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.
Figures & tables
Figure 1: Nested teacher for estimation of aleatoric uncertainty. Each row fixes one scan and varies the entire diffusion trajectory, while different rows use independent scans of the same object. Every cell is a complete final reconstruction and therefore has an image residual r(m,d)=x0−x~0(m,d) . Within-row variation estimates udiff and corrected between-row variation estimates umeas .
Figure 2: PEEL-DDPM workflow. Stage 1 trains a conventional measurement-conditioned DDPM by ε -MSE and freezes it. Stage 2 runs complete reverse trajectories and fits the identifiable Student- t coordinates (α,c) to the image residual r=x0−x~0 . Stage 3 resamples both the scanner and the complete diffusion trajectory to measure umeas and udiff , then trains an image channel head. Stage 4 combines the learned likelihood coordinates with uale=umeas+udiff to recover (β,ν) and report an image uncertainty summary.
Figure 3: Representative held-out final-image feasibility case. The nested uale reference is computed from independent repeated scans and complete reverse trajectories. The prediction uses only one scan and one final reconstruction. The displayed case has within-image Spearman correlation ρ=0.717 , close to the test-set median. The uncertainty panels share common display limits.
Figure 4: Held-out image-error behavior in the feasibility study. (a) Empirical coverage of central Student- t intervals closely follows nominal coverage, and (b) mean squared final-image residual increases strongly in the highest uncertainty quintile. The highest-to-lowest ratios are 4.89 for total predictive variance and 3.82 for the independently trained aleatoric coordinate.
Quantity
Pilot result
Final reconstruction RMSE
1.224×10−3±6.32×10−5
Final reconstruction SSIM
0.891±0.015
Student- t coverage, nominal 50/80/90/95%
49.3/80.3/90.1/95.6%
E[r2]/E[V^pred]
0.967
u^ale pooled Spearman ρ
0.785
Median within-image ρ (range)
0.723 ( 0.504 – 0.852 )
Table 1: One-dose feasibility experiment for the final-image formulation. Correlations against the nested reference are exploratory, with split-half reliability of 0.497 .
Figure 5: Multi-dose stress test of the final-image variance decomposition using the frozen middle-dose DDPM. Points show means over eight held-out objects and error bars show between-object standard deviations. The fitted log–log slopes are −1.15 for umeas and −0.01 for udiff .
I0
mean umeas
mean udiff
scanner fraction
1.50×104
2.516×10−3
1.357×10−3
0.650
2.25×104
1.531×10−3
1.349×10−3
0.532
3.50×104
0.898×10−3
1.339×10−3
0.401
5.25×104
0.581×10−3
1.349×10−3
0.301
8.00×104
0.368×10−3
1.339×10−3
0.215
Table 2: Measured final-image variance components across dose. Variances are in normalized image units and are averaged over eight held-out objects.
Normal-inverse-gamma (NIG) regression is not uniquely identifiable from its marginal Student-t likelihood: the likelihood determines three combinations of four NIG parameters and is constant along a one-dimensional fiber. We identify that fiber using independent physical measurement. As an initial embodiment, a reconstruction network receives one noisy filtered-backprojection (FBP) image and is first trained only by Student-t negative log-likelihood to estimate the three identifiable coordinates (gamma, alpha, c). The network is then frozen; repeated physical-noise realizations propagated through its reconstruction output form a Monte Carlo (MC) teacher label for output-domain aleatoric variance. An aleatoric head attached to frozen features learns this label, after which (beta, nu) are recovered algebraically. On 30 held-out simulated objects at five photon levels, one-image predictions achieved pooled Spearman correlations of 0.832-0.951 against independent 400-repeat references, median within-image correlations were 0.834-0.947, and 98.81-99.55% of evaluated pixels satisfied the algebraic admissibility condition. The method needs no KL term, reference prior, evidence regularizer, or cross-loss weight.
Ge Wang
Rensselaer Polytechnic Institute · Biomedical Imaging Center, Rensselaer Polytechnic Institute, Troy, New York, USA
Positron emission tomography (PET) seeks to balance diagnostic quality with ra-diation dose. Low-count PET noise is non-Gaussian, non-stationary, and spatial-ly dependent. It scales directly with local activity and is shaped by iterative recon-struction and physical corrections. Standard denoising diffusion probabilistic models (DDPMs) ignore these PET properties. Their forward process adds iso-tropic, homoscedastic Gaussian noise to the target. Such an approach fails to cap-ture the realistic physical degradation generated by the imaging system. To ad-dress the above limitations, this study introduces a heteroscedastic residual diffu-sion model (HDDPM) for low-count brain PET recovery in which the forward corruption is itself intensity-aware. We designed a fixed, Poisson-based variance module to generate voxel-wise noise maps. These maps naturally place stronger noise perturbation on low-activity regions than high-activity ones, meanwhile the network predicts the low-to-standard-count residual under explicit dose-fraction conditioning. We evaluated our proposed model (HDDPM) alongside generative frameworks across three different scanners, using both internal and external da-tasets at various simulated dose levels (1% to 50%). HDDPM and isotropic DDPM showed comparable overall image quality, but HDDPM stood out in the lowest-dose (1%) external scans. It is highly reliable and significantly reduces measurement errors in both high- and low-activity regions, compared to the standard model. These results support that heteroscedastic noising with the pro-posed HDDPM is feasible, and it provides a physically motivated inductive bias for quantitative low-count PET recovery by reflecting the activity-dependent noise structure of PET.
Raymond Confidence, Udunna C. Anazodo
Department of Biomedical Engineering, McGill University, Montreal, Canada · Montreal Neurological Institute, McGill University, Montreal, QC, Canada
Existing score-based methods for inverse problems often resort to approximate minimization of the KL divergence between the inversion distribution and the Bayesian posterior. Such an approximation leads to severe mode collapse and unreliable uncertainty quantification. In this paper, we propose Principled Posterior Matching (PPM), a framework that returns to the fundamentals of variational inference, rather than using tricky approximations. Instead of relying on heuristic approximations, we rigorously formulate the exact optimization of the KL divergence via the integration of Fisher divergence. We derive a tractable, equivalent gradient form of this integral, enabling precise optimization without the biases introduced by prior approximations. Our analysis clearly reveals that the mode collapse in previous methods stems directly from this approximation gap. Supported by our theoretical solution, PPM unifies two complementary paradigms: (1) In variational inference, PPM adopts mass-covering divergences that significantly improve the inversion diversity and uncertainty quantification; (2) In amortized inference, it enables the training of an efficient reconstruction network for rapid, single-step reconstruction. Furthermore, our formulation naturally extends to a broader family of divergence measures by generalizing the integral of the Fisher divergence. We validate PPM across challenging computational imaging tasks, including inpainting, super-resolution fluorescent microscopy, and radio interferometric black-hole imaging. In all experiments, PPM achieves superior reconstruction fidelity, faithful multimodal posterior recovery, and well-calibrated uncertainty estimates, establishing a robust framework for scientific imaging.