A variational Bayesian approach for frequency diverse non-linear microwave imaging

Microwave imaging of piece-wise homogeneous objects is considered as a nonlinear inverse scattering problem. The purpose is to obtain the object contrast (i.e. a map of its electromagnetic parameters: permittivity and conductivity), given measurements of the scattered fields for various object illuminations and operating frequencies. This nonlinear problem is solved by means of a joint contrast-source approach in a Bayesian framework. A Gauss-Markov-Potts prior is assumed for the contrast in order to account for the piece-wise homogeneity of the object. However, the joint posterior for contrast, sources and other hyperparameters yields a complex expression and conventional estimators (e.g. MAP or PM) do not have simple forms. Hence, the variational Bayesian approximation technique is applied to obtain a tractable estimator. An approximated posterior is calculated and shaping parameter expressions are derived. The performance of the method is estimated from experimental data collected at the Institut Fresnel (Marseille, France).