  We propose an automatic algorithm for 3D inverse electromagnetic scattering
based on the combination of topological derivatives and regularized
Gauss-Newton iterations. The algorithm is adapted to decoding digital
holograms. A hologram is a two-dimensional light interference pattern that
encodes information about three-dimensional shapes and their optical
properties. The formation of the hologram is modeled using Maxwell theory for
light scattering by particles. We then seek shapes optimizing error functionals
which measure the deviation from the recorded holograms. Their topological
derivatives provide initial guesses of the objects. Next, we correct these
predictions by regularized Gauss-Newton techniques. In contrast to standard
Gauss-Newton methods, in our implementation the number of objects can be
automatically updated during the iterative procedure by new topological
derivative computations. We show that the combined use of topological
derivative based optimization and iteratively regularized Gauss-Newton methods
produces fast and accurate descriptions of the geometry of objects formed by
multiple components with nanoscale resolution, even for a small number of
detectors and non convex components aligned in the incidence direction. The
method could be applied in general imaging set-ups involving other waves
(microwave imaging, elastography...) provided closed-form expressions for the
topological and Frechet derivatives are determined.
