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.
Figures & tables
Figure 1: PEEL workflow with a sequential teacher–student design. Stage 1 trains Fθ from noisy FBP using only Student- t NLL. Repeated physical-noise inputs propagated through frozen Fθ⋆ generate ualeMC . Stage 2 attaches Aψ to frozen features hθ⋆ and updates only ψ .
Figure 2: Two-network architectures for PEEL. (a) In Stage 1, the reconstruction network Fθ maps a single noisy FBP realization to shared features hθ and the identifiable Student- t coordinates (γ^,α^,c^) . (b) In Stage 2, with Fθ frozen, the aleatoric head Aψ uses hθ from a single noisy realization to predict the output-domain aleatoric variance u^ale , supervised by the Monte Carlo teacher ualeMC constructed from repeated physical-noise realizations. The decoupled design avoids cross-loss coupling between reconstruction and aleatoric regression.
I0
log-RMSE
Pooled ρ
Within-image ρ
Case-mean ρ
Admissible (%)
1.50×104
0.953
0.832
0.834
0.879
98.81
2.25×104
0.631
0.884
0.883
0.819
99.17
3.50×104
0.439
0.932
0.925
0.886
99.55
5.25×104
0.532
0.944
0.940
0.876
99.49
8.00×104
0.732
0.951
0.947
0.850
99.49
Table 1: One-image uale prediction against independent Rref=400 MC references on 30 held-out objects. All correlations are Spearman coefficients.
Figure 3: Representative held-out object at I0=5.25×104 . Left: the single noisy FBP supplied at inference. Middle: output-domain ualeref from 400 independent physical-noise realizations. Right: uale predicted from the single FBP. The variance maps share identical color limits.
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.
Ge Wang
Biomedical Imaging Center, Rensselaer Polytechnic Institute Troy, New York, USA
Here we introduce ELECTRIC (Evidential Learning-Enhanced CT Reconstruction via Iterative Correction), a physics-guided Bayesian formulation. An evidential neural network provides an image proposal and an error-predictive epistemic-uncertainty surrogate. The latter is converted into an adaptive precision field and inserted into a Poisson-weighted MAP update. The resulting image-evidence-precision-reconstruction loop treats prior confidence as a learned state variable of iterative reconstruction. In addition to the formulation and theoretical analysis, we report two simulation studies on image slices from the AAPM Mayo Clinic Low-Dose CT dataset: a mechanism-validation pilot using transparent surrogate estimators, and a feasibility study in which a trained Normal-Inverse-Gamma evidential network drives the full closed loop. On held-out patients, the learned prior mean reduces reconstruction error by roughly 70 percent relative to filtered back-projection, the learned epistemic uncertainty is error-predictive and supports selective trust, and the physics-guided update restores measurement consistency while the adaptive-precision reconstruction matches or exceeds a validation-tuned fixed prior and remains markedly more robust to prior-strength misspecification. Together these results demonstrate the complete ELECTRIC closed-loop pipeline, while identifying formal uncertainty calibration and joint training as the principal directions for future work.
Ge Wang
Department of Biomedical Engineering Department of Electrical, Computer and Systems Engineering Department of Computer Science Rensselaer Polytechnic Institute, Troy, New York, USA
Reconstructing 3D Computed Tomography (CT) images from a few X-ray projections is a highly ill-posed inverse problem due to the loss of volumetric information. We propose PhyDiCT, a training-free framework that integrates a differentiable Physics-based forward model, grounded in the Beer-Lambert law, with a text-conditioned Diffusion as a strong prior to reconstruct 3D lung CT images. We refer to our approach as training-free since the prior model is used without fine-tuning, and our goal is to steer the denoising procedure to generate samples consistent with X-ray observations. We guide the diffusion generation using Split Gibbs sampling to jointly optimize for projection fidelity (reward) and consistency with prior knowledge. Also, we introduce a test-time refinement step that enhances image realism and anatomical coherence. We extensively evaluate our method on publicly available 3D CT datasets using both perceptual and semantic metrics, demonstrating that it surpasses existing plug-and-play diffusion and fully trained reconstruction approaches. Our findings highlight that combining a strong generative prior with the underlying physics of image formation substantially improves reconstruction quality, e.g., 7.5% improvement on SSIM compared to full training methods. Code will be released at https://github.com/batmanlab/PhyDiCT.