We describe fresh and promising techniques for the reconsruction of small inclusions from boundary measurements. These techniques rely on accurate asymptotic expansions of the boundary perturbations due to the presence of the inclusions. The general approach we will take to derive these asymptotic expansions is based on layer potential techniques. This allows to handle inclusions with rough boundaries. In the course of deriving these asymptotic expansions we introduce new concepts of generalized polarization tensors that contain significant information on the inclusion. We then use the asymptotic expansions for designing efficient direct (non-iterative) reconstruction algorithms to detect the location and the size of the unknown inclusions. Finally, we apply these techniques in breast cancer imaging with a mathematical model based on the T-Scan trans-admittance imaging system.
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