On the Use of Wavelet to Enhance the Performance of Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is one of the widely used non-intrusive technique to reconstruct the internal properties of a medium. These properties, typically expressed in the form of electrical conductivity or resistivity, are obtained using boundary voltage measurements resulting from a known current excitation. Tomography is a prototypical inverse problem that uses a numerical model to mimic the physical response of the system in order to identify the optimal distribution of material parameters that best fit the measured data. EIT is a highly ill posed nonlinear inverse problem. Numerical regularization and prior information on the system are often a necessity to guarantee the stability of the solution. Analytical solutions are rarely available so that iterative numerical techniques are typically employed.

The main goal of this thesis is to develop the numerical techniques based on wavelet decomposition capable to enhance the computational efficiency of tomographic inverse problems while maximizing the reconstruction accuracy and resolution.

Wavelet decomposition has been widely used in signal processing. Thanks to their compact support, wavelet enables efficient identification of localized features in both the temporal and the spatial domain. A more recent development in the field of wavelet consists in wavelet adaptive mesh refinement (WAMR) algorithms. In this area, wavelet allows to adaptively adjust the spatial discretization of the numerical mesh in order to capture the subtle changes in field variables while maintaining a high computational efficiency and accuracy. The power of this algorithm lies in the multi-resolution representation of a function expanded in wavelet basis. The model parameters are represented in the wavelet space and quantified using wavelet coefficients. Then, to represent the spatial scales in the given analysis, the corresponding wavelet coefficients are filtered so to eliminated that are not significant. By reducing the design variable to be evaluated, the reconstruction procedure becomes much faster while the recovery of the calculation is preserved. A detailed discussion of WAMR algorithms and its application in solving partial differential equations (PDEs) and inverse problems will be presented, with specific application to Electrical impedance tomography (EIT).

The nature of iterative methods brings inevitably a considerable computational burden. In an effort to develop efficient computational algorithms, this thesis explores a wavelet basis function approach (WBFA) method to fully exploit the localized features of electrical conductivity distribution. The core idea of this algorithm is based on the superposition principle. The conductivity distribution is assumed to be a linear combination of a few basis 'images', i.e., the wavelet basis functions. The reconstruction of the conductivity distribution is obtained by reconstructing the unknown coefficients corresponding to these wavelet basis functions. This procedure greatly reduces the unknown parameters to be evaluated and thus accelerates the computational efficiency. Different types of wavelet functions are chosen as basis for the solution of the EIT inverse problem, and their performances are evaluated and compared. A detailed implementation of wavelet basis function approach in 1-D and 2-D will be presented. It will be shown that the WBFA has a better performance than periodic methods.