A Layer-stripping Reconstruction Algorithm Of Piecewise Constant Conductivity

An electrical impedance tomography (EIT) problem is to find the electrical conductivity and permittivity distributions inside a body from measurements made an the boundary. In this paper, we only consider the electrical conductivity. Possible applications range from non-destructive materials testing geophysical prospecting, and process control to medical imaging.One mathematical model for the direct problem of EIT is called continuum model:orAnd the other model is called complete electrode model: The impedance imaging problem is to form an image of the electromagnetic properties of an inaccessible region of space by applying patterns of electric currents to the bounding surface of the region and measuring the resulting voltages on that surface. The impedance imaging problem is nonlinear and ill-posed. And the model is: Recover the conductivity a(x) from the conductivity equation▽·σ(x)â–½u(x) = 0by the information of the DtN map (?)_σ^D-N or the NtD map (?)_σ^D-N.This paper describes the performance of a direct numerical method for approximating the conductivity in the interior. The idea behind the algorithm can be described in a few words as follows. A set of electric currents are applied to the surface of the body and the resulting voltages are measured on that surface. The algorithm proceeds via two steps: first the conductivity is found near the bounding surface of the body from the data having the highest available spatial frequency; next the boundary data on an interior surface are synthesized using a nonlinear differential equation of Riccati type. The process is then repeated, and an estimate of the conductivity is found, layer by layer.The paper is divided into two parts: We discuss the theoretical background necessary to understand the algorithm with two separate steps, namely (1) the reconstruction of the conductivity at the boundary, and (2) the synthesis of the data on an interior surface; Then describes the algorithm itself in a simple geometry, and contains a discussion of some of the numerical tests that we have performed.We only discuss the layer-stripping method on continuum model. First we consider the Neumann problemu satisfies∫_(?)Ω u|_(?)Ωds = 0, and where f and j are subject to the integral constraintThis condition is required to guarantee the existence of the solution. We denote by T_σthe mapThen in the conductivity equationwhere u,j satisfy∫_(?)Ω u|((?)Ω)ds = 0和∫∫_(?)Ωjds = 0 (conservertion of charge condition) . The NtD mapsatisfies whereγis the trace mappingProposition 1. The operator T_σadmits a decompositionwhere we write T for T_σwhenσ= 1 identically, and T_r is a mappingWe convert the decomposition of T_σinto information about R by taking f = 0 and restricting to the boundary. Thus we havewhere maps and maps to . So the principal part of the operator R is just(1/σ)R⁰, then the principal symbol of the operator R is .The following proposition shows that we can recoverσfrom the matrix elements of R in the Fourier basis, which we denote byMore specifically, we recover the Fourier coefficientsProposition 2. LetΩbe the unit disc, and assume thatσis smooth. Then Here we give a simply description of proposition 2.Proposition 2'. WhenΩis the unit disk andσis smooth and bounded away from zero,Once the conductivity is determined at the boundary from the high-frequency behavior of the mapping R, the next step is to synthesize R on a subsurface in-finitesimally close to the boundary. We do this by means of a differential equation, which is of the Riccati type, that R satisfies. For the sake of clarity, we derive the Riccati equation for R only in a spherical domain.Theorem 1. The operators R_γ, 0 < r≤1, form a left-continuous family with respect to r in the strong operator topology of L(H^s(S^n-1)), H^s+1(S^n-1), s≥3/2. Moreover, the strong limitexists in L(H^s(S^n-1), H^s(S^n-1)), and it satisfieswhereâ–½_t ?σ▽_T is the tangential part of the operator▽·σ▽along the surface |x| = r.Therefore, consider the two dimensional version of the equationWe have presented a preliminary study of a layer-stripping method. This study has demonstrated that at least on a limited class of data sets, the method is capable of making useful reconstructions. Formally, the algorithm described here could be extended to many fixed-frequency and fixed-energy inverse problems.