A multiscale, statistically based inversion scheme for linearized inverse scattering problems

The application of multiscale and stochastic techniques to the solution of a linearized inverse scattering problem is presented. This approach allows for the explicit and easy handling of many difficulties associated with problems of this type. Regularization is accomplished via the use of a multiscale prior stochastic model which offers considerable flexibility for the incorporation of prior knowledge and constraints. We use the relative error covariance matrix (RECM), introduced in [28], as a tool for quantitatively evaluating the manner in which data contribute to the structure of a reconstruction. Given a set of scattering experiments, the RECM is used for understanding and analyzing the profess of data fusion and allows us to define the space-varying optimal scale for reconstruction as a function of the nature (resolution, quality, and distribution of observation points) of the available measurement sets. Examples of our multiscale inversion algorithm are presented using the Born approximation of an inverse electrical conductivity problem formulated so as to illustrate many of the features associated with inverse scattering problems arising in fields such as geophysical prospecting and medical imaging.