Conductivity imaging with a single measurement of boundary and interior data
Abbreviated Journal Title
MAGNETIC-RESONANCE; MREIT; Mathematics, Applied; Physics, Mathematical
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 vertical bar J(x)vertical bar 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. del.vertical bar J( x)vertical bar vertical bar del u vertical bar(-1)del u = 0. We show that equipotential surfaces areminimal surfaces in the conformalmetric vertical bar J vertical bar(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.
"Conductivity imaging with a single measurement of boundary and interior data" (2007). Faculty Bibliography 2000s. 7459.