Abbreviated Journal Title
SIAM J. Appl. Math.
conductivity imaging; current density impedance imaging; minimal; surfaces; 1-Laplacian; ELECTRICAL-IMPEDANCE TOMOGRAPHY; MAGNETIC-RESONANCE; MREIT; INDUCTION; ALGORITHM; EQUATION; Mathematics, Applied
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 > = 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.
Siam Journal on Applied Mathematics
Nachman, Adrian; Tamasan, Alexandru; and Timonov, Alexandre, "Reconstruction of Planar Conductivities In Subdomains From Incomplete Data" (2010). Faculty Bibliography 2010s. 576.