A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. As previously shown, the corresponding voltage potential u in $\Omega$ is a minimizer of the weighted least gradient problem $$u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},$$ with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm

On A New Approach To Frequency Sounding Of Layered Media

Frequency sounding of layered media is modeled by a hyperbolic problem. Within the framework of this model, we formulate an inverse problem. Applying the Laplace transform and introducing the impedance function, the latter is first reduced to the inverse boundary value problem for the Riccati equation and then to the Cauchy problem for a first-order quadratic equation. The advantage of such transformations is that the quadratic equation does not contain an unknown coefficient. For a specific class of data, it is shown that the Cauchy problem is uniquely solvable. Based on the asymptotic behavior of solutions to both the Riccati and quadratic equations, a stable reconstruction algorithm is constructed. Its feasibility is demonstrated in computational experiments

Conductivity Imaging With A Single Measurement Of Boundary And Interior Data

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. © 2007 IOP Publishing Ltd

Reconstruction of Planar Conductivities In Subdomains From Incomplete Data

We consider the problem of recovering a sufficiently smooth isotropic conductivity from interior knowledge of the magnitude of the current density field vertical bar J vertical bar generated by an imposed voltage potential f on the boundary. In any dimension n \u3e = 2, we show that equipotential sets are global area minimizers in the conformal metric determined by vertical bar J vertical bar. In two dimensions, assuming the boundary voltage is almost two-to-one, we prove uniqueness of the minimization problem. This yields two results on reconstruction from incomplete data. In the first case, vertical bar J vertical bar is known in all of Omega, but the almost two-to-one f is know only on subintervals of the boundary. The second case assumes that vertical bar J vertical bar is known only in an appropriate subdomain (Omega) over tilde: our method works provided that (Omega) over tilde contains entire equipotential curves joining boundary points. Based on solving two point boundary value problems for the geodesic system, we give a procedure to determine whether (Omega) over tilde satisfies this property, to construct the equipotential curves lying entirely in the interior of (Omega) over tilde, and to obtain the conductivity in the region spanned by these curves. We also conduct a numerical study to illustrate the computational feasibility of the method

Recovering The Conductivity From A Single Measurement Of Interior Data

We consider the problem of recovering the conductivity of an object from knowledge of the magnitude of one current density field in its interior. A known voltage potential is assumed imposed at the boundary. We prove identifiability and propose an iterative reconstruction procedure. The computational feasibility of this procedure is demonstrated in some numerical experiments. © 2009 IOP Publishing Ltd

Stable Reconstruction Of Regular 1-Harmonic Maps With A Given Trace At The Boundary

We consider the numerical solvability of the Dirichlet problem for the 1-Laplacian in a planar domain endowed with a metric conformal with the Euclidean one. Provided that a regular solution exists, we present a globally convergent method to find it. The global convergence allows to show a local stability in the Dirichlet problem for the 1-Laplacian nearby regular solutions. Such problems occur in conductivity imaging, when knowledge of the magnitude of the current density field (generated by an imposed boundary voltage) is available inside. Numerical experiments illustrate the feasibility of the convergent algorithm in the context of the conductivity imaging problem