A NON-ITERATIVE SAMPLING APPROACH USING NOISE SUBSPACE PROJECTION FOR EIT C´edric Bellis joint with Andrei Constantinescu, Armin Lechleiter Applied Physics & Applied Mathematics Dept. Columbia University, New York, USA
[email protected] ABSTRACT This study concerns the problem of defect reconstruction in the context of electrical impedance tomography (EIT), which is investigated within the framework of a non-iterative sampling approach. Such identification strategy typically relies on the characteristic behavior of an indicator function that varies with coordinates of an interior sampling point. This function is constructed from the projection of a fundamental singular solution onto the space spanned by the singular eigenvectors associated with some of the smallest singular eigenvalues of the measurement operator. An introductory overlook to the forward and inverse conductivity problems is followed by the exposition of the Picard criterion as a characterization of the range of the measurement operator. The construction of the indicator function based on the noise subspace projection is then introduced and discussed in the context of classical computational schemes. Finally, the robustness of the approach is analyzed and a set of numerical results is presented to assess for its efficiency.
1. INTRODUCTION Electrical impedance tomography is an imaging technique for the reconstruction of objects embedded in a given conductive background medium Ω ⊂ Rd with d = 2, 3. Applications range over a broad spectrum such as non-destructive material testing or tumor detection in medical imaging [1]. This approach aims at determining the internal electrical conductivity map γ ∈ L∞ (Ω, R) from the application of a current density f on the boundary and measurement on ∂Ω of the corresponding voltage u. Given a mean-free current density f ∈ L2 (∂Ω), the generated potential u is the unique solution, up to a normalization constant, of the
boundary value problem ∇ · (γ∇u) = 0 (γ∇u) · n = f
in Ω on ∂Ω,
where n is the outward unit normal on the boundary. The problem considered can be stated as the question of the reconstruction of the conductivity γ given the knowledge of the Neumann-to-Dirichlet (NtD) map Λ which is defined such that Λf = u|∂Ω . However, along the lines of characterizing the lack of stability of this reconstruction towards the input data, it turns out that the inverse conductivity problem is severely ill-posed [2, 3]. 2. QUALITATIVE METHODS To provide an alternative to the customary minimization and linearization approaches, so-called qualitative methods for non-iterative reconstructions of background perturbations have been proposed and analyzed in the last years. These approaches are commonly centered around the development of an indicator function which is normally designed to reach extreme values when a featured sampling point belongs to the support of the hidden flaw (or the set thereof), thereby providing a computationally-effective platform for geometric defect reconstruction. Along these lines, this study investigates a noniterative sampling approach which aims at reconstructing geometrically the unknown set I of inclusions where γ varies from a unitary background value. On introducing the background Neumann-to-Dirichlet map Λ1 , this approach revolves around extracting the information encapsulated in the measurement operator defined as Π = Λ − Λ1 . To do so, the idea is to probe the range of this relative NtD operator with a fundamental solution of the diffusion equation in Ω which exhibits a
singular behavior at a chosen sampling point z ∈ Ω as z varies over a region of interest. 2.1. Picard criterion The operator Π is self-adjoint and compact [3], therefore there exists an eigensystem {λj , ψj } with positive eigenvalues λj sorted here in decreasing order and eigenfunctions ψj ∈ L2 (∂Ω) for j ∈ N with the subscript indicating a mean-free property over ∂Ω, such that for f ∈ L2 (∂Ω) Πf =
∞ X
λj (f, ψj )L2 (∂Ω) ψj .
j=1
In order to present a key characterization, one introduces the dipole potential Φz,d for a given point z ∈ Ω and a unit vector d ∈ Rd , which is the harmonic function in Rd \{z} given by Φz,d (ξ) =
1 (z − ξ) · d , ωd |z − ξ|d
for ξ 6= z,
where ω2 = 2π and ω3 = 4π. Now let gz,d be the test function defined, up to the normalization constant c, by gz,d = Φz,d |∂Ω − Λ1 (∇Φz,d · n) + c.
where the parameters M∗ , M ∗ have to be adequately chosen, and which is expected to provide the characterization IM∗ ,M ∗ (z e ) IM∗ ,M ∗ (z i ) for any exterior point z e ∈ Ω \ I and interior point z i ∈ I. The intended contributions of such definition are twofold. Firstly, it avoids the question of determining whether the series featured in (1) converges or not, which has been a long-standing issue for a practical use of the Picard criterion. Secondly, in noisy environments, the projection in the large set of those eigenfunctions below the noise level can indeed improve the quality and stability of the reconstruction, a result which may appear counter-intuitive at first. In connection with numerical implementations of the method and with the use of real measurements, we investigate the approximation quality of certain finite dimensional approximations of the full NtD operators. In this discrete setting and using noisy data, a stability analysis is conducted to evaluate the quality of the reconstruction employing perturbed spectral information. Finally, a set of numerical results (see e.g. Fig. 1) will be presented to illustrate the performance of the proposed method.
Then, an extended set I of inclusions is characterized [4] by the so-called Picard criterion showing that z∈I
⇔
∞ X |(gz,d , ψj )L2 (∂Ω) |2 j=1
λj
< +∞,
(1)
which provides a point-by-point binary criterion if one is able to characterize the possible “blow-up” of the above series for the points z that lie outside of the support of the inclusion. 2.2. Noise subspace projection approach The novel approach presented here is based on the alternative projection of the test function gz,d on the noise subspace of the operator Π. This approach finds its roots in the recent mathematical justification of the MUSIC algorithm for the reconstruction of extended objects in inverse scattering problems [5]. In this context, the so-called signal subspace Sδ coincides with the space spanned by the eigenfunctions associated with the largest eigenvalues of the data-to-measurement operator, i.e. those such that λj > δ for a user-chosen threshold level δ > 0. Then the noise subspace is the orthogonal complement Nδ = Sδ ⊥ . The aim of this presentation is to discuss a new indicator function defined as −1 ∗ M X |(gz,d , ψj )L2 (∂Ω) |2 , IM∗ ,M ∗ (z) = λj j=M∗
Figure 1: Normalized indicator function: L-shaped inclusion I ⊂ Ω = [0; 1]×[0; 1] with γ(ξ) = 1 − 0.99χI (ξ) 3. REFERENCES [1] W. R. B. Lionheart, Developments in EIT reconstruction algorithms: pitfalls, challenges and recent developments, Physiol. Meas., 25:125–142, 2004. [2] L. Borcea, Electrical Impedance Tomography, Inverse Problems, 18:R99–R136, 2002. [3] A. Kirsch, An introduction to the mathematical theory of inverse problems, Springer, 2011. [4] M. Br¨ uhl Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32:1327–1341, 2001. [5] T. Arens, A. Lechleiter and D. R. Luke, MUSIC for extended scatterers as an instance of the factorization method, SIAM J. Appl. Math., 70:1283–1304, 2009.
