Bayesian Formulation of Acousto-Electric Tomography and Quantified Uncertainty in Limited View

Abstract

Acousto-electric tomography (AET) is a hybrid imaging modality that combines electrical impedance tomography with focused ultrasound perturbations to obtain interior power density measurements, which provide additional information that can enhance the stability of conductivity reconstruction. In this work, we study the AET inverse problem within a Bayesian framework and compare statistical reconstruction with analytical approaches. The unknown conductivity is modeled as a random field, and inference is based on the posterior distribution conditioned on the measurements. We consider likelihood constructions based on both L1- and L2-type data misfit norms and establish Bayesian well-posedness for both formulations within the framework of Stuart (2010). Numerical experiments investigate the performance of the Bayesian method from noisy power density measurements using the L1 and L2 likelihood functions and a smooth prior and a piecewise-constant prior for different limited view configurations, including severely limited boundary access. In particular, we demonstrate that small inclusions near the accessible boundary can be reconstructed from AET data corresponding to a single EIT measurement, and we quantify reconstruction uncertainty through posterior statistics.

Figures (7)

• Figure 1: (a) True conductivity phantom consisting of several smoothed inclusions of varying sizes. (b)–(e) Corresponding noise-free electrical energy density measurements $h_{1,1}$ for full, half, quarter, and eighth boundary-view configurations, respectively. In each case, the red curve indicates the location and extent of the applied boundary input used to generate the measurements.
• Figure 2: The boundary functions $u_1\vert_{\partial \Omega}$ and $u_2\vert_{\partial \Omega}$ used for generating the power density data for the numerical examples. Each row corresponds to a different limited view setting.
• Figure 3: Diagnostics and sampling efficiency for the statistical AET inverse problem with a smooth prior. Panels (a) and (b) show trace plots for the first four components of $\boldsymbol{x}$ under the $L^1$ and $L^2$ likelihood constructions, respectively. Panels (c) and (d) display the posterior mean estimates of the first 10 KL coefficients of $\boldsymbol{x}$ together with the corresponding $95\%$ highest posterior density intervals (HDIs), illustrating the uncertainty in the coefficient estimates.
• Figure 4: Diagnostics and sampling efficiency for the statistical AET inverse problem with a piecewise constant prior. Panels (a) and (b) show trace plots for the first four components of $\boldsymbol{x}$ under the $L^1$ and $L^2$ likelihood constructions, respectively. Panels (c) and (d) display the posterior mean estimates of the first 10 KL coefficients of $\boldsymbol{x}$ together with the corresponding $95\%$ highest posterior density intervals (HDIs), illustrating the uncertainty in the coefficient estimates.
• Figure 5: Posterior mean reconstructions for different likelihood models and limited-view configurations. The first two columns correspond to reconstructions obtained with the $L^1$ likelihood (smooth and piecewise-constant priors, respectively), while the next two columns correspond to the $L^2$ likelihood with the same prior choices. The rows represent different limited-view settings, from full to increasingly restricted boundary coverage. In each panel, the red boundary curve indicates the extent of the applied boundary input.

Theorems & Definitions (14)

• Lemma 3.1: AlessandriniMagnanini94
• Lemma 3.2: Salo2022
• Lemma 5.1
• Lemma 5.2
• Lemma 6.1: Lipschitz continuity of the map $\sigma \mapsto h_{i,j}$ from $L^{\infty}$ to $L^1$
• proof
• Lemma 6.2: Regularity of $h_{i,j}$
• Remark 6.3: Sufficient conditions for Lemma \ref{['lem:regH']} in limited view
• Theorem 6.4
• proof