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PEEL-DDPM: Physics-Enabled Evidential Learning for the Denoising Diffusion Probabilistic Model

Ge Wang

arXiv:2609.31742v1cs.CV

TL;DR

NIG marginal likelihoods do not identify the aleatoric/epistemic split, motivating a physical measurement of the missing coordinate. PEEL-DDPM models final-image residuals and measures variance from repeated scans and diffusion trajectories. Held-out interval coverage and predicted variance were close to their targets, while a multi-dose test showed distinct scanner and sampler variance patterns.

  • Problem

    After marginalization, NIG regression identifies only three combinations of four parameters, leaving the aleatoric/epistemic split unidentified.

    Source: §1 · p. 1, §2.3 · p. 4

  • Method

    PEEL-DDPM freezes a measurement-conditioned DDPM, models its final-image residual, and uses repeated physical and sampling channels to identify the uncertainty split.

    Source: §1 · p. 2, §2.7 · p. 6

  • Results

    Held-out intervals were close to nominal and predicted variance closely matched residuals; across doses, umeas slope was −1.15 while udiff slope was −0.01.

    Source: §4 · p. 12

  • Takeaways & Limitations

    The framework keeps image generation, error modeling, and physical identification distinct while integrating them for reconstruction uncertainty quantification.

    Source: §4 · p. 13

    Feasibility and frozen-model multi-dose tests remain preliminary, establishing mechanism but not clinical calibration or robustness.

    Source: §4 · p. 12

Abstract

from arXiv · show

Normal-inverse-gamma (NIG) regression is not identifiable from its marginal Student-t likelihood: three combinations of four NIG parameters are determined, leaving one degree of freedom. We introduce PEEL-DDPM, a physics-enabled evidential learning framework for denoising diffusion probabilistic models. A measurement-conditioned DDPM is first trained with epsilon-MSE and then frozen. Its complete reverse trajectory produces a reconstruction, whose residual from the known training object is modeled by a final-image evidential network. The network learns only the identifiable Student-t coordinates, while repeated scanner-noise realizations and repeated diffusion trajectories provide a nested Monte Carlo estimate of final-image aleatoric variance, separated into scanner-induced and sampler-induced components. This measured variance resolves the remaining NIG ambiguity and yields a decomposition of predictive uncertainty into measurement, diffusion, and epistemic terms. The method uses sequential training without a cross-loss weighting coefficient. In a feasibility study on eight held-out objects, empirical central-interval coverages were 49.3%, 80.3%, 90.1%, and 95.6% for nominal 50%, 80%, 90%, and 95% intervals. The mean squared residual was 0.967 times the mean predicted variance, and a single-image aleatoric head achieved pooled Spearman correlation 0.785 against an independent nested reference. Across five dose levels, scanner-induced variance showed a log-log dose slope of -1.15, whereas sampler-induced variance remained nearly dose independent with slope -0.01. These results support PEEL-DDPM as a practical route to identifiable and physically interpretable uncertainty quantification in diffusion-based image reconstruction.

1 Introduction

NIG marginalization leaves the uncertainty split unidentified, so PEEL-DDPM supplies an independent physical measurement of the missing coordinate. It models final-image residuals and uses repeated scans and diffusion trajectories to separate sources of uncertainty.

1 Introduction

  • NIG regression identifies only three coordinates from four parameters after marginalization, leaving its aleatoric/epistemic split unresolved; PEEL measures the missing coordinate physically.Evidence regularizers, reference priors, and latent hierarchies [7] select a point on the unidentified fiber rather than identifying it.Source: §1 · p. 1
  • PEEL-DDPM attaches evidential learning to the residual between the known training object and the reconstruction after the complete reverse chain.Unlike an incremental ε prediction, the final-image error reflects the collective effect of initialization, score estimates, and data conditioning.Source: §1 · p. 2
  • Repeated scans and repeated diffusion trajectories provide resamplable sources of final-image variation, whose nested total variance separates scanner-induced and sampler-induced components.Their sum supplies the independently measured scalar needed to identify the NIG fiber.Source: §1 · p. 2
  • The DDPM is trained conventionally and frozen; separate networks fit final-image residual coordinates and measured channel variances without adding losses or requiring a cross-loss coefficient.The final uncertainty summary distinguishes measurement, diffusion, and NIG epistemic terms.Source: §1 · p. 2

2 Methodology

PEEL-DDPM freezes a conventionally trained conditional DDPM and models the residual of its complete reconstruction, rather than a single-step noise error. It measures scanner- and sampler-induced final-image variance to identify the NIG uncertainty split, using sequential training.

2.1 Conditional DDPM

  • A measurement-conditioned DDPM is trained with the standard noise-prediction loss and frozen before complete reverse trajectories generate final reconstructions.The conditioning image is derived from the measurements; the frozen model's complete reverse path includes initialization and ancestral Gaussian randomness.Source: §2.1 · pp. 2–3

2.2 Final-image residual

  • The evidential target is the final-image residual x0 −˜x0, because it integrates all reverse transitions and measurement conditioning, unlike a one-step image residual.At t = 0, the one-step estimator collapses to the ground-truth image and gives zero residual.Source: §2.2 · p. 3

2.3 Final-image NIG model

  • With the generated image fixed as location, the final-image Student-t likelihood identifies (α, c) but not the NIG aleatoric/epistemic split.Student-t NLL learns dispersion from the real image residual and cannot move the generated image.Source: §2.3 · p. 4

2.4 Final-image channel decomposition

  • Repeated scans and complete diffusion trajectories define uale = umeas + udiff, separating scanner-induced and sampler-induced final-image variance.The scanner component varies across repeat acquisitions of the same object; the sampler component varies across trajectories for the same scan. This stochastic variance is distinct from systematic bias.Source: §2.4 · p. 4

2.5 Unbiased nested estimators

  • Nested Monte Carlo estimates the two variance components from within-scan and corrected between-scan variation, with the stated estimators unbiased.Small negative finite-sample estimates may be clipped to zero for training, and clipping frequency is reported as a check on repeat counts.Source: §2.5 · p. 5

2.6 Image NIG identification

  • The independently measured uale supplies the missing coordinate, making the NIG representative unique once (α, c) are known.The two measured components provide more information than the one-dimensional NIG fiber; their sum identifies the NIG split without a KL term, reference prior, or evidence regularizer.Source: §2.6 · p. 5

2.7 Sequential training and inference

  • Training proceeds sequentially: freeze the DDPM, fit (α, c) to final-image residuals, train an aleatoric head from nested teachers, then combine the outputs at single-reconstruction inference.The teacher repeats train the mapping but are not available to the predictor at inference; the reported uncertainty separates measured aleatoric variance from the NIG epistemic remainder.Source: §2.7 · pp. 5–7

3 Experimental Design and Results

The study tests a deliberately reduced PEEL-DDPM feasibility setup, then examines final-image uncertainty calibration, repeated-reconstruction reference stability, and dose-dependent variance components. Its pilot findings are limited by the small one-dose experiment and preliminary nested reference.

3 Experimental Design and Results

  • The feasibility pilot used one dose, eight held-out test objects, 32 × 32 images, and a frozen DDPM with T = 30 reverse steps.The study was deliberately reduced and used a preliminary independent Monte Carlo reference; objects, rather than noise or trajectories, were separated across splits.Source: §3 · pp. 7–8, §3.1 · p. 8

3.2 Representative results

  • Empirical coverage tracked nominal Student-t interval coverage from 50% through 95%, and mean squared held-out residual was 96.7% of mean predicted variance.The feasibility study therefore produced a quantitatively coherent predictive distribution for actual reconstruction error.Source: §3.2 · p. 8
  • The one-image aleatoric head achieved pooled Spearman ρ = 0.785 against the independent nested reference; highest-to-lowest residual ratios were 3.82 for aleatoric and 4.89 for total predictive variance.The nested reference had pooled split-half correlation 0.497, so its aleatoric correlations remain exploratory rather than final estimates.Source: §3.2 · p. 8, Figure 4 · p. 10

3.3 Sizing the repeated final-reconstruction

  • Because each final-image Monte Carlo cell requires a complete reverse trajectory, repeat budgets were sized empirically using a pilot with 16 scans and four trajectories per scan.Nested-component uncertainty was assessed by bootstrap resampling scan rows and trajectories within rows.Source: §3.3 · p. 9

3.4 Additional multi-dose validation

  • Across five dose levels with the middle-dose DDPM frozen, scanner-induced variance had log-log slope −1.149 while sampler-induced variance had slope −0.006.Increasing the middle-dose reference budget from 24 × 6 to 96 × 6 raised median within-image split-half reliability from 0.449 to 0.750, but it remained below the prospective 0.90 criterion and spatial correlations remained exploratory.Source: §3.4 · pp. 11–12, Figure 5 · p. 11

3.5 Prospective full validation

  • The planned definitive study will retrain across five dose levels with a 600/100/30 object split and T = 200, increasing the reference budget adaptively toward 0.90 reliability.It will also repeat principal uncertainty-prediction results across multiple training seeds to distinguish full multi-dose predictive calibration from the reported physical-channel stress test.Source: §3.5 · p. 12

4 Discussion and Conclusion

The discussion reports pilot evidence of calibration and variance agreement, and a dose-stress experiment supporting the physical channel decomposition. It also explains how measured channel variance identifies the NIG split while noting preliminary scope and bias-related limits.

4 Discussion and Conclusion

  • Held-out Student-t intervals were close to nominal coverage, and mean predicted variance closely matched mean squared final-image residual.Both the predicted variance and independently learned aleatoric variance strongly stratified actual squared image error.Source: §4 · p. 12
  • Scanner-induced variance had a dose slope of −1.15, while sampler-induced variance had a dose slope of −0.01.The former followed an approximately inverse-dose law; the latter was essentially dose invariant.Source: §4 · p. 12
  • Median within-image split-half reliability increased from 0.449 at a 24 × 6 reference budget to 0.750 at 96 × 6, but remained below the pre-specified 0.90 target.Source: §4 · p. 12
  • The Student-t likelihood learns (α, c) but cannot separately determine (β, ν); the nested physical experiment supplies the missing scalar u_ale.Because the NIG fiber is one-dimensional, that scalar is enough to recover a unique representative; u_meas and u_diff are also measured directly.Source: §4 · p. 12
  • Student-t NLL does not train the DDPM score: the generator uses ε-MSE, is frozen, and remains unchanged by robust down-weighting.The evidential model treats the final image as a fixed location, so robust down-weighting changes uncertainty parameters but not the generated image or reverse score.Source: §4 · p. 12
  • The full-path teacher is computationally expensive, and preliminary feasibility and frozen-model stress tests cannot establish clinical calibration or robustness.Substantial systematic bias can also undermine interval centering even when variance is well learned; u_epi remains an operational NIG remainder.Source: §4 · p. 12
  • PEEL-DDPM models real reconstruction error with final-image Student-t fitting and uses repeated physical and algorithmic channels to identify the NIG uncertainty split.The framework keeps image generation, error modeling, and physical identification distinct within a unified reconstruction and uncertainty-quantification approach.Source: §4 · p. 13

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