On the resolving power of electrical impedance tomography

Sabatier, Pierre; Sebu, Cristiana

doi:10.1088/0266-5611/23/5/007

Cited by 7 publications

(3 citation statements)

References 23 publications

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“…The optimal choice for the regularization parameter, γ, as given by the L-curve criterion, is approximately 0.1. This is in agreement with the theoretical minimum spatial resolution achievable using our integral equation approach which was derived in [34].…”

Section: Numerical Examplessupporting

confidence: 90%

A regularized solution for the inverse conductivity problem using mollifiers

Maaß

Pidcock

Sebu

2009

Inverse Problems in Science and Engineering

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In this paper we present a reconstruction method for the inverse conductivity problem suitable for smooth conductivity distributions. The inverse problem is reformulated in terms of a pair of coupled integral equations, one of which is of first kind which we regularize using mollifier methods. An interesting feature of this method is that the kernel of this integral equation is not given, but can be modified for the choice of mollifier. We are able to obtain conductivity reconstructions rapidly and without relying on accurate a priori information.

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Section: Numerical Examplessupporting

confidence: 90%

A regularized solution for the inverse conductivity problem using mollifiers

Maaß

Pidcock

Sebu

2009

Inverse Problems in Science and Engineering

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“…Nevertheless it serves as a prototypical example for operator valued inverse problems andmore importantly-it has shown some relevance for certain real life applications [12,125,58,129,64]. Moreover, first results for sparsity constrained reconstructions from real data presented below show good potential for further development.…”

Section: Eit Reconstruction Methodsmentioning

confidence: 93%

Sparsity regularization for parameter identification problems

Jin¹,

Maaß²

2012

Inverse Problems

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The investigation of regularization schemes with sparsity promoting penalty terms has been one of the dominant topics in the field of inverse problems over the last years, and Tikhonov functionals with p -penalty terms for 1 p 2 have been studied extensively. The first investigations focused on regularization properties of the minimizers of such functionals with linear operators and on iteration schemes for approximating the minimizers. These results were quickly transferred to nonlinear operator equations, including nonsmooth operators and more general function space settings. The latest results on regularization properties additionally assume a sparse representation of the true solution as well as generalized source conditions, which yield some surprising and optimal convergence rates. The regularization theory with p sparsity constraints is relatively complete in this setting; see the first part of this review. In contrast, the development of efficient numerical schemes for approximating minimizers of Tikhonov functionals with sparsity constraints for nonlinear operators is still ongoing. The basic iterated soft shrinkage approach has been extended in several directions and semi-smooth Newton methods are becoming applicable in this field. In particular, the extension to more general non-convex, non-differentiable functionals by variational principles leads to a variety of generalized iteration schemes. We focus on such iteration schemes in the second part of this review. A major part of this survey is devoted to applying sparsity constrained regularization techniques to parameter identification problems for partial differential equations, which we regard as the prototypical setting for nonlinear inverse problems. Parameter identification problems exhibit different levels of complexity and we aim at characterizing a hierarchy of such problems. The operator defining these inverse problems is the parameter-to-state mapping. We first summarize some general analytic properties derived from the weak formulation of the underlying differential equation, and then analyze several concrete parameter identification problems in detail. Naturally, it is not possible

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“…A detailed survey of the mathematical studies on the EIT problem can be found in [54]. EIT has now found applications in many areas, such as oil and geophysical prospection [58], medical imaging [28], physiological measurement [29], early diagnosis of breast cancer [38,59], monitoring of pulmonary functions [31] and detection of leaks from buried pipes [37], as well as many other applications [6,24,27,53]. Our application of EIT is the detection of some embedded inclusions in a known homogeneous background, e.g., one may think of air bubbles, cracks, defects or impurities in an otherwise homogeneous medium of building material or biological tissue.…”

Section: Introductionmentioning

confidence: 99%

A direct sampling method for electrical impedance tomography

2014

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This work investigates the electrical impedance tomography (EIT) problem in the case when only one or two pairs of Cauchy data is available, which is known to be very difficult in achieving high reconstruction quality owing to its severely ill-posed nature. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneous inclusions. A new probing function based on the dipole potential is introduced to construct an indicator function for imaging the inclusions. Explicit formulae for the probing and indicator functions are derived in the case when the sampling domain is of spherical geometry in  n (n = 2, 3). This new method is easy to implement and computationally cheap. Numerical experiments are presented to demonstrate the robustness and effectiveness of the DSM, which provides a new numerical approach for solving the EIT problem.

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On the resolving power of electrical impedance tomography

Cited by 7 publications

References 23 publications

A regularized solution for the inverse conductivity problem using mollifiers

A regularized solution for the inverse conductivity problem using mollifiers

Sparsity regularization for parameter identification problems

A direct sampling method for electrical impedance tomography

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