Quantitative microwave imaging method in Lebesgue spaces with nonconstant exponents

An innovative nonlinear inverse scattering approach in variable exponent Lebesgue spaces is proposed for microwave imaging purposes. The main objective of the approach is to overcome one of the main problems associated to reconstruction procedures developed in Lebesgue spaces with constant exponent, i.e., the need for the selection of the optimum Lebesgue-space norm parameter, which is critical for obtaining accurate reconstructions, and no exact rules exist for this choice. The approach proposed in this paper is oriented to tomographic imaging applications at microwave frequencies, for which the adopted inversion procedure is based on a Gauss-Newton method, which has been reformulated in the unconventional variable exponent Lebesgue spaces. The capabilities of the approach are demonstrated by a set of numerical simulations, in which cylindrical dielectric targets are inspected. Moreover, an experimental result is reported.