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Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A likelihood schedule from MLEM guides a pretrained diffusion prior, making fully 3D PET reconstruction practical with a single critical hyperparameter.

desk verdict A practical, well-tested diffusion-PET method whose 'posterior sampling' framing is heuristic; the 2D evidence is solid, the real-3D claim needs qualification. read the letter →

arxiv 2412.04339 v2 pith:S6GWADHW submitted 2024-12-05 physics.med-ph cs.CVcs.LG

George Webber , Yuya Mizuno , Oliver D. Howes , Alexander Hammers , Andrew P. King , Andrew J. Reader This is my paper · ORCID
classification physics.med-phcs.CVcs.LG PACS 87.57.nj
keywords score-basedgenerativemodelingPETimagereconstructionlikelihoodschedulingMLEMDDIMsamplingfully3Dsliceconsistencylow-count
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

The paper proposes a way to inject a pre-trained score-based generative model—an image prior learned from high-quality scans—into PET reconstruction while keeping only one critical tuning knob. Instead of balancing prior and data terms with several regularization weights, it first runs MLEM to record a 'likelihood schedule' of Poisson log-likelihood values, then guides the reverse diffusion process to match that schedule at every denoising step. On simulated low-count [18F]DPA-714 data, the method matches or improves the NRMSE and SSIM of the previous state-of-the-art SGM method while using far fewer likelihood updates and less tuning. It also extends to real fully 3D PET data for the first time, using three perpendicular pre-trained 2D SGMs to remove the slice-inconsistency artifacts of single-plane SGM reconstruction.

What carries the argument

The mechanism is the likelihood schedule: a sequence of scalar Poisson log-likelihood values obtained by running MLEM and interpolating its per-iteration likelihoods across the diffusion time steps. At each reverse-diffusion step, the algorithm takes one Tweedie denoising step, the estimate of the clean image from the noisy iterate, then performs gradient ascent on the Poisson log-likelihood until the estimate reaches the scheduled value, and finally re-adds noise via the DDIM update. The schedule replaces the multiple regularization hyperparameters of earlier methods: the MLEM iteration count sets the balance between prior and data, leaving only the number of diffusion steps and a step-size parameter that controls schedule accuracy rather than prior-likelihood balance.

What would settle it

Insert a known lesion into a phantom, reconstruct from low-count sinograms at a fixed likelihood schedule, and compare lesion contrast in the sampled images with lesion contrast in the MLEM images used to set the schedule; if the sampled images fit the likelihood but systematically lose or displace the lesion while MLEM recovers it, the scalar-likelihood conditioning is insufficient.

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Extended reading notes

Core claim

The central claim is that a PET reconstruction can be sampled from a learned image distribution conditioned on a chosen level of data consistency, rather than on a tuned balance of regularization weights. The authors define the target as 'sample $x$ such that $x$ is drawn from the learned distribution and $L(x|m)=c$', with $c$ fixed by a clinically relevant MLEM iterate, and they solve it by following a precomputed Poisson log-likelihood schedule during DDIM-style reverse diffusion. They report that this likelihood-scheduled method matches or improves on PET-DDS, the prior SGM-based PET method, in NRMSE and SSIM, while cutting reconstruction time and lowering the critical hyperparameter count from four to one, the MLEM iteration count. They further report the first SGM-based reconstruction of real fully 3D PET data, using perpendicular pre-trained 2D SGMs whose score estimates are averaged so that no axial regularization prior is needed and slice inconsistency is eliminated.

Load-bearing premise

The method assumes that forcing each reconstruction to have the same scalar data-fit value as an MLEM image, without enforcing point-by-point agreement with the measured sinogram, is enough to keep the reconstruction faithful—and that the MLEM likelihood value is a sensible target.

Editorial extensions

If this is right

  • Clinicians can set the main hyperparameter, the MLEM iteration count, to a vendor-recommended clinical value, bringing diffusion priors into reconstruction at a familiar operating point.
  • Because samples are drawn at the same data-consistency level as MLEM, the spread among sampled reconstructions provides uncertainty information for a fixed clinical likelihood.
  • Averaging scores from three perpendicular pre-trained 2D SGMs removes axial slice inconsistency without the blurring effect of an axial regularization prior.
  • Dynamic likelihood steps reduce compute: fewer gradient updates are spent early and late in diffusion, with reported 2D reconstruction time of 13 seconds versus 35 seconds for the prior SGM method.
  • The likelihood-scheduling idea transfers to other imaging modalities that combine model-based iterative steps with pre-trained SGMs, as the paper itself notes.

Reading between the lines

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

  • A consequence the authors do not develop is that scalar-likelihood conditioning is weaker than full sinogram consistency; if the equal-likelihood manifold contains anatomically wrong images, samples at the MLEM likelihood value could be confidently wrong, and lesion-insertion phantoms across a range of MLEM iteration counts would test this.
  • The scheduling concept generalizes beyond likelihood: the same mechanism could schedule other scalar clinical objectives, such as a region-of-interest uptake constraint, without adding hyperparameters.
  • The paper notes that hyperparameter tuning could be removed entirely by using bootstrap estimates of the optimal MLEM iteration, which would make the method fully automatic given a pretrained SGM.
  • Because the score prior is fixed and the schedule is recomputed from the measured data, the approach may transfer across scanners with different forward models, a testable extension the paper does not explicitly run.
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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 / 5 minor

Summary. This manuscript proposes a likelihood-scheduled score-based generative modeling method for PET image reconstruction. The method first runs MLEM on the measured sinogram to produce a scalar Poisson log-likelihood schedule, then interleaves DDIM-style reverse diffusion steps with gradient ascent on Tweedie's estimate so that the reconstructed image's PLL matches the schedule value at each generative step. The authors compare the method with PET-DDS, OSEM, MAP-EM, and two FBSEM-net variants on simulated 2D [18F]DPA-714 data, reporting NRMSE, SSIM, and reconstruction times, and they demonstrate 3D reconstruction on real [18F]DPA-714 data using three perpendicular pre-trained 2D SGMs. The central practical claims are that the method matches or improves on PET-DDS's NRMSE/SSIM while reducing reconstruction time and the number of critical hyperparameters, and that it gives the first SGM-based fully 3D PET reconstruction from real data.

Significance. If the practical claims hold, this is a valuable contribution to unsupervised SGM-based PET reconstruction: it replaces several per-step likelihood hyperparameters with a single MLEM-iteration knob, makes 3D reconstruction tractable, and removes the axial slice-inconsistency artifacts of prior 3D approaches. The strengths of the paper include a real 3D data demonstration, perpendicular score averaging across sagittal, coronal, and transverse slices, robustness studies over the number of generative steps and the gradient step size, lesion recovery experiments, bias-variance and likelihood-variance assessments, and the use of a public forward projector. The main 2D comparison against PET-DDS in Table I and Fig. 4 is credible and supports parity or mild improvement. However, the paper's theoretical framing as posterior sampling conditioned on a likelihood value is not established: the algorithm enforces only a scalar Poisson log-likelihood, which is degenerate and does not by itself condition on the full measured sinogram.

major comments (3)
  1. [Section IV-A/IV-B, Eq. (9)] The problem formulation in Eq. (9), "Sample x such that x~D and L(x|m)=c", is asserted rather than derived as a posterior sampling scheme, yet the Introduction and Discussion claim that the method "provides samples from the posterior distribution of image reconstruction conditioned on both a likelihood value and noisy measured data." The algorithm enforces only the scalar PLL, and Section IV-A itself notes that the PLL is degenerate, so a whole manifold of images with different per-detector residuals share the same likelihood value. No argument connects the gradient-ascent trajectory on Tweedie's estimate to p(x|m, L=c) or p(x|m). The empirical gains in Table I show practical utility but do not validate the mechanism. Please either provide a formal derivation of the posterior-sampling claim or restate the contribution as constrained sampling from the learned prior at a prescribed scalar likelihood value.
  2. [Section VI-H.2, Fig. 14] The MLEM-derived target likelihood c is not guaranteed to be reachable by the SGM's image manifold. Fig. 14 shows that for N_MLEM > 36 the coefficient of variation of the proposed method exceeds that of MLEM, and the text attributes this to the absence of images with such high likelihood on the SGM's manifold. This is an acknowledged failure mode of the central tuning parameter N_MLEM. The paper should characterize the admissible range of N_MLEM or propose a reachability criterion; otherwise the claim that N_MLEM is a single intuitive knob that can be set to a clinically standard MLEM iteration count is incomplete.
  3. [Section V-F, Table I, Figs. 4 and 10] The "reduced hyperparameter tuning burden" claim is supported structurally by the reduction to N_MLEM and N_gen and by the robustness to N_gen and delta shown in Figs. 5 and 6, but the quantitative comparisons still select N_MLEM by a validation sweep to minimize NRMSE, just as the PET-DDS hyperparameters are swept. Since N_MLEM controls the prior-likelihood balance, and the paper does not validate setting it a priori to the clinically used number of MLEM iterations, the tuning-burden advantage is weaker than stated. A fixed-N_MLEM experiment, for example using N_MLEM equal to the vendor-recommended OSEM iteration count, would materially strengthen the claim.
minor comments (5)
  1. [Section IV-A, Eq. (9)] The notation "x ~ D" is informal; please define D explicitly as the learned image distribution and clarify the relationship between D and the SGM's marginal densities p_t(x_t).
  2. [Fig. 12 caption vs. Section V-D] The caption reports 2.7×10^5 counts while Section V-D states an average of 3.14×10^5 counts per reconstructed slice; please reconcile these numbers or explain the difference.
  3. [Fig. 13 caption] The caption states that all columns except "Clinical" match the PLL corresponding to 21 iterations of MLEM, but PET-DDS does not use a likelihood schedule; please specify how PET-DDS was made to match this PLL value.
  4. [Section VI-H.2] For the real 3D data there is no ground truth; the text refers to a "ground truth" when selecting a white-matter region, but this should be called a full-count clinical reference image to avoid ambiguity.
  5. [Section VII] The phrase "first methodology to investigate possible reconstructions for a fixed likelihood" should be qualified as "to our knowledge" and should specify "at a fixed scalar likelihood," in light of the degeneracy discussion in Section IV-A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MLEM-derived likelihood target is an input to the algorithm, not a prediction extracted from it, and the paper's quality claims are validated by independent metrics.

full rationale

The derivation chain in Sec. IV defines a target likelihood c = L(x_MLEM|m) and then runs gradient ascent on Tweedie's estimate until that target is exceeded. The resulting statement that the final PLL is within one gradient step of the MLEM PLL is true by construction, but the paper does not use that match as evidence for any empirical claim; it is the algorithm's specification. The central results (Table I, Figs. 4, 10, 11, and the 3D timing table) are evaluated with independent metrics (NRMSE, SSIM, lesion CRC, coefficient of variation, wall-clock time) against ground-truth or held-out data, so those claims are not forced by the MLEM-derived schedule. The posterior-sampling interpretation in Secs. IV-A and VII is asserted rather than derived, but that is an unproven modeling assumption and a correctness risk, not a circular reduction. The paper's self-citations ([9], [14]) appear only in background statements and are not load-bearing; no uniqueness theorem or ansatz is imported from the authors' own prior work to forbid alternatives. No fitted parameter is renamed as a prediction, and no known result is merely re-labeled. Therefore the circularity score is 0.

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

The method introduces no new physical entities. It relies on several standard mathematical tools and several domain assumptions that are stated or implicit in the reconstruction formulation. The most consequential assumptions are that the scalar PLL constraint is a sufficient data consistency objective and that the MLEM-derived schedule defines a clinically useful target.

free parameters (6)
  • N_MLEM = 9-18 in sweeps; 21 in 3D
    Number of MLEM iterations used to build the likelihood schedule. It controls the target likelihood c and hence the prior-likelihood balance, and was tuned on validation to minimize NRMSE.
  • N_gen = 100 (2D), 25 (3D)
    Number of generative diffusion steps. It trades speed against accuracy and is chosen by the authors, not derived.
  • delta = 0.2 (2D), 1.0 (3D)
    Gradient ascent step size. The paper shows robustness up to 2.0 in 2D and up to 4.0 in 3D, but the value still must be set and controls the accuracy of schedule matching.
  • eta = 0.1
    DDIM stochasticity, a fixed hyperparameter inherited from DDIM sampling.
  • beta(t) = 0.1 + 19.9t
    Variance-preserving noise schedule for the forward SDE. Fixed and not tuned, but a modeling choice.
  • SGM training epochs = 100
    Number of training epochs for each score network, selected by 5-fold cross-validation on the transverse training datasets.
assumptions (8)
  • standard math PET emission data follows a Poisson likelihood model, Eq. (2)
    The reconstruction objective assumes measured prompts follow independent Poisson statistics with mean Ax+b.
  • domain assumption The forward model A is known exactly, Eq. (10)
    The method requires an accurate system matrix including PSF, attenuation, normalization, and axial compression; any mismatch biases the likelihood schedule and gradient ascent.
  • domain assumption The score network s_theta approximates the true score, Eq. (5) and Eq. (8)
    DSM training assumes the network is a good surrogate for the true score, and the reconstruction uses this surrogate in place of the prior.
  • standard math Tweedie's formula gives E[x0|xt], Eq. (6)
    Used to denoise and to apply gradient ascent in pixel space.
  • standard math The DDIM update rule, Eq. (7), is a valid sampling scheme
    Assumed for the reverse diffusion process.
  • ad hoc to paper The MLEM likelihood schedule is a clinically meaningful target, Section IV-A
    The choice c = L(x_MLEM|m) is a heuristic aligned with clinical practice, not derived from the posterior.
  • ad hoc to paper A scalar PLL constraint suffices for data consistency, Section IV-B
    The method enforces only the likelihood value, ignoring the rest of the sinogram residual; this is an unproven assumption.
  • ad hoc to paper Averaging scores from three perpendicular 2D SGMs approximates the 3D score, Section IV-C
    The paper adapts Lee et al.'s perpendicular SGM idea but averages scores rather than alternating; no proof is given that this equals a valid 3D prior.

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Cite this review

Pith. "Pith review of Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction." pith.science (2026). https://pith.science/paper/S6GWADHW

@misc{pith2026241204339,
  author       = {Pith},
  title        = {Pith review of: Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6GWADHW}},
  note         = {Machine review of arXiv:2412.04339}
}
abstract

Medical image reconstruction with pre-trained score-based generative models (SGMs) has advantages over other existing state-of-the-art deep-learned reconstruction methods, including improved resilience to different scanner setups and advanced image distribution modeling. SGM-based reconstruction has recently been applied to simulated positron emission tomography (PET) datasets, showing improved contrast recovery for out-of-distribution lesions relative to the state-of-the-art. However, existing methods for SGM-based reconstruction from PET data suffer from slow reconstruction, burdensome hyperparameter tuning and slice inconsistency effects (in 3D). In this work, we propose a practical methodology for fully 3D reconstruction that accelerates reconstruction and reduces the number of critical hyperparameters by matching the likelihood of an SGM's reverse diffusion process to a current iterate of the maximum-likelihood expectation maximization algorithm. Using the example of low-count reconstruction from simulated [$^{18}$F]DPA-714 datasets, we show our methodology can match or improve on the NRMSE and SSIM of existing state-of-the-art SGM-based PET reconstruction while reducing reconstruction time and the need for hyperparameter tuning. We evaluate our methodology against state-of-the-art supervised and conventional reconstruction algorithms. Finally, we demonstrate a first-ever implementation of SGM-based reconstruction for real 3D PET data, specifically [$^{18}$F]DPA-714 data, where we integrate perpendicular pre-trained SGMs to eliminate slice inconsistency issues.

Figures

Figures reproduced from arXiv: 2412.04339 by the authors.

Figure 6
Figure 6. Reconstruction quality (SSIM) and number of like￾lihood updates for our method and PET-DDS using optimal hyperparameters chosen for 100 generative timesteps at alter￾nate step sizes δ, as evaluated on 10 2D slices. 200 on both our method and PET-DDS, with the effect on reconstructions shown in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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  2. Artificial Intelligence-Guided PET Image Reconstruction and Multi-Tracer Imaging: Novel Methods, Challenges, And Opportunities

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    AI-based PET reconstruction and resolution enhancement methods are surveyed for long axial field of view scanners and multiplexed multi-tracer imaging, with key challenges identified.

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