Direct numerical reconstruction of conductivities in three dimensions using scattering transforms

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.Mathematics subject classification (MSC2000): 35R30, 35C20

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“…Many recent works have been devoted to the reconstruction of radially symmetric conductivities. See, for instance, [17,33,37].…”

Section: Numerical Illustrationmentioning

confidence: 99%

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

Ammari

Deng

Kang

et al. 2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…A crude approximation of t is obtained by substituting the asymptotic value of ψ ζ (x) ≈ e ix·ζ for ψ ζ in (6). This was suggested for the 2D problem in [9,10] and also used in previous numerical implementations for the 3D problem [15,16]. The resulting approximation of t is denoted by t exp , i.e.…”

Section: The Reconstruction Algorithmsmentioning

confidence: 99%

“…Calderón's reconstruction method was also implemented for the 3D problem in [15,16] and in [29], where the method was considered in a 3D mammography geometry, adapted to a setting of electrode measurements taken on part of the boundary, and tested on tank data.…”

Section: The Calderón Inversionmentioning

confidence: 99%

“…To compute a matrix approximation of the Dirichlet-to-Neumann map we use the fact that the spherical harmonics are eigenfunctions of the map. The eigenvalues can then be computing as explained in [15]. The approximations of the Fourier transformq are plotted in the left panel of figure 1 and compared to the exact Fourier transform and the exact t for minimal ζ min .…”

Section: Radial Profilementioning

confidence: 99%

“…Several ideas from these implementations were used recently in a crude simplification of the reconstruction method for 3D [15,16]. More recently the Calderón problem in 2D was solved in full generality [17] and implemeted numerically in [18].…”

Section: Introductionmentioning

confidence: 99%

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Electrical impedance tomography: 3D reconstructions using scattering transforms

Delbary

Hansen

Knudsen

2012

Applicable Analysis

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In three dimensions the Calderón problem was addressed and solved in theory in the 1980s. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper a new simplification of the algorithm is suggested. The method is based on solving a boundary integral equation for the complex geometrical optics solutions, and the method is implemented numerically using a Nyström method. Convergence estimates are obtained using hyperinterpolation operators. We compare the method numerically to two other approximations by evaluation on two numerical examples. In addition a moment method for the numerical solution of the forward problem is given.

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Piecewise Polynomial Inversion of the Radon Transform in Three Space Dimensions via Plane Integration and Applications in Positron Emission Tomography

Protonotarios

Kastis

Δικαίος

et al. 2021

Nonlinear Analysis, Differential Equations, and Applications

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Direct numerical reconstruction of conductivities in three dimensions using scattering transforms

Cited by 41 publications

References 34 publications

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

Electrical impedance tomography: 3D reconstructions using scattering transforms

Piecewise Polynomial Inversion of the Radon Transform in Three Space Dimensions via Plane Integration and Applications in Positron Emission Tomography

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