Small Volume Expansions

As shown in Sect. 5.1, in its most general form EIT is severely ill-posed and nonlinear. These are the main obstacles to find non-iterative reconstruction algorithms. If, however, in advance we have additional structural information about the conductivity profile, then we may be able to determine specific features about the conductivity distribution with a satisfactory resolution. One such type of knowledge could be that the body consists of a smooth background containing a number of unknown small anomalies with a significantly different conductivity. These anomalies might represent potential tumors.

Over the last 10 years or so, a considerable amount of interesting work has been dedicated to the imaging of such low volume fraction anomalies. The method of asymptotic expansions provides a useful framework to accurately and efficiently reconstruct the location and geometric features of the anomalies in a stable way, even for moderately noisy data. Using the method of matched asymptotic expansions we formally derive the first-order perturbations due to the presence of the anomalies. These perturbations are of dipole-type. A rigorous proof of these expansions is based on layer potential techniques. The concept of polarization tensor (PT) is the basic building block for the asymptotic expansion of the boundary perturbations. It is then important from an imaging point of view to precisely characterize the PT and derive some of its properties, such as symmetry, positivity, and optimal bounds on its elements, for developing efficient algorithms to reconstruct conductivity anomalies of small volume.

We then provide the leading-order term in this asymptotic formula of the solution to the Helmholtz equation in the presence of small electromagnetic (or acoustical) anomalies.