Electromagnetic Imaging in this context refers to the identification of internal characteristics of a medium based on boundary (or near-field) measurements of the electric and/or magnetic fields. After a brief review of some of the main mathematical results in Electromagnetic- and Impedance Imaging (from the last 20 years, or so) I shall proceed to discuss some very recent, extremely efficient representation formulas that lead to a surprisingly accurate identification of the size, and the location of relatively small inhomogeneities.

These representation formulas take into account polarization effects, and they may be derived by variational techniques related to H- (or Gamma-) convergence. The magnitude of the polarization effects may be estimated in ways that are very reminiscent of effective media bounds (of the Hashin-Shtrikman type). A precise assessment of the polarization effects is very important for highly accurate size estimates.

Finally, these representation formulas lend themselves very naturally to the application of reconstruction methods of a linear sampling- or MUSIC (MUltiple SIgnal Chararcterization) character. On this matter I shall discuss some general ideas, and implementation issues, as well as provide examples of computational reconstructions.