Generalized polarization tensors for shape description

Abstract

International audienceWith 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

Model based estimation of ellipsoidal object using artificial electric sense

“…This probably arises from the analytical model outlined in section 2.2, which estimates the electrical response of an object by its leading order dipolar component. In reality, such responses are an infinite multipolar expansion Landau and Lifshitz (1984); Ammari, Garnier, Kang, Lim and Yu (2014). In particular, the non-uniformity of the basal electric field E 0 causes further components of the multipolar response of the object that we neglect in our present model.…”

Model based object localization and shape estimation using electric sense on underwater robots

“…In [Khairuddin andLionheart, 2016, Ammari et al, 2014], authors show that at leading order, the electric response of any shaped object can be described as that of an ellipsoid. Therefore, objects are described by ellipsoids and their electrical response are modeled analytically by their first order generalized polarization tensor [Ammari et al, 2014]. Due to experimental constraints which prevent us from measuring the object influence only (the aquarium walls were perceived by the robot because of its smallness), all experiments were performed twice with and without object to be able to remove the effects of the aquarium.…”

“…3.2 Expression of the ellipse tensor [Ammari et al, 2014] As shown in [Ammari et al, 2014], at the leading order the electric response of an object can be modeled by the following tensor:…”

“…Globally, the average processing time was around 10 min for an average experiment time of 3 min. [Ammari et al, 2014] In Tab. 1, we present the mean results over the 108 experiments using the dipolar tensor of [Ammari et al, 2014].…”

“…[Ammari et al, 2014] In Tab. 1, we present the mean results over the 108 experiments using the dipolar tensor of [Ammari et al, 2014]. This table summarizes errors on both x and y-axis in mm, on the orientation in degrees, and on the shape in percentage of shape error.…”