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REVIEW 4 major objections 6 minor 58 references

Rethinking Timing Residuals: Advancing PET Detectors with Explicit TOF Corrections

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper redefines the timing residual as half the measured-minus-expected time difference, enabling explicit per-event TOF corrections that improve CTR from (371±6) ps to (281±5) ps with a single source position.

desk verdict Explicit residual TOF correction genuinely improves on implicit models and enables single-position training; the 281 ps headline, however, comes from a 33×33 in-plane grid, not a true point source. read the letter →

arxiv 2502.07630 v2 pith:MAKPU3QX submitted 2025-02-11 physics.ins-det cs.LG

Stephan Naunheim , Luis Lopes de Paiva , Vanessa Nadig , Yannick Kuhl , Stefan Gundacker , Florian Mueller , Volkmar Schulz This is my paper · ORCID
classification physics.ins-detcs.LG PACS 87.57.uk29.40.Mc
keywords TOF-PETtimingcalibrationcoincidencetimeresolutiongradient-boosteddecisiontreesexplicitTOFcorrectionssingle-sourceTOFPET2ASICmachinelearning
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

This paper proposes that a PET detector's remaining timing errors, after a standard analytical skew calibration, can be corrected by machine-learning models that directly predict timestamp corrections, if the training labels are defined as $r_i = (\Delta t_{m,i} - \Delta t_E(z_i))/2$ rather than as the expected time difference itself. With this redefinition, the label distribution becomes continuous and translationally symmetric, so a model trained with a radiation source at one position along the axis between detectors also works at other positions. The authors show on a pair of LYSO:Ce,Ca/SiPM detector blocks read out by the TOFPET2 ASIC that explicit-correction gradient-boosted trees improve coincidence time resolution from $(371 \pm 6)$ ps to $(281 \pm 5)$ ps for 430–590 keV coincidences, preserve linearity over the full axial range, and shrink model size enough for FPGA implementation. If correct, this removes the need for a motorized multi-position calibration setup and makes machine-learning TOF calibration practical for full PET scanners.

What carries the argument

The central object is the redefined timing residual label $r_i = (\Delta t_{m,i} - \Delta t_E(z_i))/2 = (\Delta t_{m,i} + 2 z_i / c)/2$, used as the regression target for gradient-boosted decision trees. It carries the argument by converting the calibration problem from predicting an absolute expected time difference (implicit) to predicting a per-event skew correction (explicit), which makes the label distribution independent of source-position sampling and lets a model trained at $z = 0$ provide corrections for all $z$. The rest of the machinery is the two-stage residual physics scheme: a first-order analytical timestamp skew calibration removes position-independent offsets, and the ML model is trained only on the remaining higher-order, event-dependent deviations, with the appendix's linearity argument assuming that predictions equal labels.

What would settle it

Move a point source to several $z$-positions, train an explicit model using only the $z=0$ data, and compare its predicted corrections with those of a model trained on all positions: if the per-event residual $\delta_i = (\Delta t_{m,i} + 2 z_i / c)/2$ shifts with $z$ by more than the detector's timing resolution, the single-position assumption fails.

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Extended reading notes

Core claim

The paper's central claim is that the residual physics-based calibration concept reaches its full practical value when the timing residual is defined as $r_i = (\Delta t_{m,i} - \Delta t_E(z_i))/2$, because then a model's output is itself the timestamp correction ($t_{a,i} \leftarrow t_{a,i} - r_i$, $t_{b,i} \leftarrow t_{b,i} + r_i$) instead of a predicted corrected time difference. This restores a translational symmetry to the labeling: shifting the source shifts the expected time difference and the measured time difference by the same amount, so the label distribution no longer encodes source position. The authors demonstrate that explicit-correction gradient-boosted trees are immune to the step-width collapse observed for implicit models, remain linear across the full $\pm170$ mm test range, and improve timing resolution from $(371 \pm 6)$ ps to $(281 \pm 5)$ ps for 430–590 keV coincidences; a model trained on a single centered point source still reaches $(306 \pm 4)$ ps, and a $33 \times 33$ in-plane distribution reaches $(281 \pm 5)$ ps. They interpret the small degradation relative to the best implicit model as acceptable given the simplification in acquisition and the exponential reduction in model size (tree depth 4 instead of 18).

Load-bearing premise

The method assumes that, after the first analytical time-skew calibration, whatever timing error remains in an event does not depend on where the radiation source sits between the detectors, so a model trained with the source at one position works for all positions.

Editorial extensions

If this is right

  • Explicit-correction models trained with coarse axial sampling (50 mm or 100 mm step width) generalize to unseen 10 mm-sampled positions, whereas implicit models oscillate and fail linearity checks.
  • A single centered source position is enough to train an explicit model, removing the requirement for a motorized multi-position translation stage in calibration.
  • The explicit formulation preserves linearity over the full $\pm170$ mm test range, so TOF information remains interpretable for image reconstruction.
  • Because shallow trees (depth 4) suffice, model memory shrinks by a factor of about $2^{18-4}$, making the correction suitable for FPGA-based, high-throughput PET readout.
  • Both correction approaches correct time-walk effects well: enlarging the energy window from 430–590 keV to 300–700 keV degrades CTR only slightly (about 2\%) for explicit models, versus about 20\% without ML.

Reading between the lines

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

  • Editorial inference: the independence of the explicit label from source position is a symmetry argument, not a proof that all per-event skews are position-independent; the transferability of a $z=0$-trained model to off-center positions should be re-tested for detectors with stronger depth-of-interaction-dependent skews.
  • Editorial inference: the comparison between 1×1 and 33×33 in-plane distributions is confounded by a $33^2$ difference in training statistics, so the observation that 3×3 performs worse than 1×1 is the cleanest evidence that source arrangement itself matters; a matched-statistics experiment would separate the two effects.
  • Editorial inference: because the explicit residual is defined from the measured time difference and the geometric expected time difference, the same label construction could be applied to any coincidence pair with a known geometric time-difference model, such as dual-sided readout or monolithic detectors, provided the single-position assumption holds.
  • Editorial inference: a testable extension is to train explicitly on data from one detector stack and apply it to an unseen stack of the same design; the paper lists this as future work and expects feature-based robustness.
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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

4 major / 6 minor

Summary. The paper proposes an explicit timing-residual formulation for machine-learning-based TOF calibration in PET detectors. The label is defined as half of the difference between the measured time difference and the expected time difference from the known source position, r_i = (Δt_m,i − Δt_E(z_i))/2, which yields a continuous label distribution that is claimed to be independent of the number of source positions along the transaxial axis. The authors compare this explicit-correction approach with their earlier implicit-correction models using a three-stage evaluation (MAE, linearity/ε, and CTR) on real data from two 4×4 LYSO:Ce,Ca detector blocks coupled to Broadcom NUV-MT SiPMs and read out with TOFPET2. They report an improvement in CTR from (371±6) ps to (281±5) ps for 430–590 keV coincidences, robustness to spatial undersampling, preserved linearity over the full test range, and a large reduction in model size suitable for FPGA deployment.

Significance. If the results hold, the explicit residual formulation is a practically valuable contribution: it removes the need for dense transaxial source sampling, preserves linearity over the full test range, and yields compact tree ensembles suitable for high-throughput PET applications. The three-stage evaluation is thorough, uncertainties are reported, and the comparison with implicit models is informative. The main weaknesses are that the central transfer claim—that a model trained at a single z-plane corrects events at all z—rests on an assumption about the z-independence of residual skews that is not proven, and the headline CTR is obtained with a 33×33 in-plane grid rather than the single-point-source configuration emphasized in the abstract.

major comments (4)
  1. [Appendix, Eq. (25)] The linearity argument assumes predictions equal labels (p_i ≈ l_i) and then derives E[Δt_corr] = E[Δt]. This is circular because the property to be established is precisely that the model's predictions, which cannot depend on z since z is not a feature, equal the label l_i = (Δt_m,i − Δt_E(z_i))/2 for all z_i. For a model trained only at z=0, l_i reduces to Δt_m,i/2, and no argument in the appendix shows that this equality survives at z≠0. Please provide a non-circular derivation or a direct empirical demonstration that the residual skew δ_i is independent of z after the first-order calibration.
  2. [Abstract and Table 3] The headline improvement to (281±5) ps is obtained by model EM33x33, which uses 33×33 = 1089 in-plane source positions; the single-point-source model EM01x01 reaches (306±4) ps (Table 3). Since the paper's key simplifying claim is that acquisition reduces to a single source position, presenting the 281 ps value as the headline conflates the best-possible calibration with the single-source calibration. The abstract and Section 4 should either report the single-source value as the headline for the simplification claim or explicitly separate the two achievements.
  3. [Section 2.6.2 and Section 5] The single-position transfer result rests on the assumption that, after the analytical first-order time-skew calibration, the per-event residual skew δ_i is independent of z (DOI and incidence-angle effects in 20-mm crystals could violate this). The current evidence is limited to one detector pair, and the paper itself defers cross-stack stability to future work. To support the central claim, please add a quantitative analysis of CTR as a function of |z| for a model trained only at z=0, including the range over which the single-source model stays within a pre-specified tolerance of the full multi-position model.
  4. [Section 3.1.3 and Tables 1–3] Hyperparameters (tree depth) are selected from a grid evaluated on the same test data used to report CTR. The CTR differences among depths are small (e.g., 281±5 vs 287±5 ps), so selection-on-test can bias the reported improvement. Use a separate validation set for hyperparameter selection, or report the full distribution of CTR across depths and a correction for multiple comparisons.
minor comments (6)
  1. [Section 2.3] The sentence "we propose to redefine the residuals" should refer to redefining the labels; also "high coverade" is misspelled (should be "high coverage").
  2. [Figure 3 caption] The caption would benefit from a verb: "the implicit model IM10,12 and the explicit model EM10,4 are used" reads more clearly.
  3. [Section 2.4.2, Eq. (11)] The notation would benefit from an explicit statement that µ is the fitted mean of the prediction distribution and that ε is the parameter being estimated.
  4. [Section 4] The phrase "capable of proving good results" should be "capable of providing good results" or "showing good results".
  5. [Appendix, Eq. (26)] The expectation expression contains unmatched brackets and non-standard delimiters from the LaTeX source; please correct the typesetting.
  6. [Tables 1, 3, 5, 6] The placement of the CTR tables could be improved: Table 3 (the main in-plane result) appears before the extended energy-window tables, but the text in Section 3.2.3 refers to them in a way that is hard to follow; consider renumbering or adding in-text pointers.

Circularity Check

1 steps flagged · score 3.0 of 10

The appendix's linearity argument is tautological (predictions assumed to equal labels), and the single-source z-transfer claim presupposes a z-independent residual skew; the headline CTR improvement is nevertheless an independent measured result.

  1. self definitional [Appendix, 'Consideration of the Linearity of Explicit Correction Models', Eqs. (24)-(28)]
    "If we assume that the explicit correction model was successfully trained, we can approximate the predictions by the labels (see Eq. (6)), pi ≈ li := Δtm,i − ΔtE(z)/2. Inserting Eq. (25) into Eq. (24), yields ΔtE = E[{Δti − 2 · (Δti − E[{Δti}])/2}] = E[{E[{Δti}]}] = E[{Δti}], which is the expected time difference for the non-corrected timestamps."

    The linearity 'result' is obtained by substituting the label definition for the model prediction and simplifying; it assumes exactly what the model is supposed to provide (pi = li). The corrected mean then equals the uncorrected mean by construction, independent of any learned model property. The paper itself states 'we will not provide a formal proof' and calls it 'basic understanding', but Section 4 cites this appendix as the mathematical reasoning for the claimed source-location robustness ('The robustness of the explicit correction models is reasoned by the way the residuals are defined'). The empirical χ2/ε and CTR evaluations are independent evidence, so this is a partial, not total, circularity.

full rationale

The central empirical claims are independently measured: the CTR improvements in Tables 1-4 are obtained on held-out test data against a 'no ML' analytical baseline, and the linearity/ε evaluations in Section 3.1.2 and 3.2.2 use real source-position scans. No load-bearing self-citation chain was found: prior works [44-46,51] are used for comparison and for the implicit-correction baseline, not as the justification for the explicit-correction result. The genuine circular step is confined to the appendix, where the linearity robustness is 'derived' by assuming predictions equal labels; this reduces to the label construction and is not a model property. The z-independence/single-position transfer is an empirical assumption about post-calibration skew being independent of source position, validated here for one detector pair and explicitly deferred for cross-stack stability in the Outlook; it is not circular, but it is the load-bearing assumption behind the 'single source position' simplification. Note also that the headline 281 ± 5 ps comes from EM33x33 (1089 in-plane positions), while the true single-point model EM01x01 reaches 306 ± 4 ps, weakening the 'single source position' framing without making it circular. Overall, the only reduction-by-construction is the appendix's consistency check, so the score is moderate rather than severe.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the additive skew model, the translation invariance of the residual, the sufficiency of the chosen features, and the calibrated source-position labels. No new physical entities are postulated. The main free parameters are XGBoost hyperparameters and preprocessing/analysis cuts; the first-order skew offsets come from prior work.

free parameters (8)
  • XGBoost learning rate = 0.1
    Fixed by hand for all models; not swept.
  • Maximum number of trees = 1000
    Fixed with early stopping of 10 rounds; model size and depth interact.
  • Maximum tree depth = d in {4,8,12,16,20}; best CTR 281 ps at d=4
    Swept hyperparameter; results vary with d, and the headline model is selected from this grid.
  • Number of relative timestamps per detector = na=12, nb=9
    Chosen from prior statistical analysis of triggered SiPM counts; affects input features.
  • Energy exclusion thresholds for hits = 2.5 and 100 arbitrary units
    Empirically chosen hit-level cuts applied during clustering.
  • Cluster and coincidence windows = 8 ns cluster, 50 ns software coincidence
    Chosen empirically by temporal distribution analysis.
  • Evaluation energy window = 430-590 keV primary; 300-700 keV secondary
    Used for CTR and linearity evaluations; headline number depends on this cut.
  • First-order time skew calibration offsets = not reported
    Computed in prior work [59]; part of the baseline and of the residual definition.
assumptions (6)
  • domain assumption Timestamp decomposition td = Θ + Δts (Eq. 1)
    Assumes detected timestamps are the true time plus an additive event-dependent skew; the whole calibration framework rests on this.
  • domain assumption After first-order analytical calibration, remaining skews are event-dependent higher-order effects that ML can learn
    Residual physics concept introduced in Section 2.7 and prior work.
  • domain assumption The residual timing skew is independent of source position z (translation invariance)
    Required for single-position training to generalize; tested empirically in Section 3.2, not proved.
  • domain assumption Supervised labels from a 22Na source position give unbiased expected time differences via ΔtE = -2z/c (Eq. 3)
    Neglects source diameter, scatter, and finite detector geometry; uncertainty accounted only in fitting.
  • domain assumption Input features (SiPM timestamps, energies, IDs, light moments) contain sufficient information to predict the residual
    The GBDT has no waveform or measured time-difference input for explicit models.
  • ad hoc to paper The appendix linearity argument assumes predictions equal labels (p_i ≈ l_i)
    Stated in Eq. 25; the paper explicitly says it is not a formal proof.

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Cite this review

Pith. "Pith review of Rethinking Timing Residuals: Advancing PET Detectors with Explicit TOF Corrections." pith.science (2026). https://pith.science/paper/MAKPU3QX

@misc{pith2026250207630,
  author       = {Pith},
  title        = {Pith review of: Rethinking Timing Residuals: Advancing PET Detectors with Explicit TOF Corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAKPU3QX}},
  note         = {Machine review of arXiv:2502.07630}
}
abstract

PET is a functional imaging method that visualizes metabolic processes. TOF information can be derived from coincident detector signals and incorporated into image reconstruction to enhance the SNR. PET detectors are typically assessed by their CTR, but timing performance is degraded by various factors. Research on timing calibration seeks to mitigate these degradations and restore accurate timing information. While many calibration methods use analytical approaches, machine learning techniques have recently gained attention due to their flexibility. We developed a residual physics-based calibration approach that combines prior domain knowledge with the power of machine learning models. This approach begins with an initial analytical calibration addressing first-order skews. The remaining deviations, regarded as residual effects, are used to train machine learning models to eliminate higher-order skews. The key advantage is that the experimenter guides the learning process through the definition of timing residuals. In earlier studies, we developed models that directly predicted the expected time difference, which offered corrections only implicitly (implicit correction models). In this study, we introduce a new definition for timing residuals, enabling us to train models that directly predict correction values (explicit correction models). The explicit correction approach significantly simplifies data acquisition, improves linearity, and enhances timing performance from $371 \pm 6$ ps to $281 \pm 5$ ps for coincidences from 430 keV to 590 keV. Additionally, the new definition reduces model size, making it suitable for high-throughput applications like PET scanners. Experiments were conducted using two detector stacks composed of $4 \times 4$ LYSO:Ce,Ca crystals ($3.8\times 3.8\times 20$ mm$^{3}$) coupled to $4 \times 4$ Broadcom NUV-MT SiPMs and digitized with the TOFPET2 ASIC.

Figures

Figures reproduced from arXiv: 2502.07630 by the authors.

Figure 1
Figure 1. Visualization of the label distributions of the implicit (green) and explicit (blue) correction [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Visualization of the source distribution for a part of the transaxial study. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Examples of the prediction distribution of implicit and explicit correction models. While [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: In-plane source distributions. The red dots represent the position of a source, while the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: MAE progression of the correction models trained on a dataset using a step width of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: MAE progression of the correction models trained on a dataset using a step width of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: MAE progression of the correction models trained on a dataset using a step width of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Linearity evaluation for the implicit and explicit models trained on training data with [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Obtained timing resolutions for different implicit and explicit correction models. The [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: CTR progression along the z-axis of models, trained on 10 mm stepping, which passed the data scientific and physics-based quality checks. The upper plot visualizes the (directly/indirectly) predicted time differences. The middle plot shows the CTR value at the specifi…
Figure 11
Figure 11. Figure 11: CTR progression along the z-axis of explicit correction models, trained on 100 mm stepping. The upper plot visualizes the indirectly predicted time differences. The middle plot shows the CTR value at the specific position. The lower plot visualizes the χ 2/ndf-value r…
Figure 13
Figure 13. Figure 13: It can be seen that the MAE progression is smooth, and no outliers or oscillation effects [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 12
Figure 12. Figure 12: MAE progression of explicit correction models trained with different maximum tree [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Visualizations of the mean MAE and weighted mean MAE dependent on the maximum [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Visualization of the metrics used during the physics-based evaluation dependent on the [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Models that have been trained on extensive in-plane source distributions show the best [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 15
Figure 15. Figure 15: Obtained timing resolutions for explicit correction models trained on data with different [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]

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Pith tools

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