×

Direct electrical impedance tomography for nonsmooth conductivities. (English) Zbl 1237.78014

The paper presents a new reconstruction algorithm for electrical impedance tomography in dimension two, based on the constructive uniqueness proof for Calderón’s inverse conductivity problem given by the first and the third author in [“Calderón’s inverse conductivity problem in the plane”, Ann. Math. (2) 163, No. 1, 265–299 (2006; Zbl 1111.35004)]. The method is applicable to the class of piecewise smooth conductivities, which is important for the reconstruction of a cross-section of a pipeline or a human chest, for example. It solves the full nonlinear EIT problem in an explicitly regularized fashion directly with no iterations.
The method consists of three steps. First, the traces of complex geometric optics (CGO) solutions are recovered from the DtN map by solving an ill-posed boundary integral equation. Then one approximates the CGO solutions inside the unit disk using a low-pass transport matrix, which is computed employing a fast solver of the Beltrami equation. Finally, the approximate conductivity is computed from the CGO solutions inside the unit disk using numerical differentiation and simple algebra. To demonstrate the performance of the algorithm, four numerical examples on simulated discontinuous conductivity distributions with applications in medical imaging and process tomography are given. The results suggest that the method can recover reasonably accurate EIT images from data of realistic measurement noise.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35R30 Inverse problems for PDEs
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
30C62 Quasiconformal mappings in the complex plane
78A05 Geometric optics
78M20 Finite difference methods applied to problems in optics and electromagnetic theory

Citations:

Zbl 1111.35004
PDFBibTeX XMLCite
Full Text: DOI