An efficient hierarchical Bayesian method for the Kuopio tomography challenge 2023

Monica Pragliola; Daniela Calvetti; Erkki Somersalo

doi:10.3934/ammc.2024011

Article Contents

2024, Volume 2, Issue 2: 187-207. Doi: 10.3934/ammc.2024011

This issue Previous Article Spatial regularization and level-set methods for experimental electrical impedance tomography with partial data Next Article Multibang regularization for electrical impedance tomography

An efficient hierarchical Bayesian method for the Kuopio tomography challenge 2023

1.
Department of Mathematics and Applications, University of Naples Federico Ⅱ, Naples, Italy
2.
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH, USA

^*Corresponding author: Monica Pragliola
^*Corresponding author: Monica Pragliola

Received: April 2024

Revised: May 2024

Early access: June 2024

Published: June 2024

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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 extensively 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 (SP-IAS) algorithm, originally proposed for the solution of linear inverse problems, has been adapted for the non linear case of EIT reconstruction [10] in a computationally efficient manner. Here we introduce a hybrid version of the SP-IAS algorithms for the nonlinear EIT inverse problem, consisting of a sequence of two optimization problems. The unique solution of the first problem is a suitable initial guess for the second one, which is non-convex and is expected to more effectively promote sparsity in the distribution of the increments of the blocky targets. We provide a detailed description of the implementation details of the proposed scheme, with a specific focus on parameters selection. The SP-IAS method is applied to the 2023 Kuopio Tomography Challenge dataset. A comprehensive report of the running times for the different cases and parameter selections is presented.

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Mathematics Subject Classification: Primary: 65K10, 65Z05, 65F22, 62F15.

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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

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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

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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 $

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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

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Figure 5. Pairs of output $ \sigma $-estimates and corresponding segmentations for phantom #1 as functions of different numbers $ N_{inj} $ of injected currents

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Figure 6. Iterative estimates of the conductivity map for phantom #2 with $ N_{inj} = 76 $ computed with the hybrid IAS scheme in the first phase with $ r^{(1)} = 1 $ (top panels) and in the second with $ r^{(2)} = 1/2 $ phase (middle panels). The bottom row displays the target (left) and output segmentation (right) corresponding to the 10-th iteration of the hybrid scheme

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Figure 7. Pairs of output $ \sigma $-estimates and corresponding segmentations for phantom #2 as functions of different numbers $ N_{inj} $ of injected currents

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Figure 8. Iterative estimates of the conductivity map for phantom #3 with $ N_{inj} = 76 $ computed by the hybrid IAS scheme 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

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Figure 9. Pairs of output $ \sigma $-estimates and corresponding segmentations for phantom #3 as functions of different numbers $ N_{inj} $ of injected currents

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Figure 10. For the three phantoms, behavior of $ \delta\theta_1 $ and $ \delta\theta_2 $ as functions of the iteration number of the hybrid IAS in the case of $ L = 32 $ active electrodes and $ N_{inj} = 76 $ current injections

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Figure 11. For the three phantoms, average running times in seconds and dispersion bands of the hybrid IAS as a function of the number of injected currents

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Table 1. Values of the SSIM-based scores achieved for the three different phantoms and for different difficulty levels $ \tau $, i.e. for different numbers $ N_{inj} $ of injective currents and different number $ L $ of electrodes

Level $ \tau $	$ L $	$ N_{inj} $	phantom #1	phantom #2	phantom #3
1	32	76	0.6915	0.8978	0.7628
2	30	56	0.7031	0.8987	0.7908
3	28	52	0.6981	0.8939	0.7912
4	26	48	0.6308	0.8774	0.7651
5	24	44	0.5582	0.8987	0.8093
6	22	30	0.5781	0.6978	0.7206
7	20	27	0.6361	0.6341	0.6317

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