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Finite element-based space-time total variation-type regularization of the inverse problem in electrocardiographic imaging

Manuel Haas, Thomas Grandits, Thomas Pinetz, Thomas Beiert, Simone Pezzuto, Alexander Effland

TL;DR

The paper tackles reconstructing epicardial cardiac potentials from body-surface measurements by introducing a novel space-time total variation-type regularization within a finite element ECGI framework. It combines a convex, non-differentiable reg ularizer with a first-order primal-dual optimization scheme to recover sharp cardiac interfaces while leveraging temporal continuity. Across 2D and 3D synthetic models, the method consistently outperforms standard Tikhonov regularization, particularly when using the α=2 (L2,1) variant and space-time coupling, demonstrating improved accuracy and edge preservation. The approach shows promise for noninvasive cardiac mapping, albeit with higher computational demands, and points toward future adaptations to real human geometries and computational efficiency improvements.

Abstract

Reconstructing cardiac electrical activity from body surface electric potential measurements results in the severely ill-posed inverse problem in electrocardiography. Many different regularization approaches have been proposed to improve numerical results and provide unique results. This work presents a novel approach for reconstructing the epicardial potential from body surface potential maps based on a space-time total variation-type regularization using finite elements, where a first-order primal-dual algorithm solves the underlying convex optimization problem. In several numerical experiments, the superior performance of this method and the benefit of space-time regularization for the reconstruction of epicardial potential on two-dimensional torso data and a three-dimensional rabbit heart compared to state-of-the-art methods are demonstrated.

Finite element-based space-time total variation-type regularization of the inverse problem in electrocardiographic imaging

TL;DR

The paper tackles reconstructing epicardial cardiac potentials from body-surface measurements by introducing a novel space-time total variation-type regularization within a finite element ECGI framework. It combines a convex, non-differentiable reg ularizer with a first-order primal-dual optimization scheme to recover sharp cardiac interfaces while leveraging temporal continuity. Across 2D and 3D synthetic models, the method consistently outperforms standard Tikhonov regularization, particularly when using the α=2 (L2,1) variant and space-time coupling, demonstrating improved accuracy and edge preservation. The approach shows promise for noninvasive cardiac mapping, albeit with higher computational demands, and points toward future adaptations to real human geometries and computational efficiency improvements.

Abstract

Reconstructing cardiac electrical activity from body surface electric potential measurements results in the severely ill-posed inverse problem in electrocardiography. Many different regularization approaches have been proposed to improve numerical results and provide unique results. This work presents a novel approach for reconstructing the epicardial potential from body surface potential maps based on a space-time total variation-type regularization using finite elements, where a first-order primal-dual algorithm solves the underlying convex optimization problem. In several numerical experiments, the superior performance of this method and the benefit of space-time regularization for the reconstruction of epicardial potential on two-dimensional torso data and a three-dimensional rabbit heart compared to state-of-the-art methods are demonstrated.
Paper Structure (18 sections, 2 theorems, 61 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 61 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Forward and Inverse Problem in Electrocardiographic Imaging
  3. Forward Problem
  4. Inverse Problem
  5. Discretization
  6. Discrete Function Spaces
  7. Total Variation Operator
  8. Forward Operator
  9. Primal-Dual Optimization Algorithm
  10. Problem Formulation
  11. Numerical Results
  12. Baseline Methods
  13. Ground Truth Generation
  14. Results
  15. Two-Dimensional Model
  16. ...and 3 more sections

Key Result

Theorem 1

There exists a unique solution $v\in H^1(T,H^1(\Omega_0))$ of eq:E_forward for $u\in H^1(T,H^{1/2}(\Gamma_H))$ in a weak sense.

Figures (12)

  • Figure 1: Basic illustration of total variation-based regularization for the inverse problem in ECGI on the rabbit database. Body surface measurements $z$ are compared to the optimization variable $u$ on the electrode domain $\Sigma$ and are minimized together with prior information in the form of the space-time gradient vector $\nabla_{(\mathbf{x},t)}u$, resulting in the reconstruction $u^*$.
  • Figure 2: Two-dimensional torso-heart model with heart domain $\Omega_H$, torso domain $\Omega_0$ including lungs, epicardium $\Gamma_H$, torso boundary $\Gamma$ and $16$ body surface electrodes $\Sigma$.
  • Figure 3: Basic illustration of the gradient computation on a prismatic finite element $L\otimes J\in\mathcal{T}_h\otimes \mathcal{S}_h$ in spatial dimension $d=3$ for different gradient spaces $\mathcal{Q}_h^1$ and $\mathcal{Q}_h^2$.
  • Figure 4: Two-dimensional heart potential reconstruction displayed as a function over space (Angle [rad]) and time (Time t [ms]) with ground truth function ($\mathbf{GT}$) for $50\mathrm{dB}$ noise.
  • Figure 5: Two-dimensional heart potential (ECG [mV]) with $50\mathrm{dB}$ noise reconstructed by different methods over time (Time t [ms]) evaluated on an epicardial node of the corresponding finite element discretization.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof