Sparsity prior for electrical impedance tomography with partial data

References

• Holder DS. Electrical impedance tomography: methods, history and applications. Bristol: IOP Publishing Ltd; 2005.
   Google Scholar
• Abubakar A, Habashy TM, Li M, Liu J. Inversion algorithms for large-scale geophysical electromagnetic measurements. Inverse Prob. 2009;25:30 p. Article ID 123012.
   Web of Science ®Google Scholar
• York TA. Status of electrical tomography in industrial applications. Proc. SPIE Int. Soc. Opt. Eng. 2001;4188:175–190.
   Google Scholar
• Calderón AP. On an inverse boundary value problem. In: Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980). Rio de Janeiro: Sociedade Brasiliera de Matemática; 1980. p. 65–73.
   Google Scholar
• Sylvester J, Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2). 1987;125:153–169.
   Web of Science ®Google Scholar
• Nachman AI. Reconstructions from boundary measurements. Ann. Math. (2). 1988;128:531–576.
   Web of Science ®Google Scholar
• Novikov RG. A multidimensional inverse spectral problem for the equation -Δψ+(u(x)-Eu(x))ψ=0. Funktsional. Anal. i Prilozhen. 1988;22:11–22, 96.
   Google Scholar
• Nachman AI. Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. (2). 1996;143:71–96.
   Web of Science ®Google Scholar
• Astala K, Päivärinta L. Calderón’s inverse conductivity problem in the plane. Ann. Math. (2). 2006;163:265–299.
   Web of Science ®Google Scholar
• Haberman B, Tataru D. Uniqueness in Calderón’s problem with Lipschitz conductivities. Duke Math. J. 2013;162:496–516.
   Web of Science ®Google Scholar
• Uhlmann G. Electrical impedance tomography and Calderón’s problem. Inverse Prob. 2009;25:39 p. Article ID 123011.
   Web of Science ®Google Scholar
• Alessandrini G. Stable determination of conductivity by boundary measurements. Appl. Anal. 1988;27:153–172.
   Google Scholar
• Alessandrini G. Singular solutions of elliptic equations and the determination of conductivity by boundary measurements. J. Differ. Equ. 1990;84:252–272.
   Web of Science ®Google Scholar
• Mandache N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Prob. 2001;17:1435–1444.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Knudsen K, Lassas M, Mueller JL, Siltanen S. Regularized D-bar method for the inverse conductivity problem. Inverse Prob. Imaging. 2009;3:599–624.
   Web of Science ®Google Scholar
• Bikowski J, Knudsen K, Mueller JL. Direct numerical reconstruction of conductivities in three dimensions using scattering transforms. Inverse Prob. 2011;27:015002, 19.
   Web of Science ®Google Scholar
• Delbary F, Hansen PC, Knudsen K. Electrical impedance tomography: 3D reconstructions using scattering transforms. Appl. Anal. 2012;91:737–755.
   Web of Science ®Google Scholar
• Delbary F, Knudsen K. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Prob. Imaging. 2014;8:991–1012.
   Web of Science ®Google Scholar
• Bukhgeim AL, Uhlmann G. Recovering a potential from partial Cauchy data. Comm. Partial Differ. Equ. 2002;27:653–668.
   Web of Science ®Google Scholar
• Kenig CE, Sjöstrand J, Uhlmann G. The Calderón problem with partial data. Ann. Math (2). 2007;165:567–591.
   Web of Science ®Google Scholar
• Knudsen K. The Calderón problem with partial data for less smooth conductivities. Comm. Partial Differ. Equ. 2006;31:57–71.
   Web of Science ®Google Scholar
• Zhang G. Uniqueness in the Calderón problem with partial data for less smooth conductivities. Inverse Prob. 2012;28:105008, 18.
   Web of Science ®Google Scholar
• Isakov V. On uniqueness in the inverse conductivity problem with local data. Inverse Prob. Imaging. 2007;1:95–105.
   Web of Science ®Google Scholar
• Imanuvilov OY, Uhlmann G, Yamamoto M. The Calderón problem with partial data in two dimensions. J. Am. Math. Soc. 2010;23:655–691.
   Web of Science ®Google Scholar
• Heck H, Wang JN. Stability estimates for the inverse boundary value problem by partial Cauchy data. Inverse Prob. 2006;22:1787–1796.
   Web of Science ®Google Scholar
• Hamilton SJ, Siltanen S. Nonlinear inversion from partial EIT data: computational experiments. In: Stefanov P, Zworski M, editors. Inverse problems and applications. Vol. 615, Contemporary mathematics. Providence (RI): American Mathematical Society; 2014. p. 105–129.
   Google Scholar
• Mueller JL, Siltanen S. Linear and nonlinear inverse problems with practical applications. Vol. 10, Computational science & engineering. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2012.
   Google Scholar
• Somersalo E, Cheney M, Isaacson D. Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 1992;52:1023–1040.
   Web of Science ®Google Scholar
• Gehre M, Kluth T, Lipponen A, Jin B, Seppänen A, Kaipio JP, Maass P. Sparsity reconstruction in electrical impedance tomography: an experimental evaluation. J. Comput. Appl. Math. 2012;236:2126–2136.
   Web of Science ®Google Scholar
• Gehre M, Kluth T, Sebu C, Maass P. Sparse 3D reconstructions in electrical impedance tomography using real data. Inverse Prob. Sci. Eng. 2014;22:31–44.
   Web of Science ®Google Scholar
• Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 2004;57:1413–1457.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Bonesky T, Bredies K, Lorenz DA, Maass P. A generalized conditional gradient method for nonlinear operator equations with sparsity constraints. Inverse Prob. 2007;23:2041–2058.
   Web of Science ®Google Scholar
• Jin B, Maass P. An analysis of electrical impedance tomography with applications to Tikhonov regularization. ESAIM: Control Optim. Calcul. Varia. 2012;18:1027–1048.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Borcea L. Electrical impedance tomography. Inverse Prob. 2002;18:R99–R136.
   Web of Science ®Google Scholar
• Garde H, Knudsen K. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Submitted, 2014. Available from: http://arxiv.org/abs/1412.6288
   Google Scholar
• Adams RA, Fournier JJF. Sobolev spaces. 2nd ed. Vol. 140, Pure and applied mathematics (Amsterdam). Amsterdam: Elsevier/Academic Press; 2003.
   Google Scholar
• Neuberger JW. Sobolev gradients and differential equations. 2nd ed. Vol. 1670, Lecture notes in mathematics. Berlin: Springer-Verlag; 2010.
   Google Scholar
• Barzilai J, Borwein JM. Two-point step size gradient methods. IMA J. Numer. Anal. 1988;8:141–148.
   Web of Science ®Google Scholar
• Wright SJ, Nowak RD, Figueiredo MAT. Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 2009;57:2479–2493.
   Web of Science ®Google Scholar
• Stampacchia G. Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. lnst. Fourier. 1965;15:189–257.
   Google Scholar
• Kirsch A, Grinberg N. The factorization method for inverse problems. Vol. 36, Oxford lecture series in mathematics and its applications. Oxford: Oxford University Press; 2008.
   Google Scholar
• von Harrach B, Ullrich M. Monotonicity-based shape reconstruction in electrical impedance tomography. SIAM J. Math. Anal. 2013;45:3382–3403.
   Web of Science ®Google Scholar
• von Harrach B, Seo JK. Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J. Math. Anal. 2010;42:1505–1518.
   Web of Science ®Google Scholar
• Logg A, Mardal KA, Wells GN. Automated solution of differential equations by the finite element method. Vol. 84, Lecture notes in computational science and engineering. Heidelberg: Springer; 2012; the FEniCS book.
   Google Scholar
• Borsic A, Graham BM, Adler A, Lionheart W. In vivo impedance imaging with total variation regularization. IEEE Trans. Med. Imaging. 2010;29:44–54.
   PubMed Web of Science ®Google Scholar