Learned Finite Element-based Regularization of the Inverse Problem in Electrocardiographic Imaging

TL;DR

This work addresses the ill-posed inverse problem in electrocardiographic imaging by introducing a learned space-time regularizer that combines a spatial regularizer with a data-driven temporal Fields-of-Experts prior. The forward map is discretized via finite elements on unstructured cardiac meshes, and the regularizer is built from multivariate operators that include a temporal cross-correlation kernel, with potentials defined through the Moreau envelope to ensure differentiability. The authors establish Mosco-convergence of the discrete energies to the continuous limit and develop a scalable optimization framework with bi-level learning for the regularizer parameters, using implicit differentiation. Numerical experiments on synthetic data show that the proposed CMFoE/MFoE regularizers yield denoising and ECGI reconstructions that are both more accurate and more physiologically plausible than handcrafted baselines, demonstrating robust performance under noise.

Abstract

Electrocardiographic imaging (ECGI) seeks to reconstruct cardiac electrical activity from body-surface potentials noninvasively. However, the associated inverse problem is severely ill-posed and requires robust regularization. While classical approaches primarily employ spatial smoothing, the temporal structure of cardiac dynamics remains underexploited despite its physiological relevance. We introduce a space-time regularization framework that couples spatial regularization with a learned temporal Fields-of-Experts (FoE) prior to capture complex spatiotemporal activation patterns. We derive a finite element discretization on unstructured cardiac surface meshes, prove Mosco-convergence, and develop a scalable optimization algorithm capable of handling the FoE term. Numerical experiments on synthetic epicardial data demonstrate improved denoising and inverse reconstructions compared to handcrafted spatiotemporal methods, yielding solutions that are both robust to noise and physiologically plausible.

Figures (6)

• Figure 1: Illustration of the discretized cross-correlation computation of the matrix $\mathbf{D}(t)$ evaluated at the nodes $t_s$ of the uniform mesh with kernel defined in $\widetilde{\mathbb{T}}=[t_{-2},t_2]$. The matrix entries are computed by the overlapping areas of the hat functions.
• Figure 2: Lifting of the $\mathbf{x}$ point in a simplex at a boundary element $J\subset \mathcal{U}_h$ to the true domain by $H_h(\mathbf{x})\in J^e\subset \Omega_0$ in $\mathbb{R}^2$. We compute the linear projection $\mathbf{y}$ of $\mathbf{x}$ onto the boundary $\mathcal{T}_h^\Gamma$, take the closest-point projection $p(\mathbf{y})$ on $\Gamma$, and map $\mathbf{x}$ towards $p(\mathbf{y})$ to compute $H_h(x)$.
• Figure 3: Extracellular potential $v$ on the myocardium at three time steps with scar tissue, and a spacetime plot on the epicardium normalized to $[0,1]$. Reduced conductivity in the scar region deforms the characteristic spike-shaped potentials and slows propagation.
• Figure 4: Learned kernel functions $(k_{h,i})_{i=1}^{N_C}$ for $N_C=16$ of the FoE approaches by denoising.
• Figure 5: Denoising reconstructions and $L^2$-errors of multiple regularization approaches for an observation with a noise level of $\kappa=0.2$ applied on the ground truth visualized in a space-time plot.

Theorems & Definitions (19)

• Theorem 1
• proof
• Remark 1
• Definition 1: Lift and inverse lift
• Proposition 1: Norm equivalence under lifting
• Definition 2
• Definition 3
• Lemma 1
• proof
• Remark 2