18 research outputs found
Reconstructing conductivities with boundary corrected D-bar method
The aim of electrical impedance tomography is to form an image of the
conductivity distribution inside an unknown body using electric boundary
measurements. The computation of the image from measurement data is a
non-linear ill-posed inverse problem and calls for a special regularized
algorithm. One such algorithm, the so-called D-bar method, is improved in this
work by introducing new computational steps that remove the so far necessary
requirement that the conductivity should be constant near the boundary. The
numerical experiments presented suggest two conclusions. First, for most
conductivities arising in medical imaging, it seems the previous approach of
using a best possible constant near the boundary is sufficient. Second, for
conductivities that have high contrast features at the boundary, the new
approach produces reconstructions with smaller quantitative error and with
better visual quality
Incorporating a Spatial Prior into Nonlinear D-Bar EIT imaging for Complex Admittivities
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
filtering of nonlinear (nonphysical) Fourier transform data specific 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 filter 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.Comment: 18 pages, 10 figure
Direct EIT Reconstructions of Complex Admittivities on a Chest-Shaped Domain in 2-D
Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, hyperinflation, and pleural effusion. The method demonstrates robustness in the presence of noise. Reconstructions from trigonometric and pairwise current injection patterns are included
A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D
A direct reconstruction algorithm for complex conductivities in
, where is a bounded, simply connected Lipschitz
domain in , is presented. The framework is based on the
uniqueness proof by Francini [Inverse Problems 20 2000], but equations relating
the Dirichlet-to-Neumann to the scattering transform and the exponentially
growing solutions are not present in that work, and are derived here. The
algorithm constitutes the first D-bar method for the reconstruction of
conductivities and permittivities in two dimensions. Reconstructions of
numerically simulated chest phantoms with discontinuities at the organ
boundaries are included.Comment: This is an author-created, un-copyedited version of an article
accepted for publication in [insert name of journal]. IOP Publishing Ltd is
not responsible for any errors or omissions in this version of the manuscript
or any version derived from it. The Version of Record is available online at
10.1088/0266-5611/28/9/09500
A Hybrid Segmentation and D-bar Method for Electrical Impedance Tomography
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 D-bar 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
Reconstruction of piecewise constant layered conductivities in electrical impedance tomography
This work presents a new constructive uniqueness proof for Calder\'on's
inverse problem of electrical impedance tomography, subject to local Cauchy
data, for a large class of piecewise constant conductivities that we call
"piecewise constant layered conductivities" (PCLC). The resulting
reconstruction method only relies on the physically intuitive monotonicity
principles of the local Neumann-to-Dirichlet map, and therefore the method
lends itself well to efficient numerical implementation and generalization to
electrode models. Several direct reconstruction methods exist for the related
problem of inclusion detection, however they share the property that "holes in
inclusions" or "inclusions-within-inclusions" cannot be determined. One such
method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the
method presented here is a modified variant of the monotonicity method which
overcomes this problem. More precisely, the presented method abuses that a PCLC
type conductivity can be decomposed into nested layers of positive and/or
negative perturbations that, layer-by-layer, can be determined via the
monotonicity method. The conductivity values on each layer are found via basic
one-dimensional optimization problems constrained by monotonicity relations.Comment: 12 pages, 1 figur