Feasibility of Acousto-Electric Tomography

Abstract

In acousto-electric tomography the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is penetrated by a time-dependent acoustic wave. We explicitly model the phenomena, and we propose a complete inversion framework for acousto-electric tomography in two steps: First the interior power density is obtained from boundary measurements by solving a linear, ill-posed problem; second the interior conductivity is reconstructed from the power density by solving a non-linear, fairly well-posed problem. We perform numerical experiments on synthetic data with realistically chosen parameters. We investigate how feasibility of reconstructing the electrical conductivity from boundary measurements depends on the acousto-electric coupling constant and measurement noise. Our findings are positive, and indicate that AET is indeed feasible for interesting applications in for example medical imaging. Finally, we consider a limited angle setup and show that the conductivity is well reconstructed near the measurement boundary.

Figures (6)

• Figure 1: (a) Electric conductivity. (b) Perturbed conductivity due to the acousto-electric effect via a wave generated from transducers on the top; exaggerated for visibility.
• Figure 2: The top image contains the graph of the function $I(t)$ for the phantom seen in Figure \ref{['fig:phan_sigp']} corresponding to the illustrated propagating wave and the boundary condition $f(x,y) = x$. The red vertical lines mark times corresponding to the four instant wave positions seen in the plots below.
• Figure 3: (a) The phantom conductivity. The conductivity is 1.0 in the disc inclusion and the background is 0.1. (b) The points where the wave fronts gets focused and the transducers. Each transducer focus waves in a subset of the focus points. The red-ringed transducer at the top focuses in the cone of red-ringed points in the domain. The point distribution corresponding to each transducer is rotationally symmetric. In simulations there are no gaps between transducer arrays and the domain.
• Figure 4: True power densities.
• Figure 5: Reconstruction of the power densities $H_i$ from data with an added relative noise of (row 1) 100%, (row 2) 500% and (row 3) 1000%. Column $i$ correspond to boundary condition $f_i$. $k$ is the number of singular values used in the TSVD reconstruction for $H$. $\epsilon = 0.01$.

Theorems & Definitions (1)

• Remark 5.1