Application of constrained optimisation techniques in electrical impedance tomography

A Constrained Optimisation technique is described for the reconstruction of temporal resistivity images. The approach solves the Inverse problem by optimising a cost function under constraints, in the form of normalised boundary potentials. Mathematical models have been developed for two different data collection methods for the chosen criterion. Both of these models express the reconstructed image in terms of one dimensional (I-D) Lagrange multiplier functions. The reconstruction problem becomes one of estimating these 1-D functions from the normalised boundary potentials. These models are based on a cost criterion of the minimisation of the variance between the reconstructed resistivity distribution and the true resistivity distribution. The methods presented In this research extend the algorithms previously developed for X-ray systems. Computational efficiency is enhanced by exploiting the structure of the associated system matrices. The structure of the system matrices was preserved in the Electrical Impedance Tomography (EIT) implementations by applying a weighting due to non-linear current distribution during the backprojection of the Lagrange multiplier functions. In order to obtain the best possible reconstruction it is important to consider the effects of noise in the boundary data. This is achieved by using a fast algorithm which matches the statistics of the error in the approximate inverse of the associated system matrix with the statistics of the noise error in the boundary data. This yields the optimum solution with the available boundary data. Novel approaches have been developed to produce the Lagrange multiplier functions. Two alternative methods are given for the design of VLSI implementations of hardware accelerators to improve computational efficiencies. These accelerators are designed to implement parallel geometries and are modelled using a verification description language to assess their performance capabilities.