## Enhanced Electrical Impedance Tomography via the Mumford-Shah Functional (2001)

Venue: | ESAIM: Control, Optimization and Calculus of Variations |

Citations: | 14 - 0 self |

### BibTeX

@ARTICLE{Rondi01enhancedelectrical,

author = {Luca Rondi and Fadil Santosa},

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

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

year = {2001},

volume = {6},

pages = {517--538}

}

### OpenURL

### 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 MumfordShah 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 eective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image. 1 Introduction and formulation of the problem The purpose of this work is to demonstrate that the Mumford-Shah functional from image processing can be used eectively to regularize the classical problem of electrical impedance tomography. In electrical impedance tomogr...

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Citation Context ...( and for any 1 the following estimate holds kuk H 1;q( 1 ) C kuk H 1(s+ kfk L q(s+ khk L q( (2.1) where the constant C depends on , n, q, 1 , and . The following result, due to N.G. Meyers, [M], states that any satisfying (1.1) with a constant , 0s1, veries the Q-property for a constant Q > 2 depending on and n only. Theorem 2.2 (Meyers) Let be a bounded Lipschitz domain contained in ... |

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Citation Context ...rst time by A. P. Calderon in [C], consists in determining if the conductivity 2 is uniquely determined by the associated Neumann-to-Dirichlet map. The pioneering work of R.V. Kohn and M. Vogelius [K=-=o-V-=-], and later, the work of J. Sylvester and G. Uhlmann [Sy-U], provided the denitive uniqueness results on this problem. For an overview discussion on uniqueness we refer to a recent monograph by V. Is... |

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Citation Context ... is closed ins 2 H 1( nK) and r = 0 almost everywhere in nK. The existence of a solution to the Mumford-Shah minimization problem has been obtained in the framework of free-discontinuity problem by [D=-=G-Ca-L]-=- introducing a relaxed functional in a space of functions of bounded variation. By the direct method in the Calculus of Variations a minimum of such a relaxed functional does exist and, by a regularit... |

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Citation Context ....1)-(3.2), we will use an approximation procedure to construct the solutions. We recall here just the denition and some basic properties of -convergence. For a more detailed introduction we refer to [=-=DM-=-]. Let (X; d) be a metric space. Then a sequence F k : X 7! [1;+1] - converges as k ! 1 to a function F : X 7! [1;+1] if for every x 2 X we have (i) for every x k converging to x we have F (x) lim in... |

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Citation Context ...later, the work of J. Sylvester and G. Uhlmann [Sy-U], provided the denitive uniqueness results on this problem. For an overview discussion on uniqueness we refer to a recent monograph by V. Isakov, [=-=I-=-]. In this paper we shall deal with the reconstruction issue, that is, we are looking for a procedure which allows us to recover the unknown conductivity from the knowledge, possibly partial, of the ... |

11 |
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Citation Context ... at the cost of loss of resolution. In this work, we explore the use of techniques from image processing, which stabilizes and enhances the reconstruction. An earlier work of D. Dobson and F. Santosa =-=[D-S]-=- employed minimal total variation (TV) criterion, which is popular in image processing, to stabilize and enhance the reconstruction for electrical impedance tomography. In the present work, we also bo... |

7 |
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Citation Context ...oduces also a minimum for the original functional. Moreover they showed that if (; K) solves the variational problem, then is indeed piecewise C 1 , that is 2 C 1( nK). In a similar framework in [Co=-=-Ta]-=- the existence of a solution has been proved also for the minimal partition problem (1.6). The computational problem associated with these functionals is an active area of research. A main diculty is ... |

7 |
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Citation Context ...Dirichlet map as a function of the conductivity will be needed in the sequel to establish existence results for the minimization problems stated above. We follow some of the techniques developed in [=-=D-=-] where it is shown that higher integrability properties of the solutions to problem (1.2) are decisive for proving dierentiability properties of the Neumann-to-Dirichlet map. In order to lower as muc... |