Sparse Reconstruction of Log-Conductivity in Current Density Impedance Tomography

Abstract

A new non-linear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the interior electric field data corresponding to two boundary voltage measurements, where the conductivity and the electric potential are related through an elliptic PDE arising in electrical impedance tomography. Further, the objective functional consists of a L 1 regularization term that promotes sparsity patterns in the conductivity and a Perona-Malik anisotropic diffusion term that enhances the edges to facilitate high contrast and resolution. This framework is motivated by a similar recent approach to solve an inverse problem in acousto-electric tomography. Several numerical experiments and comparison with an existing method demonstrate the effectiveness of the proposed method for superior image reconstructions of a wide-variety of log-conductivity patterns.

A sparsity-based nonlinear reconstruction method for two-photon photoacoustic tomography

“…Instead an upper bound for L is computed at each iterative step that leads to a fixed step size in the gradient update, known as the inertial parameter. The resulting scheme is known as the variable inertial proximal method (VIP) [12] and has nice convergent properties. We summarize the VIP scheme in the algorithm below as given in [12] Algorithm 4.2 (Variable inertial proximal (VIP) method).…”

“…The resulting scheme is known as the variable inertial proximal method (VIP) [12] and has nice convergent properties. We summarize the VIP scheme in the algorithm below as given in [12] Algorithm 4.2 (Variable inertial proximal (VIP) method).…”

“…The regularization terms σ − σ b L 1 and µ − µ b L 1 in the above functional (6) implement L 1 regularization of the minimization problem that helps promote sparsity patterns in the reconstruction of absorption coefficients. The use of such L 1 regularization terms has been shown to obtain high contrast in the reconstructions [12,24]. The H…”

“…In this article, we aim at using a robust computational framework that has the ability to provide high contrast and high resolution reconstructions of objects with holes and inclusions. The framework is based on a non-linear PDEconstrained optimization technique, developed recently [1,12,24] to study the aforementioned hybrid inverse problem for 2P-PAT. We start by formulating a minimization problem where we aim to determine σ and µ given the interior acoustic wave pressure field H σ,µ .…”

Levenberg-Marquardt algorithm for acousto-electric tomography based on the complete electrode model

“…Since the mapping from σ to E(σ) is non-linear, ( 3) is a non-linear least-squares problem in L 2 (Ω) and for that reason and iterative approach such as the Levenberg-Marquardt method is suitable [6]. See [1,13,22,29] for alternative optimization approaches, and [19] for a related analysis of the limited boundary data problem.…”