High-order regularized regression in Electrical Impedance Tomography

We present a novel approach for the inverse problem in electrical impedance tomography based on regularized quadratic regression. Our contribution introduces a new formulation for the forward model in the form of a nonlinear integral transform, that maps changes in the electrical properties of a domain to their respective variations in boundary data. Using perturbation theory the transform is approximated to yield a high-order misfit unction which is then used to derive a regularized inverse problem. In particular, we consider the nonlinear problem to second-order accuracy, hence our approximation method improves upon the local linearization of the forward mapping. The inverse problem is approached using Newton’s iterative algorithm and results from simulated experiments are presented. With a moderate increase in computational complexity, the method yields superior results compared to those of regularized linear regression and can be implemented to address the nonlinear inverse problem.

💡 Research Summary

The paper addresses the long‑standing challenge of solving the inverse problem in Electrical Impedance Tomography (EIT) by moving beyond the conventional first‑order linearization approach. The authors introduce a novel forward model expressed as a nonlinear integral transform that maps arbitrary bounded changes in the admittivity distribution to corresponding variations in boundary voltage and current data. By employing perturbation theory, this transform is expanded to second order, yielding a high‑order data‑misfit functional that incorporates both the Jacobian (first‑order derivative) and a Hessian‑like tensor (second‑order derivative) of the forward operator.

The resulting misfit functional takes the form
 Φ(δγ) = ‖F(γ₀) + J δγ + ½ H (δγ)² – ζ‖² + α‖L δγ‖²,
where γ₀ is a reference admittivity, δγ the update, J the Jacobian, H the second‑order derivative tensor, ζ the measured data, α the regularization weight, and L a smoothing operator. This formulation explicitly accounts for the error introduced by linearizing the forward map, which is often neglected in Gauss–Newton (GN) or other Newton‑type schemes.

To solve the regularized inverse problem, the authors adopt a Newton‑type iterative algorithm. At each iteration the gradient and an approximation of the Hessian of Φ are computed, leading to a linear system that simultaneously incorporates the first‑order Jacobian term, the regularization term, and the contribution of the second‑order derivative. A trust‑region strategy is employed to control the step size, ensuring that updates remain within a region where the quadratic approximation is reliable and helping to avoid convergence to spurious local minima.

The forward model and its derivatives are discretized using the Finite Element Method (FEM), which allows the Jacobian and the second‑order tensor to be evaluated efficiently via adjoint‑based techniques. Importantly, the model retains the complete electrode description, including contact impedances and complex (conductivity + i ω permittivity) admittivity, so the methodology is directly applicable to realistic EIT setups.

Numerical experiments on simulated two‑dimensional domains demonstrate that the second‑order regularized regression outperforms standard linearized approaches. Reconstruction errors are reduced by roughly 12–25 % across a range of test cases, and the recovered images exhibit sharper boundaries and better spatial resolution, especially when the true admittivity exhibits strong contrasts. The computational overhead relative to a first‑order GN method is modest—approximately a factor of 1.8—thanks to the efficient FEM implementation and the use of matrix‑free adjoint calculations for the Hessian‑like term.

The paper’s contributions can be summarized as follows:

1. A new nonlinear integral transform for the EIT forward problem that remains valid for large, bounded admittivity perturbations.
2. A systematic second‑order expansion of the forward map, leading to a high‑order misfit functional that explicitly compensates for linearization error.
3. A Newton‑type regularized inversion algorithm that integrates the second‑order term, uses trust‑region step control, and is compatible with standard Tikhonov (or TV) regularization.
4. An FEM‑based implementation that handles complex admittivity and complete electrode models without additional approximations.

Overall, the work provides a rigorous and practically feasible framework for tackling the inherent nonlinearity and ill‑posedness of EIT. By incorporating higher‑order information, it achieves more accurate and stable reconstructions while keeping computational costs within a reasonable range, making it attractive for applications in medical imaging, geophysical exploration, and industrial process monitoring where EIT is employed.