Graph-Based ECG Synthesis with Activation-Consistency Certification and Diagnostics-Aware Morphology Curation

Yongjun Yoo

arxiv: 2606.26580 · v1 · pith:TN5KFSYHnew · submitted 2026-06-25 · 🧮 math.NA · cs.NA· eess.SP

Graph-Based ECG Synthesis with Activation-Consistency Certification and Diagnostics-Aware Morphology Curation

Yongjun Yoo This is my paper

Pith reviewed 2026-06-26 04:47 UTC · model grok-4.3

classification 🧮 math.NA cs.NAeess.SP

keywords synthetic ECGgraph EikonalBellman residualactivation certificationmorphology curationcardiac modelingECG generation

0 comments

The pith

A unified heart graph generates synthetic ECGs whose activation consistency is certified by the Bellman residual of a fixed-point solve.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a graph model of the heart that supports two activation backends, one based on a pure Eikonal template and one that adds a pseudo-diffusion reaction term. Activation times are obtained by solving a Bellman fixed-point equation on the graph, and the size of the residual supplies a direct numerical certificate that the times are internally consistent. Each generated ECG then passes through a two-stage diagnostics pipeline that first computes standard metrics and then applies experiment-specific acceptance rules to curate morphology. When the reaction-eikonal backend is used, the resulting activation times stay within milliseconds of the Eikonal reference, the coefficient of determination reaches 0.99876 after predecessor filtering, and the fraction of accepted samples with improved morphology coverage rises from 0.09248 to 0.09888.

Core claim

The framework provides a conservative basis for controllable and curated synthetic ECG generation, with RE-derived activation times showing near-millisecond agreement with the Eikonal backbone and R²=0.99876 after causal predecessor filtering, while increasing per-model morphology coverage from 0.09248 to 0.09888.

What carries the argument

The unified heart graph that supports an eikonal-template backend and a pseudo-diffusion reaction-eikonal backend, with activation times obtained as the solution of a Bellman fixed-point problem whose residual serves as the consistency certificate.

If this is right

RE-derived activation times agree with the Eikonal backbone to near-millisecond precision.
After causal predecessor filtering the coefficient of determination reaches 0.99876.
The RE backend accepts 658 out of 2000 samples versus 578 for the ET backend in the final balanced curation.
Endo-epicardial APD gradients set the primary T-wave morphology window while diffusion strength supplies secondary smoothing.
Per-model morphology coverage rises from 0.09248 to 0.09888.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same graph-plus-residual approach could be reused to certify activation consistency in other organ-scale electrophysiological simulations.
Because the curation stage separates metric computation from acceptance policy, the pipeline can be retargeted to different clinical or research use cases without retraining the underlying solver.
If patient-specific graphs can be constructed from imaging or sparse recordings, the framework supplies a route to individualized yet certified synthetic ECG libraries.

Load-bearing premise

The Bellman residual obtained from the graph Eikonal solve is assumed to be a meaningful and sufficient certificate that the activation times are physiologically consistent.

What would settle it

A collection of activation times that produce small Bellman residuals yet generate ECG morphologies that violate known physiological ranges or contradict independent high-resolution mapping data would falsify the claim that the residual certifies consistency.

read the original abstract

Synthetic electrocardiogram (ECG) generation can support algorithm development and robustness evaluation, but simulated signals must preserve interpretable activation, recovery, and morphology properties. We present a graph-based ECG synthesis framework that combines activation-consistency certification with diagnostics-aware morphology curation. A unified heart graph supports an eikonal-template backend (ET) and a pseudo-diffusion reaction--eikonal backend (RE). We formulate graph Eikonal activation as a Bellman fixed-point problem and use the Bellman residual as a computable certificate for activation-time consistency. Each simulated ECG is evaluated by a two-stage diagnostics pipeline that separates metric computation from experiment-specific acceptance policies. On the cardiac graph, RE-derived activation times showed near-millisecond agreement with the Eikonal backbone and achieved $R^2=0.99876$ after causal predecessor filtering. Recovery experiments showed that endo-epicardial APD gradients determined the main T-wave morphology window, whereas the diffusion strength $\kappa$ provided secondary repolarization smoothing. In final balanced multi-lead curation, RE accepted 658/2000 samples versus 578/2000 for ET and increased per-model morphology coverage from 0.09248 to 0.09888. The framework provides a conservative basis for controllable and curated synthetic ECG generation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable internal-consistency pipeline for graph-based ECG synthesis but the physiological grounding stays assumed.

read the letter

The core new element is turning the graph Eikonal problem into a Bellman fixed-point on a single heart graph, then treating the residual as an explicit certificate, while running two backends (ET and RE) and keeping the diagnostics metrics separate from the acceptance rules.

The execution is tidy. RE activation times line up with the Eikonal reference to roughly a millisecond and hit R²=0.99876 after the causal filter. The curation step raises accepted traces from 578 to 658 out of 2000 and lifts per-model morphology coverage from 0.092 to 0.099. The separation of metric calculation from policy is a practical touch that avoids post-hoc fiddling.

The soft spot is exactly where the stress-test note flags it. The Bellman residual confirms the discrete equations are solved, but it does not show that the resulting times match endo-epicardial gradients or bidomain reference solutions. All reported agreement is between RE and ET inside the same graph; there is no external anchor. The coverage gain is also small, and the single free parameter κ receives no sensitivity check in the abstract.

This is for computational cardiology groups that generate synthetic ECGs for algorithm stress-testing. The framework is explicit enough and the numbers are reproducible from the description, so it clears the bar for a serious referee even though external validation would be the next obvious step.

Referee Report

2 major / 1 minor

Summary. The paper presents a graph-based ECG synthesis framework on a unified heart graph supporting eikonal-template (ET) and pseudo-diffusion reaction-eikonal (RE) backends. Graph Eikonal activation is cast as a Bellman fixed-point problem whose residual serves as an activation-consistency certificate. A two-stage diagnostics pipeline performs metric computation separate from acceptance policies. Reported results include near-millisecond RE-ET activation agreement, R²=0.99876 after causal predecessor filtering, endo-epicardial APD gradients dominating T-wave morphology, and RE accepting 658/2000 samples versus 578/2000 for ET while raising per-model morphology coverage from 0.09248 to 0.09888.

Significance. If the internal consistency metrics and curation pipeline can be shown to correspond to physiologically grounded activation and recovery, the approach could supply a controllable, certifiable source of synthetic ECGs for algorithm robustness testing. The separation of metric computation from acceptance policies and the quantitative coverage gain are concrete strengths; however, the significance hinges on whether the reported internal agreement translates to external validity against measured or high-fidelity bidomain data.

major comments (2)

[Abstract] Abstract: the statement that the Bellman residual constitutes a 'computable certificate for activation-time consistency' is load-bearing for the central claim, yet the residual only quantifies discrete fixed-point convergence on the graph; the cited near-millisecond agreement and R²=0.99876 are internal RE-versus-ET comparisons after filtering, not comparisons against independent physiological measurements or bidomain solutions.
[Abstract] Abstract: the R²=0.99876 figure and the final acceptance counts (658/2000 vs 578/2000) are obtained after 'causal predecessor filtering' and 'balanced multi-lead curation'; the manuscript must specify the exact filtering criteria and acceptance policies and demonstrate that these steps do not introduce post-hoc selection that inflates the reported agreement and coverage metrics.

minor comments (1)

[Abstract] The abstract introduces the diffusion strength κ and the ET/RE backends without defining their governing equations or discretization; a brief parenthetical or reference to the relevant section would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We agree that the abstract language regarding the Bellman residual requires clarification to avoid implying external physiological validation, and that the curation pipeline must be described with explicit criteria and pre/post metrics. We will revise the manuscript accordingly while preserving the paper's focus on internal graph consistency and controllable synthesis.

read point-by-point responses

Referee: [Abstract] Abstract: the statement that the Bellman residual constitutes a 'computable certificate for activation-time consistency' is load-bearing for the central claim, yet the residual only quantifies discrete fixed-point convergence on the graph; the cited near-millisecond agreement and R²=0.99876 are internal RE-versus-ET comparisons after filtering, not comparisons against independent physiological measurements or bidomain solutions.

Authors: We agree that the Bellman residual certifies convergence of the fixed-point iteration on the discrete graph for the RE and ET backends, and does not constitute a direct comparison to bidomain PDE solutions or measured data. The reported agreement metrics are strictly internal. We will revise the abstract to read 'computable certificate for activation-time consistency on the graph discretization' and add a clarifying sentence in the introduction that explicitly limits the scope to internal model agreement. revision: yes
Referee: [Abstract] Abstract: the R²=0.99876 figure and the final acceptance counts (658/2000 vs 578/2000) are obtained after 'causal predecessor filtering' and 'balanced multi-lead curation'; the manuscript must specify the exact filtering criteria and acceptance policies and demonstrate that these steps do not introduce post-hoc selection that inflates the reported agreement and coverage metrics.

Authors: The full manuscript details the filtering and curation steps in Section 4.2 and Algorithm 2, but we acknowledge the need for greater transparency. We will add an appendix with the precise numerical thresholds (predecessor time difference ≤ 2 ms, per-lead morphology score cutoffs, and balanced sampling rules) together with supplementary tables reporting R², acceptance rates, and coverage before and after each stage. This will allow readers to verify that the gains are not artifacts of post-hoc selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper formulates graph Eikonal activation explicitly as a Bellman fixed-point problem and defines the residual directly from those equations as a computable certificate. This is a standard numerical construction measuring equation satisfaction rather than a fitted target or self-referential definition. Reported metrics (near-millisecond agreement, R²=0.99876 after filtering, morphology coverage increase) are internal comparisons between RE and ET backends or separate curation statistics, presented as empirical outcomes rather than predictions forced by construction. No self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the text. The central claims rest on the modeling assumptions stated in the abstract and formulation sections, which are independent of the reported numerical results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 3 invented entities

The central claim depends on the modeling assumption that a unified heart graph plus the Bellman formulation yields certifiable activation times, plus the free parameter κ for diffusion smoothing and the invented ET/RE backends whose validity is demonstrated only by internal simulation agreement.

free parameters (1)

diffusion strength κ
Described as providing secondary repolarization smoothing in recovery experiments; its value is not derived from first principles.

axioms (1)

domain assumption Graph Eikonal activation can be formulated as a Bellman fixed-point problem whose residual serves as a computable certificate for activation-time consistency
Invoked to justify the certification step in the abstract.

invented entities (3)

unified heart graph no independent evidence
purpose: Supports both ET and RE backends for ECG synthesis
Core modeling construct introduced by the framework; no independent external validation cited in abstract.
eikonal-template backend (ET) no independent evidence
purpose: Provides baseline activation times
One of two backends defined in the paper.
pseudo-diffusion reaction--eikonal backend (RE) no independent evidence
purpose: Adds diffusion and reaction for recovery simulation
Second backend introduced to improve morphology coverage.

pith-pipeline@v0.9.1-grok · 5756 in / 1628 out tokens · 44099 ms · 2026-06-26T04:47:27.783297+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 5 canonical work pages

[1]

Chaos, Solitons & Fractals 7(3):293–301

Aliev RR, Panfilov AV (1996) A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals 7(3):293–301

1996
[2]

Antzelevitch C, Viskin S, Shimizu W, et al (2007) Does tpeak-tend provide an index of transmural dispersion of repolarization? Heart rhythm: the official journal of the Heart Rhythm Society 4(8):1114

2007
[3]

Quarterly of applied mathematics 16(1):87–90

Bellman R (1958) On a routing problem. Quarterly of applied mathematics 16(1):87–90

1958
[4]

Journal of theoretical biology 253(3):544–560

Bueno-Orovio A, Cherry EM, Fenton FH (2008) Minimal model for human ven- tricular action potentials in tissue. Journal of theoretical biology 253(3):544–560

2008
[5]

Medical & biological engineering & computing 47(1):11–20

van Dam PM, Oostendorp TF, van Oosterom A (2009) Application of the fastest route algorithm in the interactive simulation of the effect of local ischemia on the ecg. Medical & biological engineering & computing 47(1):11–20

2009
[6]

In: Edsger Wybe Dijkstra: his life, work, and legacy

Dijkstra EW (2022) A note on two problems in connexion with graphs. In: Edsger Wybe Dijkstra: his life, work, and legacy. Association for Computing Machinery, 18 p 287–290

2022
[7]

Pharmacology & therapeutics 139(2):260–288

Dobrzynski H, Anderson RH, Atkinson A, et al (2013) Structure, function and clinical relevance of the cardiac conduction system, including the atrioventricular ring and outflow tract tissues. Pharmacology & therapeutics 139(2):260–288

2013
[8]

Circulation 41(6):899–912

Durrer D, Van Dam RT, Freud G, et al (1970) Total excitation of the isolated human heart. Circulation 41(6):899–912

1970
[9]

Journal of mathematical biology 29(7):629–651

Keener JP (1991) An eikonal-curvature equation for action potential propagation in myocardium. Journal of mathematical biology 29(7):629–651

1991
[10]

Circulation 115(10):1306–1324

Kligfield P, Gettes LS, Bailey JJ, et al (2007) Recommendations for the standard- ization and interpretation of the electrocardiogram: part i: the electrocardiogram and its technology: a scientific statement from the american heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american college of cardiology...

2007
[11]

Journal of Biomedical Informatics 172:104948

Lin Y, Ma J, Dong S, et al (2025) Transdiffecg: Semantically controllable ecg synthesis via transformer-based diffusion modeling. Journal of Biomedical Informatics 172:104948. https://doi.org/10.1016/j.jbi.2025.104948

work page doi:10.1016/j.jbi.2025.104948 2025
[12]

Journal of computational physics 346:191–211

Neic A, Campos FO, Prassl AJ, et al (2017) Efficient computation of electro- grams and ecgs in human whole heart simulations using a reaction-eikonal model. Journal of computational physics 346:191–211

2017
[13]

Pacing and Clinical Electrophysiology 44(1):15–25

Padala SK, Cabrera JA, Ellenbogen KA (2021) Anatomy of the cardiac conduc- tion system. Pacing and Clinical Electrophysiology 44(1):15–25

2021
[14]

Circulation 119(10):e241–e250

Rautaharju PM, Surawicz B, Gettes LS (2009) Aha/accf/hrs recommendations for the standardization and interpretation of the electrocardiogram: part iv: the st segment, t and u waves, and the qt interval: a scientific statement from the amer- ican heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american c...

2009
[15]

Heart Rhythm 5(8):1213–1215

Roden DM (2008) Keep the qt interval: it is a reliable predictor of ventricular arrhythmias. Heart Rhythm 5(8):1213–1215

2008
[16]

Rudy Y (2015) The forward problem of electrocardiography revisited

2015
[17]

Radiographics 37(5):1330–1351 19

Saremi F, S´ anchez-Quintana D, Mori S, et al (2017) Fibrous skeleton of the heart: anatomic overview and evaluation of pathologic conditions with ct and mr imaging. Radiographics 37(5):1330–1351 19

2017
[18]

proceedings of the National Academy of Sciences 93(4):1591–1595

Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. proceedings of the National Academy of Sciences 93(4):1591–1595

1996
[19]

Scientific Data 10(1):279

Strodthoff N, Mehari T, Nagel C, et al (2023) PTB-XL+, a comprehensive elec- trocardiographic feature dataset. Scientific Data 10(1):279. https://doi.org/10. 1038/s41597-023-02153-8

2023
[20]

Journal of the American College of Cardiology 53(11):976–981

Surawicz B, Childers R, Deal BJ, et al (2009) Aha/accf/hrs recommendations for the standardization and interpretation of the electrocardiogram: part iii: intraven- tricular conduction disturbances a scientific statement from the american heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american college o...

2009
[21]

Zhongwei Wan, Che Liu, Mi Zhang, Jie Fu, Benyou Wang, Sibo Cheng, Lei Ma, César Quilodrán- Casas, and Rossella Arcucci

Wagner P, Strodthoff N, Bousseljot RD, et al (2020) PTB-XL, a large publicly available electrocardiography dataset. Scientific Data 7(1):154. https://doi.org/ 10.1038/s41597-020-0495-6

work page doi:10.1038/s41597-020-0495-6 2020
[22]

Biomedical Signal Processing and Control 82:104587

Xia Y, Wang W, Wang K (2023) Ecg signal generation based on conditional generative models. Biomedical Signal Processing and Control 82:104587. https: //doi.org/10.1016/j.bspc.2023.104587

work page doi:10.1016/j.bspc.2023.104587 2023
[23]

Computers in Biology and Medicine 184:109453

Zanchi B, Monachino G, Fiorillo L, et al (2025) Synthetic ecg signals generation: A scoping review. Computers in Biology and Medicine 184:109453. https://doi. org/10.1016/j.compbiomed.2024.109453

work page doi:10.1016/j.compbiomed.2024.109453 2025
[24]

Journal of Electrocardiology 69:6–14

Zhang YH, Babaeizadeh S (2021) Synthesis of standard 12-lead electrocar- diograms using two-dimensional generative adversarial networks. Journal of Electrocardiology 69:6–14. https://doi.org/10.1016/j.jelectrocard.2021.08.019 20 Supplementary Material S1 Proofs of main results For convenience, let X:={x:V→R∪ {∞}:x(s) = 0 for alls∈S}, letN:=|V|, letFdenote...

work page doi:10.1016/j.jelectrocard.2021.08.019 2021

[1] [1]

Chaos, Solitons & Fractals 7(3):293–301

Aliev RR, Panfilov AV (1996) A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals 7(3):293–301

1996

[2] [2]

Antzelevitch C, Viskin S, Shimizu W, et al (2007) Does tpeak-tend provide an index of transmural dispersion of repolarization? Heart rhythm: the official journal of the Heart Rhythm Society 4(8):1114

2007

[3] [3]

Quarterly of applied mathematics 16(1):87–90

Bellman R (1958) On a routing problem. Quarterly of applied mathematics 16(1):87–90

1958

[4] [4]

Journal of theoretical biology 253(3):544–560

Bueno-Orovio A, Cherry EM, Fenton FH (2008) Minimal model for human ven- tricular action potentials in tissue. Journal of theoretical biology 253(3):544–560

2008

[5] [5]

Medical & biological engineering & computing 47(1):11–20

van Dam PM, Oostendorp TF, van Oosterom A (2009) Application of the fastest route algorithm in the interactive simulation of the effect of local ischemia on the ecg. Medical & biological engineering & computing 47(1):11–20

2009

[6] [6]

In: Edsger Wybe Dijkstra: his life, work, and legacy

Dijkstra EW (2022) A note on two problems in connexion with graphs. In: Edsger Wybe Dijkstra: his life, work, and legacy. Association for Computing Machinery, 18 p 287–290

2022

[7] [7]

Pharmacology & therapeutics 139(2):260–288

Dobrzynski H, Anderson RH, Atkinson A, et al (2013) Structure, function and clinical relevance of the cardiac conduction system, including the atrioventricular ring and outflow tract tissues. Pharmacology & therapeutics 139(2):260–288

2013

[8] [8]

Circulation 41(6):899–912

Durrer D, Van Dam RT, Freud G, et al (1970) Total excitation of the isolated human heart. Circulation 41(6):899–912

1970

[9] [9]

Journal of mathematical biology 29(7):629–651

Keener JP (1991) An eikonal-curvature equation for action potential propagation in myocardium. Journal of mathematical biology 29(7):629–651

1991

[10] [10]

Circulation 115(10):1306–1324

Kligfield P, Gettes LS, Bailey JJ, et al (2007) Recommendations for the standard- ization and interpretation of the electrocardiogram: part i: the electrocardiogram and its technology: a scientific statement from the american heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american college of cardiology...

2007

[11] [11]

Journal of Biomedical Informatics 172:104948

Lin Y, Ma J, Dong S, et al (2025) Transdiffecg: Semantically controllable ecg synthesis via transformer-based diffusion modeling. Journal of Biomedical Informatics 172:104948. https://doi.org/10.1016/j.jbi.2025.104948

work page doi:10.1016/j.jbi.2025.104948 2025

[12] [12]

Journal of computational physics 346:191–211

Neic A, Campos FO, Prassl AJ, et al (2017) Efficient computation of electro- grams and ecgs in human whole heart simulations using a reaction-eikonal model. Journal of computational physics 346:191–211

2017

[13] [13]

Pacing and Clinical Electrophysiology 44(1):15–25

Padala SK, Cabrera JA, Ellenbogen KA (2021) Anatomy of the cardiac conduc- tion system. Pacing and Clinical Electrophysiology 44(1):15–25

2021

[14] [14]

Circulation 119(10):e241–e250

Rautaharju PM, Surawicz B, Gettes LS (2009) Aha/accf/hrs recommendations for the standardization and interpretation of the electrocardiogram: part iv: the st segment, t and u waves, and the qt interval: a scientific statement from the amer- ican heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american c...

2009

[15] [15]

Heart Rhythm 5(8):1213–1215

Roden DM (2008) Keep the qt interval: it is a reliable predictor of ventricular arrhythmias. Heart Rhythm 5(8):1213–1215

2008

[16] [16]

Rudy Y (2015) The forward problem of electrocardiography revisited

2015

[17] [17]

Radiographics 37(5):1330–1351 19

Saremi F, S´ anchez-Quintana D, Mori S, et al (2017) Fibrous skeleton of the heart: anatomic overview and evaluation of pathologic conditions with ct and mr imaging. Radiographics 37(5):1330–1351 19

2017

[18] [18]

proceedings of the National Academy of Sciences 93(4):1591–1595

Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. proceedings of the National Academy of Sciences 93(4):1591–1595

1996

[19] [19]

Scientific Data 10(1):279

Strodthoff N, Mehari T, Nagel C, et al (2023) PTB-XL+, a comprehensive elec- trocardiographic feature dataset. Scientific Data 10(1):279. https://doi.org/10. 1038/s41597-023-02153-8

2023

[20] [20]

Journal of the American College of Cardiology 53(11):976–981

Surawicz B, Childers R, Deal BJ, et al (2009) Aha/accf/hrs recommendations for the standardization and interpretation of the electrocardiogram: part iii: intraven- tricular conduction disturbances a scientific statement from the american heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american college o...

2009

[21] [21]

Zhongwei Wan, Che Liu, Mi Zhang, Jie Fu, Benyou Wang, Sibo Cheng, Lei Ma, César Quilodrán- Casas, and Rossella Arcucci

Wagner P, Strodthoff N, Bousseljot RD, et al (2020) PTB-XL, a large publicly available electrocardiography dataset. Scientific Data 7(1):154. https://doi.org/ 10.1038/s41597-020-0495-6

work page doi:10.1038/s41597-020-0495-6 2020

[22] [22]

Biomedical Signal Processing and Control 82:104587

Xia Y, Wang W, Wang K (2023) Ecg signal generation based on conditional generative models. Biomedical Signal Processing and Control 82:104587. https: //doi.org/10.1016/j.bspc.2023.104587

work page doi:10.1016/j.bspc.2023.104587 2023

[23] [23]

Computers in Biology and Medicine 184:109453

Zanchi B, Monachino G, Fiorillo L, et al (2025) Synthetic ecg signals generation: A scoping review. Computers in Biology and Medicine 184:109453. https://doi. org/10.1016/j.compbiomed.2024.109453

work page doi:10.1016/j.compbiomed.2024.109453 2025

[24] [24]

Journal of Electrocardiology 69:6–14

Zhang YH, Babaeizadeh S (2021) Synthesis of standard 12-lead electrocar- diograms using two-dimensional generative adversarial networks. Journal of Electrocardiology 69:6–14. https://doi.org/10.1016/j.jelectrocard.2021.08.019 20 Supplementary Material S1 Proofs of main results For convenience, let X:={x:V→R∪ {∞}:x(s) = 0 for alls∈S}, letN:=|V|, letFdenote...

work page doi:10.1016/j.jelectrocard.2021.08.019 2021