Conductivity imaging with a single measurement of boundary and interior data

Abstract

We consider the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside. The internal data are obtained from magnetic resonance measurements. The problem is reduced to a boundary value problem with partial data for the equation ∇ ⋅ |J(x)||∇u|−1∇u = 0. We show that equipotential surfaces are minimal surfaces in the conformal metric |J|2/(n−1)I. In two dimensions, we solve the Cauchy problem with partial data and show that the conductivity is uniquely determined in the region spanned by the characteristics originating from the part of the boundary where measurements are available. We formulate sufficient conditions on the Dirichlet data to guarantee the unique recovery of the conductivity throughout the domain. The proof of uniqueness is constructive and yields an efficient algorithm for conductivity imaging. The computational feasibility of this algorithm is demonstrated in numerical experiments.

Avaliação de 3 diferentes aproximações para a solução do problema direto da tomografia de impedância elétrica

“…However as far as authors know, the existence and the stability are obtained for the first time. One can find conditional stability in [19] for a equipotential line method, which contains certain stability structure obtained in the theorem. The proof of Theorem 2 is given in Section 3.…”

“…First r • x (s, t) is differentiable with respect to the variable t by (19). Also, x(s, t) is Lipschitz and r 0 (s) is Hölder continuous on the boundary Γ − with respect to the variable s, hence their composition map s → r(x(s, 0)) is also Hölder continuous with respect to s. Similarly, the map s → e t 0 0 ∇×F(x(s,τ ))dτ is Hölder continuous and hence r in (20) is Hölder continuous with respect to s because it is given by the product of those two maps.…”

“…MREIT has been developed rather recently. The uniqueness and conditional stability have been obtained using a single set of current density (see [7,9,19]). In addition, many numerical algorithms have been suggested (see [4,21,22,24]).…”

Stable reconstruction of regular 1-Harmonic maps with a given trace at the boundary

“…The name is motivated by the geometric property that level sets of regular solutions of (1) are geodesics in the ambient plane endowed with the metric g = a 2 I , [1] (where I denotes the identity metric), thus generalizing the Euclidean case a ≡ 1.…”

“…From [2] is known that, if it exists, a regular solution to (1) is unique in the class of functions in W 1,1 ( ) with a negligible set of singular points (where the gradient vanishes). However, there are examples of Dirichlet problems for the 1-Laplacian which may not have regular solutions.…”