Linearized internal functionals for anisotropic conductivities

TL;DR

This paper develops conditions under which the linearized inverse problem for reconstructing anisotropic conductivities from power density measurements is elliptic and injective, potentially reducing the number of measurements needed.

Contribution

It introduces new sufficient conditions for the linearized problem's ellipticity and injectivity, improving upon existing methods by requiring fewer measurements.

Findings

Conditions for ellipticity of the linearized system

Conditions for injectivity of the linearized system

Reduced measurement requirements compared to nonlinear methods

Abstract

This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic tomography. We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are found to hold for a lesser number of measurements than those required in recently established explicit reconstruction procedures for the nonlinear problem.