Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-20T20:30:07.455Z Has data issue: false hasContentIssue false
  • English
  • Français

Enhanced Electrical ImpedanceTomography via the Mumford–Shah Functional

Published online by Cambridge University Press:  15 August 2002

Luca Rondi  and
Fadil Santosa
Luca Rondi
Affiliation:
Institut für Industriemathematik/SFB , Johannes Kepler Universität Linz, Austria. : School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Fadil Santosa
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.; santosa@math.umn.edu.
Get access

Abstract

We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford–Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an effective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image.

Keywords

Electrical impedance tomographyinverse problems for elliptic equationsregularization of illposed problemimage enhancement.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 517 - 538
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosio, L., A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B 3 (1989) 857-881.
Ambrosio, L., Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111 (1990) 291-322. CrossRef
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Clarendon Press, Oxford (2000).
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$ -convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. CrossRef
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problem. Boll. Un. Mat. Ital. B 6 (1992) 105-123.
A. Blake and A. Zisserman, Visual Reconstruction. The MIT Press, Cambridge Mass, London (1987).
Bonnetier, E. and Vogelius, M., An elliptic regularity result for a composite medium with ``touching'' fibers of circular cross-section. SIAM J. Math. Anal. 31 (2000) 651-677. CrossRef
A. Braides, Approximation of Free-Discontinuity Problems. Springer-Verlag, Berlin Heidelberg New York (1998).
A.P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics. Sociedade Brasileira de Matemática, Rio de Janeiro (1980) 65-73.
Congedo, G. and Tamanini, I., On the existence of solutions to a problem in multidimensional segmentation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 175-195. CrossRef
G. Dal Maso, An Introduction to $\Gamma$ -convergence. Birkhäuser, Boston Basel Berlin (1993).
De Giorgi, E., Carriero, M. and Leaci, A., Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195-218. CrossRef
D.C. Dobson, Stability and Regularity of an Inverse Elliptic Boundary Value Problem, Ph.D. Thesis. Rice University, Houston (1990).
Dobson, D.C. and Santosa, F., An image-enhancement technique for electrical impedance tomography. Inverse Problems 10 (1994) 317-334. CrossRef
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton Ann Arbor London (1992).
V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, New York Berlin Heidelberg (1998).
E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston Basel Stuttgart (1984).
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York London Toronto (1980).
Kohn, R.V. and Vogelius, M., Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37 (1984) 289-298. CrossRef
Li, Y.Y. and Vogelius, M., Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Rational Mech. Anal. 153 (2000) 91-151. CrossRef
Meyers, N.G., An L p -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963) 189-205.
D. Mumford and J. Shah, Boundary detection by minimizing functionals, I, in Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Press/North-Holland, Silver Spring Md./Amsterdam (1985) 22-26.
Mumford, D. and Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577-685. CrossRef
Sylvester, J. and Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125 (1987) 153-169. CrossRef
G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York London (1987).