A frequency based constraint for a multi-frequency linear sampling method

The linear sampling method (LSM) has become a well established non-iterative technique for a variety of inverse scattering problems. The method offers a number of advantages over competing inverse scattering methods, mainly it is based on solving a linear problem while being able to account for multi-path effects. Unfortunately under the current framework the method is only effective when using a large number of multi-static data, and therefore may be impractical for many imaging applications. While primarily developed under a single frequency framework, recently the extension of the method to multi-banded data sets has been considered. It is known in general that the availability of multi-frequency data should compensate for reduced spatial diversity, but it is not clear how this can be accomplished for the LSM. In this work we take a step in this direction by considering a frequency based partial variation approach. We first establish that on bands absent of any corresponding Dirichlet eigenvalues the Herglotz density exhibits bounded variation. We then consider a regularization method incorporating this prior knowledge. The proposed approach exhibited a good estimate of the unknown Dirichlet eigenvalues of the obstacle in question when using reduced data. This observation also correlated with higher quality 3D reconstructions.