Galerkin Strategy for Level Set Shape Analysis: Application to Geodesic Tube

Abstract

Abstract. In this paper, we consider the geodesic tube characterization using a Galerkin-Level Set strategy. The first section is devoted to the analysis of a geodesic tube construction between two sets through the definition of the shape metric. In the second section, we define the Galerkin-Level Set strategy in shape analysis. This new variational formulation associated to a Hilbert space metric for shape identification problem consists in parameterizing the level set function in a finite dimensional subspace spanned by linear independent functions. Consequently, this method is more focused on topological changes than on high accuracy for the boundary evaluation as in a traditional level set formulation. In the third section, we use the Galerkin-Level Set formulation applied to a geodesic tube construction between two sets, through the calculus of the shape derivative of the normal speed. Finally, this geodesic tube construction is validated by a numerical experiment. Tube Formulation Using Moving DomainIn this section, we briefly recall the concept of connecting tube, introduced in [6]. Let us consider D as a bounded universe in R n and two open sets domains Ω 0 , Ω 1 ⊂ D. We denote the initial domain by Ω 0 and the final domain by Ω 1 , and consider the tube connecting Ω 0 with Ω 1 defined by the n + 1 dimensional graph of an n-dimensional moving domain: see Fig. 1. Consequently, considering the time interval I = [0, 1], we define the tube evolution Q by product space, using the cylinder I × Ω as follows:Moreover, we denote by Σ the lateral boundary of the tube, defined by the following expression: Σ = 0≤t≤1 {t} ×Γ t , where Γ t denotes the boundary of Ω t . The characteristic function of the tube is defined by ζ (t, x) de f = χ Ω t (x) and verifies ζ 2 = ζ . Following [4,5], the set of connecting tubes between Ω 0 and Ω 1 is defined by: