A data-driven edge-preserving D-bar method for electrical impedance tomography

Hamilton, Sarah J.; Hauptmann, Andreas; Siltanen, Samuli

doi:10.3934/ipi.2014.8.1053

Cited by 9 publications

(14 citation statements)

References 38 publications

(64 reference statements)

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“…This motivates the use of a scattering transform computed from the forward problem for the prior in an annulus outside the disk of the experimental scattering data. This approach differs significantl from the methods based on post-processing D-bar conductivity images [45], [46]. …”

Section: Methodsmentioning

confidence: 99%

Incorporating a Spatial Prior into Nonlinear D-Bar EIT Imaging for Complex Admittivities

Hamilton

Mueller

Alsaker

2017

IEEE Trans. Med. Imaging

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Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filterin of nonlinear (nonphysical) Fourier transform data specifi to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass filte to include higher frequency components in the reconstruction, in particular, features that are known with high confidence This information is additionally included in the system of D-bar equations with an independent regularization parameter from that of the extended scattering transform. In this paper, this approach is used in the 2-D D-bar method for admittivity (conductivity as well as permittivity) EIT imaging. Noise-robust reconstructions are presented for simulated EIT data on chest-shaped phantoms with a simulated pneumothorax and pleural effusion. No assumption of the pathology is used in the construction of the prior, yet the method still produces significant enhancements of the underlying pathology (pneumothorax or pleural effusion) even in the presence of strong noise.

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Section: Methodsmentioning

confidence: 99%

Incorporating a Spatial Prior into Nonlinear D-Bar EIT Imaging for Complex Admittivities

Hamilton

Mueller

Alsaker

2017

IEEE Trans. Med. Imaging

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“…The CGO solutions can be of particular interest, since they contain nonlinear information of the inverse conductivity problem and can be used as a data fidelity term as shown in [17]. If one is instead just interested in computing the reconstruction, the approximate scattering transform can be directly obtained from…”

Section: 3mentioning

confidence: 99%

Direct inversion from partial-boundary data in electrical impedance tomography

2017

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In Electrical Impedance Tomography (EIT) one wants to image the conductivity distribution of a body from current and voltage measurements carried out on its boundary. In this paper we consider the underlying mathematical model, the inverse conductivity problem, in two dimensions and under the realistic assumption that only a part of the boundary is accessible to measurements. In this framework our data are modeled as a partial Neumann-to-Dirichlet map (ND map). We compare this data to the full-boundary ND map and prove that the error depends linearly on the size of the missing part of the boundary. The same linear dependence is further proved for the difference of the reconstructed conductivities -from partial and full boundary data. The reconstruction is based on a truncated and linearized D-bar method. Auxiliary results include an extrapolation method to estimate the full-boundary data from the measured one, an approximation of the complex geometrical optics solutions computed directly from the ND map as well as an approximate scattering transform for reconstructing the conductivity. Numerical verification of the convergence results and reconstructions are presented for simulated test cases.Neumann-to-Dirichlet data. This is not true any more for partial-boundary data, where the graphs of the two operators represent different subsets of the Cauchy data. This can be further emphasized by the fact that the partialboundary Dirichlet problem is nonphysical in many applications. That is, given a noninsulating body (e.g. a human), applied voltages on a part of the boundary will immediately distribute to the full boundary. Thus, partially supported Dirichlet data do not represent a common physical problem. On the other hand, if one injects current only on a subset Γ ⊂ ∂Ω, the current will stay zero on ∂Ω \Γ. We stress that even in this setting the resulting voltage distribution will be supported on the full-boundary. This is an essential problem for the measurements: we need in our representation the measurement information on ∂Ω. This limitation is overcome by an extrapolation procedure of the measured data, as we will discuss in Section 2.2.Theoretical results for the inverse conductivity problem with partial boundary data mainly concentrate on the uniqueness question. That means, does it follow from infinite-precision data that the conductivities are equal? Uniqueness has been proved in several cases for the Dirichlet-to-Neumann problem, including the important works [4,12,25,28,30]. A thorough survey of these results can be found in [29]. For the more physical Neumann-to-Dirichlet problem there are just a few uniqueness results published. In particular for C 2 conductivities and coinciding measurement and input domains in R 2 by [26], in higher dimensions in [19], and for different input and measurement domains in [9]. A more pratical case with bisweep data has been addressed in [22]. We would like to note that these results are of great importance for the theoretical understanding, but are so far not readily appl...

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“…The method proposed in this paper is very different from all of these approaches as it uses a proven regularized D-bar image as its starting point and futher uses the inverse scattering of the D-bar methodology to guide the image segmentation. The proposed method differs from [33], where the CGO sinogram was introduced, by instead focusing on enlarging the scattering radius to produce more accurate conductivity reconstructions.…”

Section: Mathematical Model and Literature Reviewmentioning

confidence: 99%

“…The Beltrami CGO sinogram. We extend the concept of the CGO sinogram, introduced in [33], to discontinuous conductivities. Set…”

Section: Data-driven Contrast Adjustmentsmentioning

confidence: 99%

See 1 more Smart Citation

A Hybrid Segmentation and D-Bar Method for Electrical Impedance Tomography

Hamilton¹,

Reyes²,

Siltanen

et al. 2016

SIAM J. Imaging Sci.

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Abstract. The Regularized D-bar method for Electrical Impedance Tomography provides a rigorous mathematical approach for solving the full nonlinear inverse problem directly, i.e. without iterations. It is based on a low-pass filtering in the (nonlinear) frequency domain. However, the resulting D-bar reconstructions are inherently smoothed leading to a loss of edge distinction. In this paper, a novel approach that combines the rigor of the Dbar approach with the edge-preserving nature of Total Variation regularization is presented. The method also includes a data-driven contrast adjustment technique guided by the key functions (CGO solutions) of the D-bar method. The new TV-Enhanced D-bar Method produces reconstructions with sharper edges and improved contrast while still solving the full nonlinear problem. This is achieved by using the TV-induced edges to increase the truncation radius of the scattering data in the nonlinear frequency domain thereby increasing the radius of the low pass filter. The algorithm is tested on numerically simulated noisy EIT data and demonstrates significant improvements in edge preservation and contrast which can be highly valuable for absolute EIT imaging.

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A data-driven edge-preserving D-bar method for electrical impedance tomography

Cited by 9 publications

References 38 publications

Incorporating a Spatial Prior into Nonlinear D-Bar EIT Imaging for Complex Admittivities

Incorporating a Spatial Prior into Nonlinear D-Bar EIT Imaging for Complex Admittivities

Direct inversion from partial-boundary data in electrical impedance tomography

A Hybrid Segmentation and D-Bar Method for Electrical Impedance Tomography

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