Imaging cardiac activity by the D-bar method for electrical impedance tomography

Isaacson, David; Mueller, Jennifer L.; Newell, J.C.; Siltanen, Samuli

doi:10.1088/0967-3334/27/5/s04

Cited by 100 publications

(139 citation statements)

References 26 publications

Supporting

Mentioning

136

Contrasting

Unclassified

Order By: Relevance

“…We develop a re-formulation of the (Fourier transformed) ∂ equation based on the support properties of the solution for large k, which yields a once iterated variant of (8) and handles disparate support issues via careful shifting within the computational domain. Remarkably, this new equation is equivalent (up to a Fourier transform) of the iterated system of equations which Perry studies in ( [26]), and in fact can be effectively used for all values of k.…”

Section: (6)mentioning

confidence: 99%

See 1 more Smart Citation

Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

Abstract. We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.

show abstract

Section: (6)mentioning

confidence: 99%

“…In section 3 we present a short review on the existence theory for equation (8). In section 4 we discuss the numerical implementation of a regularization approach for |ξ| → 0 of the integrand in (8).…”

Section: (6)mentioning

confidence: 99%

Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…Isaacson et al demonstrated in [IMNS04,IMNS06] that the D-bar method performs well on practical data measured from laboratory phantoms and from human subjects. The mandatory regularization step was provided by low-pass filtering in the nonlinear frequency domain.…”

Section: (B)mentioning

confidence: 99%

“…2009(a): The numerical evidence from practical imaging experiments [IMNS04,IMNS06] inspired a rigorous regularization proof of convergence in the form of (1.4), see [KLMS09]. This gave an answer to (iii) and outlined a method for choosing the regularization parameter as the inverse of the nonlinear cutoff frequency.…”

Section: (B)mentioning

confidence: 99%

Nonlinear Inversion from Partial EIT Data:Computational Experiments

Hamilton¹,

Siltanen²

2014

Contemporary Mathematics

View full text Add to dashboard Cite

Abstract. Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small k frequencies.

show abstract

“…See Figure 2.2 for an example of an image of the heart and lungs obtained from an actual impedance imaging system. The articles [5] and [11] provide a good general overview of the subject.…”

Section: Impedance Tomographymentioning

confidence: 99%

Impedance Imaging, Inverse Problems, and Harry Potter's Cloak

Bryan¹,

Leise²

2010

SIAM Rev.

View full text Add to dashboard Cite

Abstract. In this article we provide an accessible account of the essential idea behind cloaking, aimed at nonspecialists and undergraduates who have had some vector calculus, Fourier series, and linear algebra. The goal of cloaking is to render an object invisible to detection from electromagnetic energy by surrounding the object with a specially engineered "metamaterial" that redirects electromagnetic waves around the object. We show how to cloak an object against detection from impedance tomography, an imaging technique of much recent interest, though the mathematical ideas apply to much more general forms of imaging. We also include some exercises and ideas for undergraduate research projects.

show abstract

Imaging cardiac activity by the D-bar method for electrical impedance tomography

Cited by 100 publications

References 26 publications

Spectral Approach to D‐bar Problems

Spectral Approach to D‐bar Problems

Nonlinear Inversion from Partial EIT Data:Computational Experiments

Impedance Imaging, Inverse Problems, and Harry Potter's Cloak

Contact Info

Product

Resources

About