A piecewise-constant binary model for electrical impedance tomography

Inverse Problems and Imaging

Volumn 1, Issue 2, 2007, Pages 423-435

  Tanushev, Nicolay M ^a   Vese, Luminita A ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Electrical impedance tomography; Functional minimization; Implicit curve evolution; Inverse problem; Regularization; Segmentation

Indexed keywords

EID: 70350258691     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2007.1.423     Document Type: Article

References (31)

1. L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems, Bol-lettino della Unione Matematica Italiana, 6B (1992), 105–123.
2. K. Astala and L. Paivarinta, Calderón’s inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265–299.
3. L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136.
4. L. Borcea, G.A. Gray and Y. Zhang, Variationally constrained numerical solution of electrical impedance tomography, Inverse Problems, 19 (2003), 1159–1184.
5. L. Borcea, J.G. Berryman, G.C. Papanicolaou, High-contrast impedance tomography, 12 (1996), 835–858.
6. R.M. Brown and G.A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Communications in Partial Differential Equations, 2 (1997), 1009–1027.
7. M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327–1355.
8. A.-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat. (1980), 65–73.
9. T.F. Chan and L. Vese, An active contour model without edges, LNCS, 1682 (1999), 141–151.
10. T.F. Chan and L.A. Vese, Active Contours Without Edges, IEEE Transactions on Image Processing, 10 (2001), 266–277.
11. T.F. Chan and X.-C. Tai, Level Set and Total Variation Regularization for Elliptic Inverse Problems With Discontinuous Coefficients,, UCLA CAM Report, 03-15, 2003.
12. E.T. Chung, T.F. Chan and X.C. Tai, Electrical impedance tomography using level set representation and total variational regularization, Journal of Computational Physics, 205 (2005), 357–372.
13. A. Dervieux and F. Thomasset, A finite element method for the simulation of Rayleigh-Taylor instability, Lecture Notes in Mathematics, 771 (1979), 145–159.
14. A. Dervieux and F. Thomasset, Multifluid incompressible flows by a finite element method, Lecture Notes in Physics, 141 (1980), 158–163.
15. R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Communications on Pure and Applied Mathematics, 37 (1984), 289–298.
16. R. Kohn and M. Vogelius, Determining conductivity by boundary measurements 2. Interior results, Communications on Pure and Applied Mathematics, 38 (1985), 643–667.
17. B. Kortschak, H. Wegleiter and B. Brandstatter, Formulation of cost functionals for different measurement principles in nonlinear capacitance tomography, Measurement Science & Technology, 18 (2007), 71–78.
18. B. Kortschak and B. Brandstatter, A FEM-BEM approach using level-sets in electrical capacitance tomography, COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 24 (2005), Sp. Iss. SI, 591–605.
19. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577–685.
20. A.I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71–96.
21. L.K. Nielsen, X.C. Tai, S.I. Aanonsen SI and M. Espedal, A binary level set model for elliptic inverse problems with discontinuous coefficients, International Journal of Numerical Analysis and Modeling, 4 (2007), 74–99.
22. S. Osher and J.A. Sethian, Fronts propagating with curvature-dependent speed - Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12–49.
23. L. Paivarinta, A. Panchenko and G. Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Revista Matematica Iberoamericana, 19 (2003), 57–72.
24. L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM-Control Optimisation and Calculus of Variations, 6 (2001), 517–538.
25. C. Samson, L. Blanc-Féraud, G. Aubert and J. Zerubia, A level set model for image classification, International Journal of Computer Vision, 40 (2000), 187–197.
26. M. Soleimani, O. Dorn and W.R.B. Lionheart, A narrow-band level set method applied to EIT in brain for cryosurgery monitoring, IEEE Transactions on Biomedical Engineering, 53 (2006), 2257–2264.
27. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary-value problem, Annals of Mathematics, 125 (1987), 153–169.
28. P. Neittaanmaki, J. Sprekels, and D. Tiba, “Optimization of Elliptic Systems: Theory and Applications,” Springer Monographs in Mathematics, 2005.
29. A. Tsai, A. Yezzi, and A.S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification, IEEE Transactions on Image Processing, 10 (2001), 1169–1186.
30. L.A. Vese and T.F. Chan, A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model, International Journal of Computer Vision, 50 (2002), 271–293.
31. H.K. Zhao, T. Chan, B. Merriman, and S. Osher, Variational level set approach to multiphase motion, Journal of Computational Physics, 127 (1996), 179–195.