# Enhanced Electrical Impedance Tomography via the Mumford–Shah Functional

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 6, page 517-538
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topRondi, Luca, and Santosa, Fadil. "Enhanced Electrical Impedance Tomography via the Mumford–Shah Functional." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 517-538. <http://eudml.org/doc/116576>.

@article{Rondi2010,

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.
},

author = {Rondi, Luca, Santosa, Fadil},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Electrical impedance tomography; inverse problems for elliptic equations; regularization of
illposed problem; image enhancement.; electrical impedance tomography; regularization; image enhancement},

language = {eng},

month = {3},

pages = {517-538},

publisher = {EDP Sciences},

title = {Enhanced Electrical Impedance Tomography via the Mumford–Shah Functional},

url = {http://eudml.org/doc/116576},

volume = {6},

year = {2010},

}

TY - JOUR

AU - Rondi, Luca

AU - Santosa, Fadil

TI - Enhanced Electrical Impedance Tomography via the Mumford–Shah Functional

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 6

SP - 517

EP - 538

AB -
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.

LA - eng

KW - Electrical impedance tomography; inverse problems for elliptic equations; regularization of
illposed problem; image enhancement.; electrical impedance tomography; regularization; image enhancement

UR - http://eudml.org/doc/116576

ER -

## References

top- L. Ambrosio, A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B3 (1989) 857-881. Zbl0767.49001
- L. Ambrosio, Existence theory for a new class of variational problems. Arch. Rational Mech. Anal.111 (1990) 291-322. Zbl0711.49064
- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Clarendon Press, Oxford (2000). Zbl0957.49001
- L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via$\Gamma $-convergence. Comm. Pure Appl. Math.43 (1990) 999-1036. Zbl0722.49020
- L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problem. Boll. Un. Mat. Ital. B6 (1992) 105-123. Zbl0776.49029
- A. Blake and A. Zisserman, Visual Reconstruction. The MIT Press, Cambridge Mass, London (1987).
- E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with ``touching'' fibers of circular cross-section. SIAM J. Math. Anal.31 (2000) 651-677. Zbl0947.35044
- A. Braides, Approximation of Free-Discontinuity Problems. Springer-Verlag, Berlin Heidelberg New York (1998). Zbl0909.49001
- 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.
- G. Congedo and I. Tamanini, On the existence of solutions to a problem in multidimensional segmentation. Ann. Inst. H. Poincaré Anal. Non Linéaire8 (1991) 175-195. Zbl0729.49003
- G. Dal Maso, An Introduction to $\Gamma $-convergence. Birkhäuser, Boston Basel Berlin (1993). Zbl0816.49001
- E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal.108 (1989) 195-218. Zbl0682.49002
- D.C. Dobson, Stability and Regularity of an Inverse Elliptic Boundary Value Problem, Ph.D. Thesis. Rice University, Houston (1990).
- D.C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography. Inverse Problems10 (1994) 317-334. Zbl0805.35149
- L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton Ann Arbor London (1992). Zbl0804.28001
- V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, New York Berlin Heidelberg (1998). Zbl0908.35134
- E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston Basel Stuttgart (1984). Zbl0545.49018
- D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York London Toronto (1980). Zbl0457.35001
- R.V. Kohn and M. Vogelius, Determining conductivity by boundary measurements. Comm. Pure Appl. Math.37 (1984) 289-298. Zbl0586.35089
- Y.Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Rational Mech. Anal.153 (2000) 91-151. Zbl0958.35060
- N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci.17 (1963) 189-205. Zbl0127.31904
- 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.
- D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42 (1989) 577-685. Zbl0691.49036
- J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem. Ann. Math.125 (1987) 153-169. Zbl0625.35078
- G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York London (1987). Zbl0655.35002

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.