Generalized polarization tensors for shape description

With each domain, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain. In the recent paper (Ammari et al. in Math. Comput. 81, 367-386, 2012), a recursive optimal control scheme to recover fine shape details of a given domain using GPTs is proposed. In this paper, we show that the GPTs can be used for shape description. We also show that high-frequency oscillations of the boundary of a domain are only contained in its high-order GPTs. Indeed, we provide an original stability and resolution analysis for the reconstruction of small shape changes from the GPTs. By developing a level set version of the recursive optimization scheme, we make the change of topology possible and show that the GPTs can capture the topology of the domain. We also propose an indicator of topology which could be used in some particular cases to test whether we have the correct number of connected components in the reconstructed image. We provide analytical and numerical evidence that GPTs can capture topology and high-frequency shape oscillations. The results of this paper clearly show that the concept of GPTs is a very promising new tool for shape description.