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Enhanced electrical impedance tomography via the Mumford-Shah functional. (English) Zbl 0989.35136

Summary: 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 ill-posed. 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 by several numerical examples. Our results indicate that this is an effective approach for overcoming the ill-posedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image.

MSC:

35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35R25 Ill-posed problems for PDEs
68U10 Computing methodologies for image processing
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