In Electrical Impedance Tomography the conductivity of an object is estimated from
boundary measurements. An array of electrodes is attached to the surface of the object
and current stimuli are applied via these electrodes. The resulting voltages are measured.
The process of estimating the conductivity as a function of space inside the object from
voltage measurements at the surface is called reconstruction. Mathematically the ElT
reconstruction is a non linear inverse problem, the stable solution of which requires regularisation
methods. Most common regularisation methods impose that the reconstructed image should
be smooth. Such methods confer stability to the reconstruction process, but limit the
capability of describing sharp variations in the sought parameter.
In this thesis two new methods of regularisation are proposed. The first method, Gallssian
anisotropic regularisation, enhances the reconstruction of sharp conductivity changes
occurring at the interface between a contrasting object and the background. As such
changes are step changes, reconstruction with traditional smoothing regularisation techniques
is unsatisfactory. The Gaussian anisotropic filtering works by incorporating prior
structural information. The approximate knowledge of the shapes of contrasts allows us
to relax the smoothness in the direction normal to the expected boundary. The construction
of Gaussian regularisation filters that express such directional properties on the basis
of the structural information is discussed, and the results of numerical experiments are
analysed. The method gives good results when the actual conductivity distribution is in
accordance with the prior information. When the conductivity distribution violates the
prior information the method is still capable of properly locating the regions of contrast.
The second part of the thesis is concerned with regularisation via the total variation
functional. This functional allows the reconstruction of discontinuous parameters. The
properties of the functional are briefly introduced, and an application in inverse problems
in image denoising is shown. As the functional is non-differentiable, numerical difficulties
are encountered in its use. The aim is therefore to propose an efficient numerical implementation
for application in ElT. Several well known optimisation methods arc analysed,
as possible candidates, by theoretical considerations and by numerical experiments. Such
methods are shown to be inefficient. The application of recent optimisation methods
called primal- dual interior point methods is analysed be theoretical considerations and
by numerical experiments, and an efficient and stable algorithm is developed. Numerical
experiments demonstrate the capability of the algorithm in reconstructing sharp conductivity profiles.
Department of Mechanical Engineering and Mathematical SciencesFaculty of Technology, Design and Environment
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