauhkonen PJ, Vauhkonen M, Savolainen T, Kaipio JP. Three-dimensional electrical impedance tomography based on the complete electrode model. IEEE Trans. Biomed. Eng., 1999, 46(9): 1150-1160

[63] Wright SJ, Nowak RD, Figueiredo MA. Sparse reconstruction by separable approximation. IEEE Trans. Signal Process., 2009, 57(7): 2479-2493.

[64] Zou Y, Guo Z. A review of electrical impedance techniques for breast cancer detection. Med. Eng. Phys., 2003, 25: 79-90

Rights & permissions

Shanghai University