An efficient hierarchical Bayesian method for the Kuopio tomography challenge 2023

TL;DR

This work addresses nonlinear Electrical Impedance Tomography (EIT) reconstruction by leveraging a sparsity-promoting Bayesian framework and a hybrid IAS algorithm to encourage blocky conductivities. The authors extend the Sparsity Promoting IAS (SP-IAS) to nonlinear forward models, employing two hyperpriors to balance convexity and sparsity and performing alternating updates between conductivity increments and hyperparameters. They provide detailed implementation guidance, parameter selection rules, and extensive numerical tests on the 2023 Kuopio Tomography Challenge (KTC23) dataset, including runtime analyses and segmentation quality quantified by SSIM. The results demonstrate efficient, robust reconstruction with sharp object delineation and potential runtime gains when reducing boundary measurements, supporting the method’s practical applicability for fast, accurate EIT imaging.

Abstract

The aim of Electrical Impedance Tomography (EIT) is to determine the electrical conductivity distribution inside a domain by applying currents and measuring voltages on its boundary. Mathematically, the EIT reconstruction task can be formulated as a non-linear inverse problem. The Bayesian inverse problems framework has been applied expensively to solutions of the EIT inverse problem, in particular in the cases when the unknown conductivity is believed to be blocky. Recently, the Sparsity Promoting Iterative Alternating Sequential (PS-IAS) algorithm, originally proposed for the solution of linear inverse problems, has been adapted for the non linear case of EIT reconstruction in a computationally efficient manner. Here we introduce a hybrid version of the SP-IAS algorithms for the nonlinear EIT inverse problem, providing a detailed description of the implementation details, with a specific focus on parameters selection. The method is applied to the 2023 Kuopio Tomography Challenge dataset, with a comprehensive report of the running times for the different cases and parameter selections.

Figures (11)

• Figure 1: Overview of the hybrid IAS algorithm for the EIT nonlinear inverse problem. The apricot boxes contain the input and the output stages. The red and reddish boxes are related to the process of selection of parameters, either in an automatic or manual fashion. The purple and purplish boxes contain the actual body of the algorithm, i.e. the sequence of the two phases and, within each phase, the alternation between the $\zeta$- and the $\theta$-update.
• Figure 2: Tessellation used for solving the inverse problem, with the $L=32$ circular arcs representing the areas on the boundary of the circular domain spanned by the electrodes.
• Figure 3: Output reconstruction obtain with the single hyperprior IAS algorithm with $r=1$ (left), $r=1/2$ (middle), and by the hybrid IAS (right) for phantom #1 and $N_j=76$.
• Figure 4: Iterative estimates of the conductivity map for phantom #1 with $N_{inj}=76$ obtained with the hybrid IAS in the first phase with $r^{(1)}=1$ (top panels) and in the second with $r^{(2)}=1/2$ phase (middle panels). In the bottom, target (left) and output segmentation (right) corresponding to the 10-th iteration of the overall hybrid scheme.
• Figure 5: Pairs of output $\sigma$-estimates and corresponding segmentations for phantom #1 as functions of different numbers $N_{inj}$ of injected currents.