Implicit Neural Shape Optimization for 3D High-Contrast Electrical Impedance Tomography

TL;DR

This work tackles 3D electrical impedance tomography under high-contrast interfaces, where traditional methods struggle due to ill-posedness. It introduces an implicit neural shape optimization framework that parameterizes interfaces with a neural signed distance function and performs latent-space optimization while enforcing topology preservation via a neural diffeomorphic flow. A rigorous gradient formula from shape calculus and adjoint methods enables efficient updates in a compact latent space, and convergence guarantees are established for the stochastic optimization. Numerical results on pancreas and cardiac reconstruction demonstrate robust, high-fidelity geometries even under noisy measurements, highlighting the method's potential for practical medical and industrial EIT applications.

Abstract

We present a novel implicit neural shape optimization framework for 3D high-contrast Electrical Impedance Tomography (EIT), addressing scenarios where conductivity exhibits sharp discontinuities across material interfaces. These high-contrast cases, prevalent in metallic implant monitoring and industrial defect detection, challenge traditional reconstruction methods due to severe ill-posedness. Our approach synergizes shape optimization with implicit neural representations, introducing key innovations including a shape derivative-based optimization scheme that explicitly incorporates high-contrast interface conditions and an efficient latent space representation that reduces variable dimensionality. Through rigorous theoretical analysis of algorithm convergence and extensive numerical experiments, we demonstrate substantial performance improvements, establishing our framework as promising for practical applications in medical imaging with metallic implants and industrial non-destructive testing.

Figures (14)

• Figure 1: Geometric configuration of the 3D EIT problem (2D cross-sectional view): The conducting domain $\Omega = D \setminus \overline{S}$ (shaded) is bounded externally by $\Sigma := \partial D$ and internally by the interface $\Gamma := \partial S$. Current injection electrodes and voltage measurement points are distributed on $\Sigma$.
• Figure 2: Illustration of boundary integral coupling: The operators $\mathcal{S}_{AB}$ and $\mathcal{K}_{AB}$ capture the interaction between boundaries $A$ and $B$ through the fundamental solution $G(\bm{x},\bm{y})$.
• Figure 3: Integration of NDF with EIT reconstruction: (a) Pre-trained NDF model encodes anatomical shapes into a low-dimensional manifold $\mathcal{M}$ via diffeomorphic flows. (b) Our physics-informed optimization in $\mathcal{M}$ minimizes the EIT data misfit while inheriting NDF's topology preservation properties.
• Figure 4: The reconstructions for Scenario I: the first, second, and third rows refer to the initial, the optimal recovery, and the ground truth target, respectively.
• Figure 5: Convergence of the algorithm for Scenario I in terms of the loss $\mathcal{L}(z)$ (upper left), the indicator error $e$ (upper left), the volume error (lower left) and the Hausdorff error (lower right).

Theorems & Definitions (11)

• Theorem 3.1: Gradient Representation
• proof
• Theorem 4.1: Shape Derivative Bounds
• Lemma 4.2: Energy Estimate
• proof
• Lemma 4.3
• proof
• proof : Proof of Theorem \ref{['thm:hessian']}
• Theorem 4.4
• proof