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# Radial EIT with the D-bar method

## EIT with the D-bar method: smooth and radial case

Linear and Nonlinear Inverse Problems with Practical Applications
written by Jennifer Mueller and Samuli Siltanen and published by SIAM in 2012.

### Introduction

The D-bar method is a reconstruction method for the nonlinear inverse conductivity problem arising from
Electrical Impedance Tomography. This page contains Matlab routines implementing the D-bar method
for a smooth and rotationally symmetric conductivity.

Note carefully that although we use the rotational symmetry of the conductivity to speed up some computations,
the reconstruction process including the solution of the D-bar equation is two-dimensional, not one-dimensional.

### Definition of the example

The first example concerns a rotationally symmetric and smooth conductivity that equals one near the unit circle.
Outside the unit disc the conductivity has value 1.

The following file defines a rotationally symmetric and smooth conductivity in the unit disc: sigma.m.

Furthermore, this file implements the Schrödinger potential related to the conductivity: poten.m.

The Laplace operator appearing in the definition of the potential is implemented by finite differences in poten.m.
The following routines plot the conductivity and the potential, respectively: sigma_plot.mpoten_plot.m.
Please run the plot commands before continuing to make sure that everything is working properly.
You should see something like this:  ### Computation of the scattering transform via the Lippmann-Schwinger equation

Next we define a set of points in the k-plane for evaluating the scattering transform t(k).

Because of the rotational symmetry of this example, it is enough to choose k-values along the positive real axis:
the scattering transform is known in this case to be rotationally symmetric and real-valued.

The above file kvec_comp.m defines a set of k-points and saves them to a file called 'data/kvec.mat'.
(Note that kvec_comp.m creates a subdirectory called 'data'. If you already created it before, Matlab will show
a warning. However, you don't need to care about the warning.)

When running the example for the first time you might just use the file kvec_comp.m as it is.
Later you might want to modify it to choose a different set of k-values.

Now that we have decided on the k-points, it's time to evaluate the scattering transform. Here we do it first by
'cheating', or by knowing the actual conductivity, because then there are no ill-posed steps involved.
Later we will compute the scattering transform also honestly from (simulated) EIT measurements.
This is the file that evaluates the scattering transform t(k) at the k-points:

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