  Inverse scattering is the process of estimating the spatial distribution of
the scattering potential of an object by measuring the scattered wavefields
around it. In this paper, we consider reflection tomography of high contrast
objects that commonly occurs in ground-penetrating radar, exploration
geophysics, terahertz imaging, ultrasound, and electron microscopy. Unlike
conventional transmission tomography, the reflection regime is severely
ill-posed since the measured wavefields contain far less spatial frequency
information of the target object. We propose a constrained incremental
frequency inversion framework that requires no side information from a
background model of the object. Our framework solves a sequence of regularized
least-squares subproblems that ensure consistency with the measured scattered
wavefield while imposing total-variation and non-negativity constraints. We
propose a proximal Quasi-Newton method to solve the resulting subproblem and
devise an automatic parameter selection routine to determine the constraint of
each subproblem. We validate the performance of our approach on synthetic
low-resolution phantoms and with a mismatched forward model test on a
high-resolution phantom.
