Generalized polarization tensors for shape description

Habib Ammari, Josselin Garnier, Hyeonbae Kang, Mikyoung Lim, Sanghyeon Yu

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)199-224
Number of pages26
JournalNumerische Mathematik
Volume126
Issue number2
DOIs
Publication statusPublished - 2014 Feb
Externally publishedYes

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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