PaperlayerView exact revision on arXiv

Source-linked AI summary

PEEL: Physics-Enabled Evidential Learning for Identifiable Uncertainty in CT Imaging

Ge Wang

arXiv:2609.29599v1cs.AIphysics.med-ph

TL;DR

The marginal Student-t likelihood leaves a one-dimensional NIG parameter fiber unidentified. PEEL uses output-domain aleatoric variance measured from repeated physical-noise realizations to resolve it. In synthetic CT, one-image predictions correlated strongly with independent 400-repeat references, and most evaluated pixels met the algebraic admissibility condition.

  • Problem

    The marginal Student-t likelihood identifies only (γ, α, c) and is invariant along a one-dimensional (β, ν) fiber, leaving full NIG identification unresolved.

    Source: §4 · p. 9

  • Method

    PEEL first trains and freezes the likelihood model, then uses repeated physical-noise realizations to supervise an aleatoric head before algebraically recovering (β, ν).

    Source: §2.4 · p. 6, §4 · p. 9

  • Results

    Across five doses, pooled ρ increased from 0.832 at I0 = 1.5 × 10^4 to 0.951 at 8.0 × 10^4, while median within-image ρ increased from 0.834 to 0.947.

    Source: §3.2 · p. 8, §4 · p. 9

  • Takeaways & Limitations

    The synthetic CT results support resolving NIG non-identifiability with additional physical information, without a KL term, reference prior, evidence regularizer, or cross-loss coefficient.

    Source: §4 · p. 10

    Evidence is limited to in-distribution synthetic data, one fixed training seed, and Monte Carlo rather than clinical ground truth.

    Source: §3.2 · p. 9, §4 · p. 10

Abstract

from arXiv · show

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.

1 Introduction

The introduction frames NIG uncertainty estimation as a non-identifiability problem that calls for new information rather than a selected regularizer or reference prior. PEEL proposes repeated physical-noise measurements as the missing observation, learned after the likelihood model is frozen.

1 Introduction

  • The likelihood and aleatoric objectives are optimized sequentially: the reconstruction network is frozen before the aleatoric head learns from MC supervision, then (β, ν) are recovered algebraically.The losses are never combined as a weighted sum.Source: §1 · p. 1
  • Deep evidential regression adds evidence regularization, whereas prior-network approaches specify a reference distribution.The earlier ELVAE formulation placed an input-dependent NIG hierarchy at a variational latent bottleneck and distinguished its hierarchy-level decomposition from the marginalized Student-t law [4].Source: §1 · pp. 1–2
  • The MC teacher is intended to supply a physical observation missing from regularized approaches that may select a point on the unidentified fiber.The aleatoric head is trained after the reconstruction network is frozen.Source: §1 · pp. 1–2

2 Methodology

PEEL separates likelihood-based estimation from physics-based uncertainty identification: it freezes an NLL-trained reconstruction network, learns output-domain aleatoric variance from repeated physical-noise realizations, then recovers the full NIG parameterization algebraically.

2.5 Stage 3: algebraic recovery

  • Stage 3 combines the learned likelihood parameters with independently estimated aleatoric variance to recover the full NIG parameters, reporting pixels outside the admissible range as failures.The design keeps aleatoric regression decoupled from reconstruction through separate sequential losses.Source: §2 · p. 2, §2.5 · pp. 6–7

2.3 Stage 1: reconstruction network under NLL

  • Stage 1 trains the reconstruction network from noisy FBP using Student-t NLL alone, estimates (γ, α, c), and freezes all network parameters afterward.The proof-of-concept network uses shared features and separate 1 × 1 convolutional heads for the three outputs.Source: Figure 1 · p. 2, §2.3 · p. 5

2.1 NIG parameterization and identifiability

  • The NLL identifies (γ, α, c) but not β and ν separately; an independent aleatoric-variance measurement permits unique algebraic recovery without a KL term or reference prior.The method first learns and freezes the likelihood model, then supplies the independent variance measurement.Source: §2.1 · p. 3

2.2 CT data generation

  • The setup uses 32 × 32-pixel images, 36 parallel-beam views over [0, 180°), five dose levels, and object-separated training, validation, and test splits.It includes 100 training, 20 validation, and 30 test objects; reference repeats are 400.Source: §2.2 · pp. 3–4

2.4 Stage 2: Monte Carlo teacher and aleatoric head

  • With the reconstruction network frozen, repeated physical-noise inputs produce an output-domain Monte Carlo variance target for an aleatoric head trained with a separate loss.The teacher is the Monte Carlo target, not a second trainable teacher network; the two losses are never combined.Source: §2.4 · pp. 5–6

3 Experimental Design and Results

Across five photon levels, the study evaluates one-image aleatoric-variance prediction on held-out objects against independent Monte Carlo references. Predictions recover spatial variance ordering strongly, though magnitude mismatch remains.

3.1 Overall design

  • The evaluation asks whether one noisy FBP predicts output-domain aleatoric variance for an unseen in-distribution object, using 400 independent noise realizations only as the reference.The evaluation input and reference realizations were not used in training.Source: §3.1 · p. 7

3.2 Representative results

  • 0.832 to 0.951 pooled Spearman ρ and 0.834 to 0.947 median within-image ρ span the five doses for one-image predictions against 400-repeat references.The experiment covers 30 held-out objects; all reported correlations are Spearman coefficients.Source: Table 1 · p. 8, §3.2 · p. 8

3.1 Overall design

  • More than 98.8% of evaluated pixels were admissible for algebraic NIG recovery, and no separate ground truth was assigned to u_epi, ν, or β.Those parameters follow algebraically from the learned likelihood coordinates and independently supervised u_ale.Source: §3.1 · p. 8, §3.2 · p. 8

3.2 Representative results

  • Case-mean correlations were 0.819–0.886, while log-RMSE was lowest at the middle dose and residual calibration errors were larger at the dose extremes.Spatial ordering was more stable than absolute log-scale calibration at the dose extremes.Source: §3.2 · p. 8
  • For a representative held-out object at I0 = 5.25 × 10^4, Figure 3 compares the single noisy FBP, variance reference from 400 independent physical-noise realizations, and single-FBP prediction; maps share identical color limits.Source: Figure 3 · p. 9, §3.2 · p. 9
  • The findings establish only in-distribution prediction in a synthetic proof-of-concept, not out-of-distribution generalization, clinical performance, or ground-truth calibration of u_epi, ν, or β.The experiment was run once with a fixed seed and needs broader stability testing.Source: §3.2 · p. 9

4 Discussion and conclusion

The discussion frames NIG non-identifiability as a missing-measurement problem: an independently measured aleatoric quantity resolves the likelihood’s unidentified fiber. The synthetic CT results support this approach within a limited evaluation scope.

4 Discussion and conclusion

  • The marginal Student-t likelihood identifies only (γ, α, c) and remains invariant along a one-dimensional (β, ν) fiber; regularizers can select a point but add no physical observation.PEEL uses repeated physical-noise realizations to measure output-domain aleatoric variance, after which (β, ν) follow algebraically.Source: §4 · p. 9
  • High pooled and within-image Spearman correlations indicate reliable spatial ordering, but nonzero log-RMSE and larger errors at dose extremes show that absolute magnitude agreement remains imperfect.Log-RMSE was lowest at the middle dose; case-mean correlations were 0.819–0.886.Source: §4 · p. 9
  • The evidence is limited to in-distribution synthetic data, one fixed training seed, and Monte Carlo rather than clinical ground truth.The paper calls for broader stability tests and evaluation across object distributions, acquisition geometries, reconstruction architectures, and real CT measurements.Source: §4 · p. 10
  • PEEL combines likelihood-identifiable Student-t coordinates with independently measured aleatoric variance to recover the full NIG parameterization without a KL term, reference prior, or evidence regularizer.The conclusion also reports no cross-loss coefficient is needed.Source: §4 · pp. 9–10
  • The conclusion supports the premise that additional physical information can resolve NIG non-identifiability, while generalization beyond the in-distribution setting remains to be determined.Future work should test robustness across seeds, object distributions, acquisition geometries, reconstruction architectures, and real CT measurements.Source: §4 · p. 10
Loading 2609.29599v1…