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REVIEW 3 major objections 4 minor 58 references

Particle Filtering for Non-Deterministic Electrocardiographic Imaging

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper makes Bayesian particle filtering tractable for electrocardiographic imaging by compressing activation fronts into a few geodesic balls, yielding probability maps rather than deterministic reconstructions.

desk verdict A credible proof-of-concept for particle-filtering ECGI with honest caveats; the UQ claims need calibration checks, but the low-dimensional state representation is a genuine step forward. read the letter →

arxiv 2509.19404 v2 pith:X2TNNL46 submitted 2025-09-23 stat.ME math.PRphysics.data-anstat.AP

Emma Lagracie (UB , CARMEN) , Luc de Montella (ASTRAL) This is my paper · ORCID
classification stat.MEmath.PRphysics.data-anstat.AP MSC 62F1565C0565N2192C55
keywords ECGIparticlefilteringsequentialMonteCarloBayesianinverseproblemactivationprobabilityearliestsiteconductionlineofblockgeodesicballrepresentation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Electrocardiographic imaging (ECGI) reconstructs heart activation from torso potentials, but it is ill-posed and usually returns a single deterministic map. This paper argues that the activation sequence can be compressed into a few geodesic balls — each with a center and a radius — so that the hidden state has only a handful of parameters. That compression makes particle filtering computationally feasible, and the filter then returns a full posterior over activation fronts rather than one answer. From that posterior, the authors build activation maps, time-resolved activation probability maps, pseudo-probability maps for earliest activation sites, and a probability that a candidate conduction line of block is real. In simulated tests with deliberate conductivity mismatch, reconstructed activation maps remain above 0.95 correlation with reference, and the uncertainty regions stay informative.

What carries the argument

The load-bearing machinery is the low-dimensional state representation in Eqs. (6)-(8): the state is X_k = (c^1_k,...,c^l_k, r^1_k,...,r^l_k), where centers live on heart-mesh nodes and radii are positive reals; the transmembrane voltage is v(x) = max_i V(r_i - d(x,c_i)) with V a smoothed Heaviside and d a conductivity-weighted geodesic distance. This reduces the state dimension from the full nodal voltage vector to fewer than eight parameters, making a standard sequential importance resampling particle filter feasible, and defines the nonlinear observation map Y_k = O v_{X_k}|_{Gamma_T} + noise whose likelihood drives the particle weights.

What would settle it

Simulate or record an activation sequence whose wavefront is visibly non-ball-like — for example, a re-entrant circuit, a U-shaped block, or two fronts colliding — then run the filter on the resulting torso potentials; the central claim is falsified if the true activated regions fall outside the high-probability maps or if the reconstructed activation map misses the wavefront.

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Extended reading notes

Core claim

On its own terms, the paper claims that Bayesian particle filtering can solve the non-deterministic ECGI problem if the cardiac activation front is parameterized as l (typically fewer than four) growing geodesic balls on the heart mesh, each described by a center and a radius, with a smoothed-Heaviside transmembrane voltage profile. The forward map from these parameters to torso potentials is nonlinear, and the authors propagate a particle cloud through it with a deliberately simple random-walk state model. The resulting posterior supports quantities that deterministic solvers cannot provide: per-point activation probability over time, pseudo-probability of being an earliest activation site,

Load-bearing premise

The load-bearing premise is the Section 3.1 state parameterization: an activation front is always a union of a few growing geodesic balls with a fixed smoothed step profile; if a real front is fragmented or shaped by complex block geometry, the posterior and probability maps are misleading, and the paper itself notes in the conclusion that an unmodeled line of block will not be detected.

Editorial extensions

If this is right

  • Activation maps no longer need to be single deterministic outputs: every point on the heart gets a time-dependent activation probability, so clinicians can see where the reconstruction is trustworthy.
  • Earliest activation sites, relevant for localizing premature ventricular contractions, are returned as pseudo-probability regions rather than point estimates, and the paper reports that true sites fall inside those regions in most anisotropic test cases.
  • Candidate lines of block from any deterministic ECGI method can be tested by giving the filter a geodesic metric with and without the block; the filter's mode probability indicates which is supported by the data.
  • The forward and backward filter runs can be combined to compensate for the poor early-time behavior of the forward run and the poor late-time behavior of the backward run.
  • Because the state stays low-dimensional, adding more activation centers or discrete model choices is computationally straightforward, which opens the door to more complex priors or fiber-aware distances when such information is available.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same ball-and-radius state could be used as a proposal or initialization layer inside existing deterministic ECGI pipelines, since the posterior gives a natural prior over activation centers and speeds for a subsequent optimization.
  • The forward/backward asymmetry suggests a formal two-filter smoothing pass might yield a single calibrated posterior, rather than the averaged pseudo-probabilities the paper displays.
  • A natural testable extension is the low-electrode-count clinical setting (148-255 electrodes): the method's uncertainty maps should be evaluated on whether the probability regions enlarge gracefully as electrode density drops, since the paper's simulated data use around 1000 torso points.
  • The discrete-mode trick for lines of block generalizes beyond block detection: any discrete biophysical hypothesis (scar location, fiber orientation, or ionic model variant) could be made a competing mode in the same particle filter and scored by its posterior weight.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a particle-filtering approach to electrocardiographic imaging (ECGI) in which the cardiac activation front is represented by a small number of growing geodesic balls (centers and radii), thereby reducing the state dimension enough to make sequential Monte Carlo tractable. The method outputs, in addition to deterministic activation maps, activation probability maps, pseudo-probability maps for earliest activation sites, and a discrete-mode extension that assigns probabilities to candidate conduction lines of block. The numerical study uses simulated bidomain data, including isotropic and anisotropic cases, deliberately mismodeled conductivity in the inverse model, and a separate coarser mesh to avoid inverse crime. The reported activation-map correlations exceed 0.95 in the anisotropic mismatched cases, and the block-line mode probabilities correctly favor the true block in most tested scenarios.

Significance. If the probabilistic outputs are taken at face value, the paper contributes a genuinely new capability to ECGI: posterior inference over activation sequences, with clinically interpretable confidence maps and a mechanism to assess whether a detected line of block is physiological or an artifact. The design is thoughtful: the low-dimensional state parameterization is the key enabling idea, the numerical experiments deliberately include model mismatch, and the paper is appropriately cautious about the heuristic nature of pseudo-probabilities. The strongest empirical results, activation correlations above 0.95 under anisotropic mismodeling, support the feasibility claim. However, the central added value is uncertainty quantification, and the paper does not yet demonstrate that the reported probabilities are calibrated or robust to the hand-tuned observation covariance. That gap prevents the current version from fully supporting the probabilistic claims.

major comments (3)
  1. [§4.1.2, Eq. (12)] The observation noise covariance Σ_w is fine-tuned 'to allow sufficient diversity among particles while avoiding excessive resampling.' The likelihood scale set by Σ_w directly controls the spread of every probabilistic output (Eqs. 16–18), yet no sensitivity analysis or calibration check is provided. The paper's central claim is uncertainty-aware inference; for this, the posterior probabilities should be shown to be meaningful (e.g., via reliability diagrams, coverage of activation times, or a sensitivity study over a range of Σ_w). Without such evidence, the reported probability maps may be visually suggestive but are not established as calibrated uncertainty statements.
  2. [§4.3, Eq. (18)] The block-line extension is described as letting each particle choose among 'a predefined set of geodesic distances and corresponding transfer matrix O.' However, the experimental section only states that two distance maps were provided; it never specifies how O is modified for each mode. If O is in fact identical across modes, then the mode probability (18) only compares alternative activation-front geometries under the same forward model, not the conductivity changes that a true block would induce. The paper should either document the mode-dependent O or explicitly state that only the geodesic distance varies, and justify that this is sufficient to discriminate true from artificial block lines.
  3. [Figures 7–8 and 11] Several headline results are averages of forward and backward filter outputs, e.g., (P_fwd + P_bwd)/2 and the block-mode probabilities 'averaged across 50 forward and 50 backward filter outputs.' No justification is given for this averaging, and the forward and backward filters have different bias patterns (the paper itself notes the forward method is inaccurate early and the backward method is inaccurate late). A simple average is not a posterior quantity and may hide systematic disagreement. The authors should either provide a decision-theoretic justification for the combination or report forward and backward results separately, with a measure of agreement.
minor comments (4)
  1. [Abstract and Conclusion] Typos: 'a conduction lines of block' and 'a priori block' (twice). Please also check 'visulization' and 'F or' in Algorithm 1.
  2. [§4.2] The activation probability maps in Figures 4–6 would benefit from colorbars and a consistent scale across time frames and methods; currently the visual comparison is difficult.
  3. [§4.3] The sentence 'we ran 50 forward and 50 backward filters' is not followed by a description of how the 50 runs differ (only particle initialization randomness is implied). Please clarify whether these are independent SMC runs and how variance across runs was assessed.
  4. [References] Reference [51] duplicates [37]; [52] is cited for SBDF2 but may be better attributed to the original method; please verify citation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: probabilistic outputs are genuine data-conditioned posterior quantities; self-citations are not load-bearing.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The state X_k (Eq. 6) parameterizes activation fronts by l geodesic-ball centers and radii; Eq. 8 defines v_{X_k}(x) from that state; Eqs. 11-12 link this to torso observations via a likelihood; and the SIR filter (Algorithm 1) produces the posterior p(X_k|Y_{1:k}). The reported activation probability map (Eq. 16) is P(v(x,t_k)>0.5|Y_{1:k}), i.e., a posterior predictive under that explicit generative model. The earliest-activation pseudo-probability (Eq. 17) is the posterior of the center parameters, honestly labeled 'pseudo' because it is not normalized; it is an inference about latent variables, not a fitted constant restated as a prediction. The line-of-block probability (Eq. 18) is a Bayesian model posterior over a discrete set of candidate geodesic distances/transfer matrices; providing the candidate block maps as inputs is standard model selection, not circular. No fitted parameter is renamed as a prediction: Σ_w is a hand-tuned observation-noise covariance, not an output claimed to be predicted. Self-citations (refs. [47],[48],[57],[58] and [23],[26]) appear, but they support peripheral choices (alternative surface source models, front-width sensitivity, activation-map post-processing, contextual SMC references) and are not load-bearing for the central claim. Concerns about calibration of the probability maps or misspecification of the ball-model are correctness/model-risk issues, not circularity. The numerical experiments use a different inverse mesh and deliberately mismodeled conductivity in anisotropic cases, so the verification is not built into the derivation. No step reduces by construction to its inputs.

Assumptions & free parameters 8 free parameters · 7 assumptions · 2 invented entities

The central claim rests on the adequacy of the low-dimensional ball parameterization, the simplified isotropic observation model, and the random-walk dynamics; all reported results depend on several hand-chosen or fine-tuned hyperparameters (Sigma_w, Sigma_r, lambda, width, initial radii, l, threshold, mode-switching probability) with no sensitivity analysis. The numerical study against an independent bidomain forward model provides the only validation; no external benchmark or dataset is used to fix the parameters.

free parameters (8)
  • number of activation centers l = 3
    Hand-set to 3, strictly greater than the maximum number of true stimulation sites (2), so it acts as an upper bound; extra centers can have null radius.
  • radius evolution noise std Sigma_r = 10 mm
    Chosen deliberately large to allow the filter to track the signal; no selection criterion is given beyond heuristics.
  • center displacement parameter lambda = 5 mm
    Chosen as a compromise between center mobility and stability; hand-tuned.
  • observation noise covariance Sigma_w = (0.02)^2 I_d
    Tuned to the data by estimating the Frobenius norm of residual between observed and reconstructed torso potentials from random particles, then fine-tuned to balance particle diversity and resampling.
  • action potential front width = 5 mm
    A priori value from classical bidomain front width; not fitted here but a free modeling choice that affects the likelihood.
  • initial radii r_0 = 1 mm (fwd), 150 mm (bwd)
    Chosen so that the initial activated region is near-minimal in the forward method or covers most of the heart in the backward method.
  • activation threshold = 0.5
    Threshold on transmembrane voltage used to define a point as 'activated' in the probability maps.
  • mode-switching probability = 0.99 keep, 0.01 switch
    Hand-set to allow rare switches among candidate geodesic distance maps in the line-of-block extension.
assumptions (7)
  • domain assumption The activation sequence can be represented by a small number of growing geodesic balls with a predefined smoothed-Heaviside shape (Eqs. 7-8).
    This is the representational premise of the entire state-space model (Section 3.1). If the true activation front is not ball-like, the posterior is misspecified.
  • domain assumption The torso potentials are adequately modeled by the electrostatic equilibrium equations (9) with isotropic conductivities, even when the true data were generated with anisotropic fibers.
    Used in constructing the transfer matrix O (Section 4.1.2); the authors intentionally introduce a modeling error, relying on the premise that it is tolerable.
  • domain assumption The random-walk dynamics (Eq. 10 plus exponential-uniform center proposal) contain enough information, combined with the likelihood, to reconstruct the activation sequence without physical priors.
    Section 3.2. The state equation does not encode propagation speed or direction; the authors rely on the data to compensate.
  • domain assumption SIR with N=1000 particles gives a sufficiently accurate posterior approximation for the reported probability estimates.
    No convergence diagnostics or effective-sample-size traces are reported; finite-N behavior is assumed adequate.
  • ad hoc to paper Averaging forward and backward filter estimates yields a meaningful combined estimate.
    Section 4.2 uses (Pfwd + Pbwd)/2 for pseudo-probability maps and averages mode probabilities over 50 forward and 50 backward runs; no theory justifies this as Bayesian inference.
  • ad hoc to paper For line-of-block discrimination, the user can pre-specify candidate distance maps whose blocks correspond to plausible physiological or artificial lines of block.
    Section 4.3. The filter only selects among provided candidates; it cannot discover a block not represented in any candidate distance map.
  • domain assumption The bidomain model with Mitchell-Schaeffer ionic model produces realistic ground truth for validating the filter.
    Data generation in Section 4.1.1; the filter is validated against this model, not against clinical data.
invented entities (2)
  • Geodesic-ball activation state (centers and radii)
    purpose: Low-dimensional state representation to make particle filtering tractable for ECGI
    Introduced in Section 3.1 as a modeling construct rather than a physical entity; the only evidence is the paper's own bidomain simulations, so it lacks an external handle.
  • Discrete distance-mode state for line-of-block discrimination
    purpose: Enable the filter to estimate the probability that a candidate conduction block is real
    Section 4.3. The mode is a new discrete state variable; its usefulness is tested only internally on the Block1/Block2 simulations, not against independent external data.

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Cite this review

Pith. "Pith review of Particle Filtering for Non-Deterministic Electrocardiographic Imaging." pith.science (2026). https://pith.science/paper/X2TNNL46

@misc{pith2026250919404,
  author       = {Pith},
  title        = {Pith review of: Particle Filtering for Non-Deterministic Electrocardiographic Imaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2TNNL46}},
  note         = {Machine review of arXiv:2509.19404}
}
read the original abstract

Electrocardiographic imaging (ECGI) aims to non-invasively reconstruct activation maps of the heart from temporal body surface potentials. While most existing approaches rely on inverse and optimization techniques that may yield satisfactory reconstructions, they typically provide a single deterministic solution, overlooking the inherent uncertainty of the problem stemming from its very ill-posed nature, the poor knowledge of biophysical features and the unavoidable presence of noise in the measurements. The Bayesian framework, which naturally incorporates uncertainty while also accounting for temporal correlations across time steps, can be used to address this limitation. In this work, we propose a low-dimensional representation of the activation sequence that enables the use of particle filtering, a Bayesian filtering method that does not rely on predefined assumptions regarding the shape of the posterior distribution, in contrast to approaches like the Kalman filter. This allows to produce not only activation maps but also probabilistic maps indicating the likelihood of activation at each point on the heart over time, as well as pseudo-probability maps reflecting the likelihood of a point being part of an earliest activation site. Additionally, we introduce a method to estimate the probability of the presence of a conduction lines of block on the heart surface. Combined with classical reconstruction techniques, this could help discriminate artificial from true lines of block in activation maps. We support our approach with a numerical study based on simulated data, demonstrating the potential of our method.

Figures

Figures reproduced from arXiv: 2509.19404 by the authors.

Figure 1
Figure 1. Schematic representation of the domain Ω = Ω [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Reconstructed activation maps for the first four isotropic propagation test cases. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Reconstructed activation maps for the first four anisotropic propagation test cases. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Probability of being activated at different times for stimulation [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Probability of being activated at different times for stimulation [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Probability of being activated at different times for stimulation [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Pseudo-probability of belonging to an initial activation site for the first four isotropic [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Pseudo-probability of belonging to an initial activation site for the first four anisotropic [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Activation maps obtained with only one homogeneous geodesic distance. The conduc [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Example of the computed geodesic distance from the points in the red zone, with an [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Evolution of the probability distribution over geodesic distance modes, averaged [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Activation maps obtained with an extended state which includes the choice between [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

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Reference graph

Works this paper leans on

58 extracted references

  1. [1]

    The inverse problem of electrocardiography

    Andrew J Pullan, Leo K Cheng, Martyn P Nash, Alireza Ghodrati, Rob MacLeod, Dana H Brooks, et al. The inverse problem of electrocardiography. Comprehensive electrocardiology, 1:299–344, 2010

    2010

  2. [2]

    Introduction to noninvasive cardiac mapping

    Laura Bear, Phillip S Cuculich, Olivier Bernus, Igor Efimov, and R´ emi Dubois. Introduction to noninvasive cardiac mapping. Cardiac electrophysiology clinics, 7(1):1–16, 2015

    2015

  3. [3]

    Noninvasive re- construction of cardiac electrical activity: update on current methods, applications and challenges

    Matthijs J M Cluitmans, RLM Peeters, RL Westra, and PGA Volders. Noninvasive re- construction of cardiac electrical activity: update on current methods, applications and challenges. Netherlands Heart Journal , 23:301–311, 2015

    2015

  4. [4]

    Reducing line-of-block artifacts in cardiac activation maps estimated using ecg imaging: A comparison of source models and estimation methods

    Steffen Schuler, Matthias Schaufelberger, Laura R Bear, Jake A Bergquist, Matthijs JM Cluitmans, Jaume Coll-Font, ¨Onder N Onak, Brian Zenger, Axel Loewe, Rob S MacLeod, et al. Reducing line-of-block artifacts in cardiac activation maps estimated using ecg imaging: A comparison of source models and estimation methods. IEEE Transactions on Biomedical Engin...

  5. [5]

    Why is the cauchy problem severely ill-posed? Inverse problems , 23(2):823, 2007

    Faker Ben Belgacem. Why is the cauchy problem severely ill-posed? Inverse problems , 23(2):823, 2007

    2007

  6. [6]

    Solutions of ill-posed problems

    AN Tikhonov and V Arsenin. Solutions of ill-posed problems. (No Title) , 1977

    1977

  7. [7]

    Cam- bridge university press, 2023

    Simo S¨ arkk¨ a and Lennart Svensson.Bayesian filtering and smoothing , volume 17. Cam- bridge university press, 2023

    2023

  8. [8]

    Phillips, R.M

    J.W. Phillips, R.M. Leahy, and J.C. Mosher. Meg-based imaging of focal neuronal current sources. IEEE Transactions on Medical Imaging , 16(3):338–348, 1997. 24

    1997

Show all 58 references
  1. [9]

    Baillet and L

    S. Baillet and L. Garnero. A bayesian approach to introducing anatomo-functional priors in the eeg/meg inverse problem. IEEE Transactions on Biomedical Engineering, 44(5):374– 385, 1997

    1997

  2. [10]

    Bayesian inference applied to the electromagnetic inverse problem

    David M Schmidt, John S George, and Chris C Wood. Bayesian inference applied to the electromagnetic inverse problem. Human brain mapping , 7(3):195–212, 1999

    1999

  3. [11]

    Umit Aydin and Yesim Serinagaoglu Dogrusoz. A kalman filter-based approach to reduce the effects of geometric errors and the measurement noise in the inverse ecg problem.Medical & Biological Engineering & Computing , 49(9):1003–1013, 2011

    2011

  4. [12]

    Time-recursive solution to the inverse problem of electrocardiography

    Yves Goussard, Diane Joly, and Pierre Savard. Time-recursive solution to the inverse problem of electrocardiography. IEEE Transactions on Biomedical Engineering , 1993

    1993

  5. [13]

    Kalman filter with augmented measurement model: an ecg imaging simulation study

    Walther HW Schulze, Francesc Elies Henar, Danila Potyagaylo, Axel Loewe, Matti Sten- roos, and Olaf D¨ ossel. Kalman filter with augmented measurement model: an ecg imaging simulation study. In International Conference on Functional Imaging and Modeling of the Heart, pages 200...

    2013

  6. [14]

    Statistical estimation applied to electrocardiographic imag- ing

    Yesim Serinagaoglu Dogrusoz. Statistical estimation applied to electrocardiographic imag- ing. In 2019 12th International Conference on Measurement , pages 2–9. IEEE, 2019

    2019

  7. [15]

    The unscented kalman filter for nonlinear estimation

    Eric A Wan and Rudolph Van Der Merwe. The unscented kalman filter for nonlinear estimation. In Proceedings of the IEEE 2000 adaptive systems for signal processing, com- munications, and control symposium (Cat. No. 00EX373) , pages 153–158. Ieee, 2000

    2000

  8. [16]

    Physiological- model-constrained noninvasive reconstruction of volumetric myocardial transmembrane po- tentials

    Linwei Wang, Heye Zhang, Ken CL Wong, Huafeng Liu, and Pengcheng Shi. Physiological- model-constrained noninvasive reconstruction of volumetric myocardial transmembrane po- tentials. IEEE Transactions on Biomedical Engineering , 57(2):296–315, 2009

    2009

  9. [17]

    Noninvasive estimation of global activation sequence using the extended kalman filter

    Chenguang Liu and Bin He. Noninvasive estimation of global activation sequence using the extended kalman filter. IEEE Transactions on Biomedical Engineering , 58(3):541–549, 2010

    2010

  10. [18]

    Bayesian solutions and performance analysis in bioelectric inverse problems

    Yesim Serinagaoglu, Dana H Brooks, and Robert S MacLeod. Bayesian solutions and performance analysis in bioelectric inverse problems. IEEE Transactions on Biomedical Engineering, 52(6):1009–1020, 2005

    2005

  11. [19]

    Particle filtering for large-dimensional state spaces with multimodal observation likelihoods

    Namrata Vaswani. Particle filtering for large-dimensional state spaces with multimodal observation likelihoods. IEEE Transactions on Signal Processing , 56(10):4583–4597, 2008

    2008

  12. [20]

    Sequential monte carlo methods for dynamic systems

    Jun S Liu and Rong Chen. Sequential monte carlo methods for dynamic systems. Journal of the American statistical association , 93(443):1032–1044, 1998

    1998

  13. [21]

    Genetic algorithms in electromagnetics

    Randy L Haupt and Douglas H Werner. Genetic algorithms in electromagnetics . John Wiley & Sons, 2007

    2007

  14. [22]

    Particle filters and bayesian inference in financial econometrics

    Hedibert F Lopes and Ruey S Tsay. Particle filters and bayesian inference in financial econometrics. Journal of Forecasting, 30(1):168–209, 2011

    2011

  15. [23]

    Target motion analysis using angular measurements and underwater sound pressure levels

    Enzo Igl´ esis and Luc de Montella. Target motion analysis using angular measurements and underwater sound pressure levels. working paper or preprint, 2025

    2025

  16. [24]

    On the stability of measure valued processes with ap- plications to filtering

    Pierre Del Moral and Alice Guionnet. On the stability of measure valued processes with ap- plications to filtering. Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics, 329(5):429–434, 1999

    1999

  17. [25]

    An introduction to sequential Monte Carlo, volume 4

    Nicolas Chopin, Omiros Papaspiliopoulos, et al. An introduction to sequential Monte Carlo, volume 4. Springer, 2020. 25

    2020

  18. [26]

    On the mathematical foundations of diffusion monte carlo

    Michel Caffarel, Pierre Del Moral, and Luc de Montella. On the mathematical foundations of diffusion monte carlo. Journal of Mathematical Physics , 66(1), 2025

    2025

  19. [27]

    Non-invasive imaging of cardiac activation and recovery

    Peter M Van Dam, Thom F Oostendorp, Andr´ e C Linnenbank, and Adriaan Van Oosterom. Non-invasive imaging of cardiac activation and recovery. Annals of biomedical engineering, 37:1739–1756, 2009

    2009

  20. [28]

    A parameter optimiza- tion to solve the inverse problem in electrocardiography

    Gwladys Ravon, R´ emi Dubois, Yves Coudi` ere, and Mark Potse. A parameter optimiza- tion to solve the inverse problem in electrocardiography. In International Conference on Functional Imaging and Modeling of the Heart , pages 219–229. Springer, 2017

    2017

  21. [29]

    Source and metric estimation in the eikonal equation using optimization on a manifold

    J´ erˆ ome Fehrenbach and Lisl Weynans. Source and metric estimation in the eikonal equation using optimization on a manifold. Inverse Problems and Imaging , 17(2):419–440, 2023

    2023

  22. [30]

    Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field

    Piero Colli Franzone, Luciano Guerri, and Stefania Tentoni. Mathematical modeling of the excitation process in myocardial tissue: Influence of fiber rotation on wavefront propagation and potential field. Mathematical biosciences, 101(2):155–235, 1990

    1990

  23. [31]

    Geasi: Geodesic-based earliest activation sites identification in cardiac mod- els

    Thomas Grandits, Alexander Effland, Thomas Pock, Rolf Krause, Gernot Plank, and Si- mone Pezzuto. Geasi: Geodesic-based earliest activation sites identification in cardiac mod- els. International journal for numerical methods in biomedical engineering , 37(8):e3505, 2021

    2021

  24. [32]

    Ml and map estimation of parameters for the kalman filter and smoother applied to electrocardiographic imaging

    Taha Erenler and Yesim Serinagaoglu Dogrusoz. Ml and map estimation of parameters for the kalman filter and smoother applied to electrocardiographic imaging. Medical & Biological Engineering & Computing , 57:2093–2113, 2019

  25. [33]

    Use of simulated data for the estimation of prior models in kalman filter-based ecgi

    Yesim Serinagaoglu Dogrusoz and Taha Erenler. Use of simulated data for the estimation of prior models in kalman filter-based ecgi. In 2020 Computing in Cardiology , pages 1–4. IEEE, 2020

    2020

  26. [34]

    Modeling the electrical activity of the heart: a bidomain model of the ventricles embedded in a torso.Computing and Visualization in Science, 5:195–213, 2003

    Glenn Terje Lines, Per Grottum, and Aslak Tveito. Modeling the electrical activity of the heart: a bidomain model of the ventricles embedded in a torso.Computing and Visualization in Science, 5:195–213, 2003

    2003

  27. [35]

    A bi-domain model for describing ischemic myocardial dc potentials

    Leslie Tung. A bi-domain model for describing ischemic myocardial dc potentials. PhD thesis, Massachusetts Institute of Technology, 1978

    1978

  28. [36]

    Solving the inverse problem of electrocardiography for cardiac digital twins: A survey

    Lei Li, Julia Camps, Blanca Rodriguez, and Vicente Grau. Solving the inverse problem of electrocardiography for cardiac digital twins: A survey. IEEE Reviews in Biomedical Engineering, 2024

    2024

  29. [37]

    Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models

    Jeanne van der Waal, Veronique Meijborg, Ruben Coronel, R´ emi Dubois, and Thom Oos- tendorp. Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models. Frontiers in physiology, 14:1295103, 2023

    2023

  30. [38]

    Performance and limitations of noninvasive cardiac activation mapping

    Josselin Duchateau, Fr´ ed´ eric Sacher, Thomas Pambrun, Nicolas Derval, Judit Chamorro- Servent, Arnaud Denis, Sylvain Ploux, M´ el` eze Hocini, Pierre Ja ¨ ıs, Olivier Bernus, et al. Performance and limitations of noninvasive cardiac activation mapping. Heart rhythm , 16(3):...

    2019

  31. [39]

    Localization of activation origin on patient-specific epicardial surface by empirical bayesian method

    Shijie Zhou, John L Sapp, Fady Dawoud, and B Milan Hor´ aˇ cek. Localization of activation origin on patient-specific epicardial surface by empirical bayesian method. IEEE Transac- tions on Biomedical Engineering , 66(5):1380–1389, 2018

    2018

  32. [40]

    A com- parison of monodomain and bidomain reaction-diffusion models for action potential prop- agation in the human heart

    Mark Potse, Bruno Dub´ e, Jacques Richer, Alain Vinet, and Ramesh M Gulrajani. A com- parison of monodomain and bidomain reaction-diffusion models for action potential prop- agation in the human heart. IEEE Transactions on Biomedical Engineering , 53(12):2425– 2435, 2006. 26

    2006

  33. [41]

    Mathematical mod- els and numerical methods for the forward problem in cardiac electrophysiology

    GT Lines, ML Buist, P Grottum, AJ Pullan, J Sundnes, and A Tveito. Mathematical mod- els and numerical methods for the forward problem in cardiac electrophysiology. Computing and Visualization in Science , 5:215–239, 2003

    2003

  34. [42]

    Beyond the Kalman filter: Particle filters for tracking applications

    Branko Ristic, Sanjeev Arulampalam, and Neil Gordon. Beyond the Kalman filter: Particle filters for tracking applications . Artech house, 2003

    2003

  35. [43]

    Particle filters: A hands-on tutorial

    Jos Elfring, Elena Torta, and Ren´ e Van De Molengraft. Particle filters: A hands-on tutorial. Sensors, 21(2):438, 2021

    2021

  36. [44]

    Novel approach to nonlinear/non- gaussian bayesian state estimation

    Neil J Gordon, David J Salmond, and Adrian FM Smith. Novel approach to nonlinear/non- gaussian bayesian state estimation. In IEE proceedings F (radar and signal processing) , volume 140, pages 107–113. IET, 1993

    1993

  37. [45]

    An eikonal-curvature equation for action potential propagation in my- ocardium

    James P Keener. An eikonal-curvature equation for action potential propagation in my- ocardium. Journal of mathematical biology , 29(7):629–651, 1991

    1991

  38. [46]

    A two-current model for the dynamics of cardiac membrane

    Colleen C Mitchell and David G Schaeffer. A two-current model for the dynamics of cardiac membrane. Bulletin of mathematical biology , 65(5):767–793, 2003

    2003

  39. [47]

    A depth-averaged heart model for the inverse problem of cardiac electrophysiology

    Emma Lagracie, Yves Bourgault, Yves Coudi` ere, and Lisl Weynans. A depth-averaged heart model for the inverse problem of cardiac electrophysiology. Inverse Problems, 41(2):025002, 2025

    2025

  40. [48]

    Effects of the insertion of epicardial anisotropy versus isotropy in the inverse problem of electrocardiography

    Emma Lagracie, Lisl Weynans, and Yves Coudi` ere. Effects of the insertion of epicardial anisotropy versus isotropy in the inverse problem of electrocardiography. InFIMH 2025-13th Functional Imaging and Modeling of the Heart International Conference , number 15672, pages 83–93...

    2025

  41. [49]

    Source-field relationships for cardiac generators on the heart surface based on their transfer coefficients

    Yasuo Yamashita and David B Geselowitz. Source-field relationships for cardiac generators on the heart surface based on their transfer coefficients. IEEE transactions on biomedical engineering, (11):964–970, 1985

    1985

  42. [50]

    Description of cardiac sources in anisotropic cardiac muscle: applica- tion of bidomain model

    David B Geselowitz. Description of cardiac sources in anisotropic cardiac muscle: applica- tion of bidomain model. Journal of electrocardiology, 25:65–67, 1992

    1992

  43. [51]

    Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models

    Jeanne van der Waal, Veronique Meijborg, Ruben Coronel, R´ emi Dubois, and Thom Oos- tendorp. Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models. Frontiers in Physiology, Volume 14 - 2023, 2023

    2023

  44. [52]

    Semi-implicit time-discretization schemes for the bido- main model

    Marc Ethier and Yves Bourgault. Semi-implicit time-discretization schemes for the bido- main model. SIAM Journal on Numerical Analysis , 46(5):2443–2468, 2008

    2008

  45. [53]

    A novel rule- based algorithm for assigning myocardial fiber orientation to computational heart models

    Jason D Bayer, Robert C Blake, Gernot Plank, and Natalia A Trayanova. A novel rule- based algorithm for assigning myocardial fiber orientation to computational heart models. Annals of biomedical engineering , 40:2243–2254, 2012

    2012

  46. [54]

    Scalable and accurate ecg simulation for reaction-diffusion models of the human heart

    Mark Potse. Scalable and accurate ecg simulation for reaction-diffusion models of the human heart. Frontiers in physiology, 9:370, 2018

    2018

  47. [55]

    A fast marching level set method for monotonically advancing fronts

    J A Sethian. A fast marching level set method for monotonically advancing fronts. Pro- ceedings of the National Academy of Sciences , 93(4):1591–1595, 1996

    1996

  48. [56]

    The heat method for distance computation

    Keenan Crane, Clarisse Weischedel, and Max Wardetzky. The heat method for distance computation. Communications of the ACM , 60(11):90–99, 2017

    2017

  49. [57]

    Comparison of two formulations for computing body surface potential maps

    Emma Lagracie, Lisl Weynans, and Yves Coudi` ere. Comparison of two formulations for computing body surface potential maps. In 2023 Computing in Cardiology (CinC) , vol- ume 50, pages 1–4. IEEE, 2023. 27

    2023

  50. [58]

    Assessment of a threshold method for computing activation maps from reconstructed transmembrane voltages

    Emma Lagracie, Lisl Weynans, and Yves Coudi` ere. Assessment of a threshold method for computing activation maps from reconstructed transmembrane voltages. In 51st interna- tional Computing in Cardiology conference-CinC 2024 , 2024. 28

    2024

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.