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

作者：Moradifam Amir^*; Nachman Adrian; Timonov Alexandre

来源：Inverse Problems, 2012, 28(8): 084003.

DOI：10.1088/0266-5611/28/8/084003

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 vertical bar J vertical bar of one current for a given voltage f on the boundary partial derivative Omega. As previously shown, the corresponding voltage potential u in Omega is a minimizer of the weighted least gradient problem %26lt;br%26gt;u = argmin {integral(Omega) a(x)vertical bar del u vertical bar : u is an element of H-1 (Omega), u vertical bar partial derivative(Omega) = f}, %26lt;br%26gt;with a(x) = vertical bar J(x)vertical bar. In this paper, we present an alternating split Bregman algorithm for treating such least gradient problems, for a is an element of L-2 (Omega) non-negative and f is an element of H-1/2 (partial derivative 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 (which is a re-interpretation of the alternating direction method of multipliers adapted to L-1 problems). 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.

• 出版日期2012-8