Efficient Gradient-based Optimization for Reconstructing Binary Images in Applications to Electrical Impedance Tomography

TL;DR

A novel and highly efficient computational framework for reconstructing binary-type images suitable for models of various complexity seen in diverse biomedical applications is developed and validated and its high potential for improving the overall quality of the EIT-based procedures is demonstrated.

Abstract

A novel and highly efficient computational framework for reconstructing binary-type images suitable for models of various complexity seen in diverse biomedical applications is developed and validated. Efficiency in computational speed and accuracy is achieved by combining the advantages of recently developed optimization methods that use sample solutions with customized geometry and multiscale control space reduction, all paired with gradient-based techniques. The control space is effectively reduced based on the geometry of the samples and their individual contributions. The entire 3-step computational procedure has an easy-to-follow design due to a nominal number of tuning parameters making the approach simple for practical implementation in various settings. Fairly straightforward methods for computing gradients make the framework compatible with any optimization software, including black-box ones. The performance of the complete computational framework is tested in applications to 2D inverse problems of cancer detection by electrical impedance tomography (EIT) using data from models generated synthetically and obtained from medical images showing the natural development of cancerous regions of various sizes and shapes. The results demonstrate the superior performance of the new method and its high potential for improving the overall quality of the EIT-based procedures.

Figures (10)

• Figure 1: (a) EIT model #1: true electrical conductivity $\sigma_{true}(x)$ and equispaced geometry of electrodes $E_{\ell}$ placed over boundary $\partial \Omega$. (b) Electrical currents $I_l$ (positive in red, negative in blue) induced at electrodes $E_{\ell}$. Black arrows show the distribution of flux $\sigma(x) \boldsymbol{\nabla} u(x)$ of electrical potential $u$ in the interior of domain $\Omega$.
• Figure 2: 8 first sample solutions $\bar{\sigma}_i(x), \, i = 1, \ldots, 8$, from the set $\mathcal{C}(10,000)$.
• Figure 3: The behavior of (a) $\kappa (\epsilon)$ and (b) $\log_{10} \lvert \kappa(\epsilon) - 1 \rvert$ as functions of $\epsilon$ while checking the consistency of used gradients computed for model #1. In both graphs, red circles show results for $\kappa_{\alpha}(\epsilon)$ and $N_{\Omega} = 7,726$, while the rest relates to the application of the $\kappa$-test to both controls $\alpha$ and $\mathcal{P}$ in domain $\Omega$ with different spatial discretizations.
• Figure 4: The behavior of (a) $\kappa_{\mathcal{P}} = \kappa (\epsilon)$ and (b) $\log_{10} \lvert \kappa(\epsilon) - 1 \rvert$ as functions of $\epsilon$ while checking the consistency of gradients $\boldsymbol{\nabla}_{\mathcal{P}} \mathcal{J}$ computed for model #1 with different values of parameter $\delta \mathcal{P}$.
• Figure 5: Optimization results for model #1: (a) cost functionals $\mathcal{J}^k$ as functions of iteration count $k$ and (b) solution errors $\| \sigma^k - \sigma_{true} \|_{L_2}$ as functions of a number of cost functional evaluations evaluated while employing different optimizers (CD, MMA, IPOPT, and SNOPT). Pink dots in (a) represent Step 1 solutions ($k = 1, 2, \ldots, 10$), and empty blue circles in (a) and (b) show solutions obtained after Step 2 and 3 phases are complete.