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The Noise Plane: A Diffusion Bridge Across Exponent and Dispersion

Xuran Hu, Yuru Jia, Tengxi Wang, Miloš Daković, Ljubiša Stanković, Wufan Zhao

arXiv:2610.37576v1cs.CV

TL;DR

Sensor noise varies across imaging mechanisms, making a noise-level-only description insufficient. p-Bridge connects Tweedie measurement models through a shared, exponent-conditioned restoration framework, and exponent conditioning improves robustness across noise exponents, including with estimated exponents.

  • Problem

    Sensor noise varies substantially across sensing mechanisms, so noise level alone is insufficient to characterize imaging degradation.

    Source: §I · p. 1

  • Method

    p-Bridge indexes a Tweedie measurement bridge by exponent p and accumulated measurement ν, connecting distributions through independent increments and using a shared exponent-conditioned network.

    Source: §I · p. 2

  • Results

    Exponent conditioning improves cross-exponent robustness over fixed-exponent models, and using estimated exponents leaves robustness comparable to oracle-selected specialists.

    Source: §V · pp. 12–13

  • Takeaways & Limitations

    Results on synthetic images, fluorescence microscopy, and SAR support the usefulness of the formulation across imaging regimes.

    Source: §V · p. 13

Abstract

from arXiv · show

Diffusion-based denoising commonly uses Gaussian perturbations, whereas sensor noise follows diverse signal-dependent statistical laws. In this paper, we propose $p$-Bridge, a unified measurement-indexed diffusion framework for Tweedie noise. The noise exponent $p$ describes the variance--intensity relation, while accumulated measurement $\nu$ controls the noise level. Together, they define a noise plane that includes Gaussian, Poisson, and Gamma models. The stochastic bridge connects measurement distributions through independent increments, giving denoising a physical interpretation as progressive measurement accumulation. Both the starting and stopping indices can be specified, allowing restoration from an observed measurement level to a prescribed higher level. Measurement superposition yields a shared reverse-mean update, paired with a deviance-based objective for learning the required conditional mean. A latent compound Poisson--Gamma construction enables sampling at intermediate exponents. A single exponent-conditioned network adapts to different Tweedie noise models without separate training for each exponent. Experiments show that exponent conditioning improves cross-exponent robustness over fixed-exponent models. Using estimated exponents, the shared network achieves robustness comparable to an oracle-selected set of five specialist models. Experiments on synthetic images, FMD fluorescence microscopy, and multi-sensor SAR images further demonstrate its effectiveness across different signal-dependent noise regimes.

I. INTRODUCTION

Sensor noise varies across imaging mechanisms, while existing diffusion constructions often target particular laws or discrete families. The introduction motivates p-Bridge as a measurement-indexed framework spanning Tweedie noise models.

I. INTRODUCTION

  • Imaging measurements require an appropriate noise description because sensing mechanisms produce different statistics, and noise level alone cannot characterize these regimes.Classical and learned denoising methods rely on both image priors and observation statistics.Source: §I · p. 1

A. Diffusion models, bridges and inverse problems

  • Prior restoration bridges include I2SB [21], Poisson-likelihood DPS, unified Gaussian, Gamma, and Poisson diffusion [23], and the measurement-indexed Gamma Bridge [24].These constructions address particular noise laws or discrete families, motivating a common construction for continuously varying variance–intensity relations.Source: §I · p. 2

III. A UNIFIED MEASUREMENT-INDEXED BRIDGE

  • p-Bridge connects Tweedie measurement distributions through independent increments, using exponent p for the variance–intensity relation and accumulated measurement ν for noise level.The framework permits specified starting and stopping measurement levels and uses a shared exponent-conditioned network.Source: §I · p. 2

B. Signal-dependent denoising and noise estimation

  • Noise estimation and adaptation are often tied to a specified noise family, whereas p-Bridge uses estimated p to select its measurement law and estimated ν to set its starting noise level.The two parameters are fitted jointly because the scaling of ν depends on p.Source: B. Signal-dependent denoising and… · p. 3

A. Preliminaries and notation

  • For Tweedie models, variance follows V(µ) = µ^p, while the measured intensity has variance V(µ)/ν and larger ν corresponds to less noise.The exponent p sets the variance–intensity relation; ν represents accumulated measurement rather than noise amplitude.Source: A. Preliminaries and notation · p. 4

B. Measurement-indexed formulation

The formulation treats denoising as adding independent measurements of the same scene, moving from an observed index to a larger one. Its stochastic transition has a shared conditional-mean update, while the deterministic version does not preserve bridge marginals.

B. Measurement-indexed formulation

  • The clean image x⋆ and measurement xν are related at an index ν that represents exposure, dose, or number of looks; additional dispersion is absorbed into ν.The measurement index is expressed in the units of the noise model.Source: B. Measurement-indexed formulation · p. 4
  • Independent measurements of the same scene combine within the same family at index ν1 + ν2, with their intensity average weighted by the indices.The closure holds for the Tweedie models described and includes Poisson superposition and Gamma addition at a common rate.Source: B. Measurement-indexed formulation · p. 4
  • The bridge schedules indices from νmax down to νobs, so the final marginal matches the observation model while the finite initial state remains noisy.The clean image is approached in conditional mean square as ν tends to infinity; the construction uses independent increments with the same underlying mean.Source: B. Measurement-indexed formulation · p. 4

1) Superposition and the reverse mean:

  • A reverse step increases the measurement index by adding an independent increment with amount ∆t = ν(t−1) − ν(t).This superposition coupling recovers the Gamma Bridge update at p = 2 and the corresponding Poisson construction at p = 1.Source: 1) Superposition and the reverse mean · p. 4
  • Replacing the clean image with a network prediction yields a reverse update whose conditional mean uses a mixing coefficient αt determined by the measurement schedule, not p.The update is the conditional mean of the plug-in transition.Source: 1) Superposition and the reverse mean · p. 4
  • With an oracle prediction and correct initialization, independent increments reproduce the superposition coupling’s joint law; a fixed observation instead defines its conditional trajectory.This distinguishes the bridge-distributed stochastic construction from a trajectory conditioned on one observed input.Source: 1) Superposition and the reverse mean · p. 4
  • The deterministic update preserves the oracle mean but not bridge marginals, and its conditional variance is αtV(x⋆)/ν(t−1) rather than V(x⋆)/ν(t−1).Physical measurement marginals and oracle distributional exactness apply to the stochastic construction; the deterministic sampler follows its reverse mean.Source: 1) Superposition and the reverse mean · pp. 4–5

2) The unit deviance:

The unit deviance supplies a reconstruction objective aligned with the conditional mean used by the reverse update. For intermediate exponents, latent compound Poisson–Gamma thinning provides a measurement-transition construction.

2) The unit deviance:

  • Under the stated integrability conditions, the unit-deviance reconstruction is uniquely optimized by the conditional mean, aligning its reconstruction term with the mean used in Equation (8).The paper states that its bounded clean-image range X = [ϵ, 1] satisfies these conditions.Source: 2) The unit deviance · p. 5
  • The power-variance deviance has a shared gradient and yields squared error at p = 0, Poisson deviance at p = 1, and twice the Itakura–Saito divergence at p = 2.Intermediate exponents have corresponding intermediate deviance forms.Source: 2) The unit deviance · p. 5

D. Exponent-specific thinning

  • At p = 0, 1, and 2, the thinning kernels are Gaussian, Binomial, and scaled Beta, respectively; for 1 < p < 2, thinning uses a two-stage construction.The intermediate-exponent construction is based on a compound Poisson–Gamma representation.Source: D. Exponent-specific thinning · p. 5
  • The construction combines Poisson thinning of counts with Beta–Gamma splitting of amplitudes; simulator-retained counts need no count inference, whereas an observed total alone requires sampling the count from its posterior.Source: D. Exponent-specific thinning · p. 5
  • The finite truncation used for real observations makes the thinning implementation approximate, and the synthetic severity-transfer experiment uses simulator counts unavailable at deployment.That simulator-count information is used when thinning the conditioning channel.Source: D. Exponent-specific thinning · p. 5

E. Output parameterization and measurement calibration

The section specifies how p-Bridge parameterizes outputs and calibrates measurement coordinates across its supported exponent range, then describes its training losses and their qualifications.

E. Output parameterization and measurement calibration

  • For 1 ≤ p < 2, direct log-intensity prediction addresses higher relative variability in dark regions; at p = 2, the model uses residual anchoring.Positive means use an exponential output map for p ≥ 1, while the Gaussian branch uses an identity map.Source: E. Output parameterization and… · p. 5
  • The sampler clips predicted clean images to X = [ϵ, 1], not sampled states, while the observation support may extend beyond that range.The paper distinguishes the mean domain, normalized clean-image range, and observation support; compound observations may be zero or exceed the clean-image range.Source: E. Output parameterization and… · p. 5
  • Only at p = 2 is the estimated measurement index invariant to multiplicative gain; other exponents require a specified intensity scale and joint estimation of (p, ν).The section motivates intensity-scale calibration because radiometric rescaling changes the estimated index at other exponents.Source: E. Output parameterization and… · p. 5

F. Exponent range and Gaussian special case

  • The formulation covers p ∈ {0} ∪, with p = 0 as a separate Gaussian branch and p ∈ [1, 2] handled by the conditional network.The p = 0 marginals use the VE-SDE / EDM noise parameterization, but the deterministic update differs from an EDM probability-flow Euler step.Source: F. Exponent range and Gaussian special… · p. 6
  • Independent Gaussian read noise generally leaves the single-exponent family, and extending the bridge requires a cumulative-noise model with compatible independent increments.For real data, the fitted exponent summarizes the variance–intensity relation over the calibration range.Source: F. Exponent range and Gaussian special… · p. 6

G. Training objective

  • The reconstruction loss replaces a time-dependent weight proportional to ν(t) with the training-step mean of ν(t)/νobs, preserving average weight and pointwise reconstruction optimum.This distributes reconstruction weight uniformly across steps.Source: G. Training objective · p. 6
  • The moment term regularizes residual statistics, but for learned predictions it can shift the reconstruction optimum.Under oracle predictions, residuals have zero mean and unit variance; higher-order moments depend on the noise model.Source: G. Training objective · p. 6
  • The trajectory term supervises a network evaluation at a state generated by one deterministic update, with stop-gradient blocking differentiation through the generating prediction.Reconstruction optimality on the forward training distribution need not imply stationarity on the induced distribution, even when their supports overlap.Source: G. Training objective · p. 6

IV. EXPERIMENTS

The experiments examine exponent mismatch and conditioning, compare restoration methods, and assess noise-parameter estimation across synthetic images, fluorescence microscopy, and multi-sensor SAR.

A. Experimental setup

  • Evaluation spans five synthetic noise exponents and three severities, plus FMD fluorescence microscopy configurations and three SAR sensors.The synthetic set uses 64 held-out crops; real-data restoration comparisons use three FMD configurations and all three SAR sensors.Source: A. Experimental setup · p. 6, §IV · p. 6
  • Synthetic and FMD restoration are evaluated with PSNR and SSIM, while real SAR uses the ratio-image measures M and RIS alongside visual comparisons.For FMD, evaluation uses one held-out field per configuration; real SAR has no clean references.Source: A. Experimental setup · p. 7
  • For real-image evaluation, calibrated (p, ν) coordinates configure the bridge and choose the denoising start point; severity transfer is tested without retraining.Unless otherwise stated, inference uses deterministic sampling with five network evaluations; severe-noise models are also evaluated at moderate and mild noise.Source: A. Experimental setup · p. 8

B. Behaviour along the exponent axis

Exponent mismatch can substantially reduce restoration quality, while conditioning a shared model on the exponent improves robustness across synthetic, microscopy, and SAR data. Ablations identify benefits from selected training and inference components.

B. Behaviour along the exponent axis

  • At severe noise, the largest exponent mismatch costs 4.40 dB PSNR, and mismatched pairs in the p ∈ {1, 1.5, 2} subgrid lose 2.00 dB on average.The reverse Poisson-to-Gamma mismatch costs approximately 3.3 dB, so the two mismatch directions are not equally harmful.Source: B. Behaviour along the exponent axis · p. 8
  • The matched fixed-exponent model achieves the highest PSNR at every tested data exponent and noise level; p = 1.5 gives the best worst-case PSNR at moderate and severe noise.At mild noise, the worst-case optimum shifts to p = 1.75.Source: B. Behaviour along the exponent axis · p. 8
  • Using estimated exponents, the conditional network improves worst-case PSNR over the best fixed-exponent model by 0.921, 1.525, and 1.099 dB at severe, moderate, and mild noise.Its minimum PSNR differs from oracle-selected specialists by at most 0.006 dB across the three noise levels.Source: B. Behaviour along the exponent axis · p. 8

C. Comparison

  • Synthetic results favor conditional p-Bridge: it leads in PSNR in eight of nine settings and in SSIM in all nine, with less PSNR variation across exponents.At severe noise, its PSNR drops 0.13 dB from pdata = 1 to 2, versus 0.94 dB for Noise2VST.Source: C. Comparison · p. 9
  • On FMD, conditional p-Bridge has the highest mean SSIM (0.912) and second-highest mean PSNR (31.76 dB), with configuration-dependent PSNR rankings.It leads on TP-MICE and trails VIRNet on CF-FISH and CF-MICE.Source: C. Comparison · p. 9
  • Across three SAR sensors, p-Bridge has the lowest mean M and RIS (19.4/35.3), ranking first on miniSAR and TerraSAR-X and second on FARAD-X.On FARAD-X, NAFNet scores lower by 0.6 and 2.7, respectively.Source: C. Comparison · p. 9

D. Ablation studies

  • At five network evaluations, removing consistency regularization or replacing the reverse mixture with direct prediction reduces PSNR by 0.210 or 0.186 dB; losses grow at 25 evaluations.At 25 evaluations, the respective losses are 0.452 and 1.654 dB; at one evaluation, the samplers are identical because no reverse update is applied.Source: D. Ablation studies · p. 11, Table VI · p. 11
  • Removing output-range projection changes matched-exponent PSNR by at most 0.001 dB, but reduces it by 0.139–0.223 dB when conditioning underestimates the data exponent.The comparison uses Gamma data (pdata = 2) conditioned at p = 1.Source: D. Ablation studies · p. 11

E. Noise parameter estimation and robustness

The paper estimates noise parameters from noisy images and tests how exponent-estimation errors affect conditional restoration. Its conclusion emphasizes robustness across imaging regimes, while the supplementary material specifies the modeled Tweedie family and measurement-index calibration.

E. Noise parameter estimation and robustness

  • Exponent errors up to 0.25 preserve most conditioning gains, whereas unconstrained errors make the conditional model underperform the fixed baseline.At δ = 0.25, gains over fixed p = 1.5 are 0.830–1.311 dB; without an error constraint, the model falls below baseline by 1.707–1.863 dB.Source: E. Noise parameter estimation and… · p. 11
  • A robust regression of log noise variance on log mean estimates p from the slope and ν from the intercept of the variance–mean relation.The method takes logarithms of Var(xobs | µ) = µ^p/ν.Source: E. Noise parameter estimation and… · p. 12
  • Across the evaluation grid, the worst-case absolute error in p is 0.0914, and median relative error in ν ranges from 2.8% to 4.1%.Mild noise is harder to estimate because ν is 16 times its severe-noise value at fixed exponent, leaving a weaker noise signal.Source: E. Noise parameter estimation and… · p. 12
  • For real data, fitted variance–mean slopes differ across detector classes and configurations, supporting configuration-specific calibration rather than one fixed exponent.FMD checks compare real and simulated noise at matched clean signal, exponent, and index, then vary the candidate exponent while refitting the index.Source: E. Noise parameter estimation and… · p. 12, Figure 11 · p. 12, Figure 12 · p. 12

V. CONCLUSION

  • The conclusion reports cross-exponent robustness comparable to oracle-selected specialists using estimated exponents, with moderate conditioning errors preserving most of the benefit.The formulation is evaluated on synthetic images, fluorescence microscopy, and SAR.Source: §V · pp. 12–13

Supplementary Material

  • The Tweedie models considered use p ∈ {0} ∪, covering Gaussian, Poisson, compound Poisson–Gamma, and Gamma distributions.When physical measurement amount a gives variance ϕV(µ)/a, the index ν = a/ϕ recovers V(µ)/ν.Source: A. Accumulated signal and intensity · p. 16

B. Conditions for the bridge construction

The bridge construction requires an additive measurement family with suitable index and mean domains, and connects measurement levels through independent increments. Its reverse-mean update uses an exponent-independent mixing coefficient.

B. Conditions for the bridge construction

  • The bridge assumes an infinitely divisible family over positive measurement indices, an invertible mean map, a decreasing schedule with positive increments, and conditionally independent increments sharing the same natural parameter.Distributional identities and variances are interpreted pixelwise.Source: B. Conditions for the bridge… · p. 16

C. Clean limit and intensity domains

  • At finite νmax, the initial bridge state has variance V(x*)/νmax and is not the clean image; the conditional mean-square limit holds as measurement accumulates.The clean-image implementation range is [ϵ, 1], but sampled states are not clipped; for 1 < p < 2, observations may be zero or exceed that range.Source: C. Clean limit and intensity domains · p. 16

A. Closure under measurement accumulation

  • Independent accumulated signals with the same natural parameter remain in the family when summed, with their measurement indices added; different parameters generally do not preserve this form.This closure under measurement accumulation motivates the superposition bridge.Source: A. Closure under measurement… · p. 17

B. The superposition coupling

  • The superposition coupling moves to a larger measurement level by adding an independent increment with index Δt, producing the prescribed marginal at the earlier reverse-time step.The intensity form follows by normalizing the accumulated signal by the new measurement index.Source: B. The superposition coupling · p. 17

C. Exponent-independent mixing coefficient

  • The plug-in reverse-mean update blends the current intensity and predicted clean mean, with a mixing coefficient determined only by measurement indices, not by the noise exponent.The increment distribution and network prediction may still depend on p.Source: C. Exponent-independent mixing… · p. 17

D. Measurement-weighted intensity averaging

Measurement-weighted averaging preserves the Tweedie family under conditional independence. The deviance objective targets the conditional mean used by the reverse update, while oracle stochastic and deterministic updates preserve different properties.

D. Measurement-weighted intensity averaging

  • Independent measurements of a fixed scene, weighted by their measurement indices, average to a member of the same family with unchanged mean and summed index.Exposure, dose, or number of looks can serve as the index when the sensor model and calibration meet the assumptions.Source: D. Measurement-weighted intensity… · p. 17

E. Oracle preservation properties

  • With oracle predictions, the stochastic update preserves bridge marginals, whereas the deterministic update preserves their mean along the schedule but reduces variance rather than preserving the distribution.The mean-preservation result extends by induction.Source: E. Oracle preservation properties · p. 17
  • The stochastic sampler reproduces the superposition coupling's joint trajectory when initialized from its observation model; fixing an observation instead yields the trajectory conditional on that observation.The marginals alone do not specify a unique joint law across time.Source: E. Oracle preservation properties · p. 17

A. Bregman representation

B. Mean-optimal reconstruction

  • The deviance objective uniquely minimizes at the conditional mean when that mean lies in the domain and expected deviance is finite, matching the target required by the reverse update.The expected-deviance decomposition separates the value at the mean from a nonnegative divergence penalty.Source: B. Mean-optimal reconstruction · p. 18

C. Tweedie deviance formulas

The appendix gives Tweedie deviance special cases and derives thinning transitions for supported exponents. The Gaussian and Poisson cases have standard kernels, while Gamma thinning scales the accumulated total by a Beta-distributed fraction.

C. Tweedie deviance formulas

  • The Tweedie deviance formulas follow by integrating the variance-function gradient; Gaussian, Poisson, and Gamma cases arise at p = 0, p → 1, and p → 2, respectively.The latter two limits yield logarithmic terms.Source: C. Tweedie deviance formulas · p. 18

S4. MEASUREMENT THINNING KERNELS

  • The forward transition reduces measurement through a thinning kernel, with Gaussian, Poisson, and Gamma forms and a latent construction for intermediate exponents.The derivation also relates the compound model to the excess-noise factor.Source: S4. MEASUREMENT THINNING KERNELS · p. 18

A. Independence from the clean-signal parameter

  • Conditioning on the accumulated total removes dependence on the clean-signal parameter from the transition kernel for supported exponents, including 1 < p < 2.For positive totals, the kernel includes a continuous component and atoms at both endpoints; zero totals remain zero.Source: A. Independence from the clean-signal… · p. 18

B. Gaussian, Poisson, and Gamma kernels

  • For Gaussian noise, the retained accumulated signal is Gaussian with mean αtzt−1 and variance σ²αtΔt.The schedule determines both the retained mean fraction and transition variance.Source: B. Gaussian, Poisson, and Gamma kernels · p. 19
  • For Poisson noise, thinning retains each count independently with probability αt, yielding a Binomial(zt−1, αt) transition.The Poisson kernel follows from the p = 1 model.Source: B. Gaussian, Poisson, and Gamma kernels · p. 19
  • For Gamma noise, the fraction of the accumulated total retained by thinning follows Beta(ν(t), Δt), independently of the total.The Beta form follows after changing variables to the retained fraction.Source: B. Gaussian, Poisson, and Gamma kernels · p. 19

C. Latent thinning for intermediate exponents

For intermediate exponents, thinning combines count thinning with amplitude thinning in a compound Poisson–Gamma representation. Real observations require posterior count inference, and the exponent–gain relation applies within the Tweedie compound parameterization.

C. Latent thinning for intermediate exponents

  • For 1 < p < 2, latent thinning combines Binomial thinning of the jump count with Beta–Gamma thinning of the accumulated amplitude.The two layers are exact conditional on the count; boundary counts recover the kernel's endpoint atoms.Source: C. Latent thinning for intermediate… · p. 19

D. Latent-count inference for real observations

  • For real observations, the latent count must be sampled from its posterior; the implementation truncates the count weights at nmax = 64, and omitted posterior mass equals the total-variation error.For synthetic degradation, generated counts can be retained instead of inferred.Source: D. Latent-count inference for real… · pp. 19–20
  • Severity-transfer results use simulator-retained counts for intermediate-exponent thinning, so transfer using inferred counts remains unevaluated.Gamma thinning uses the exact Beta kernel.Source: D. Latent-count inference for real… · p. 20

E. Relation to the excess-noise factor

  • Within the Tweedie compound construction, the jump shape determines p through the excess-noise factor; assigning a sensor exponent also requires variance–intensity calibration.With fixed gain parameters and varying event rate, the compound Poisson variance-to-mean ratio is independent of that rate, unlike the scaled Poisson law at p = 1.Source: E. Relation to the excess-noise factor · p. 20
  • The excess-noise mapping is specific to the full Tweedie compound parameterization, whose endpoints recover Poisson and Gamma marginals under fixed mean and measurement index.The mapping is strictly increasing in the squared excess-noise factor and spans exponents in (1, 2).Source: E. Relation to the excess-noise factor · p. 20

S5. OUTPUT PARAMETERIZATION AND INDEX CALIBRATION

The output parameterization reflects how relative noise changes with exponent and how radiometric gain affects measurement-index calibration. The framework covers selected Tweedie exponents and applies its shared reverse-mean update to Gaussian marginals.

A. Relative noise and residual anchoring

B. Effect of radiometric gain

  • Gamma index estimation is gain-invariant because p − 2 = 0 at p = 2; for other exponents, calibration requires a fixed intensity-normalization convention.The estimated index depends on intensity units for exponents other than 2.Source: B. Effect of radiometric gain · p. 20

C. Prediction domain

  • For supported p ≥ 1 models, predictions are positive; at p = 0, the Gaussian mean domain is R and the output uses an identity map.Projection onto the bounded working range X is a separate implementation choice.Source: C. Prediction domain · p. 20

A. Admissible exponents

B. Gaussian marginals and reverse update

  • At p = 0, the Gaussian reparameterization has conditional variance independent of clean intensity, and the shared reverse-mean update applies directly.The marginal is written as x_t = x⋆ + σ_tξ, with ξ ∼ N(0, I).Source: B. Gaussian marginals and reverse update · p. 21
  • The shared update uses a variance-ratio coefficient, unlike the standard-deviation-ratio coefficient of an Euler step on the EDM probability-flow ODE.For a step from σ = 2 to σ = 1, the update retains one quarter of the displacement, versus one half under the Euler update; the coefficient follows from the conditional mean of the superposition coupling.Source: B. Gaussian marginals and reverse update · p. 21

S7. ROLE OF THE CONSISTENCY OBJECTIVE

The supplementary material examines the consistency objective, sampling budgets, exponent evaluation, estimator sensitivity, and implementation choices. It reports that longer sampling reduces exponent-mismatch losses but does not improve matched-exponent PSNR in the tested comparison.

S7. ROLE OF THE CONSISTENCY OBJECTIVE

  • The consistency term addresses a possible distribution shift: model-generated states can have a different optimal conditional-mean predictor from forward-process states, even when reconstruction is pointwise optimal.The stop-gradient removes derivatives through generated states while the network is still evaluated on them.Source: S7. ROLE OF THE CONSISTENCY OBJECTIVE · p. 21

A. Number of function evaluations

  • From one to 25 evaluations, the fixed p = 1.5 model’s minimum PSNR across tested data exponents rises by 0.855 dB, while mean mismatch penalty falls from 1.601 to 0.924 dB; matched-exponent mean PSNR is highest at five evaluations.At five evaluations, matched-exponent mean PSNR is 22.330 dB, versus 22.326 dB at one and 22.225 dB at 25; Table S2 reports these comparisons.Source: A. Number of function evaluations · p. 21

B. Evaluation across noise exponents

  • At severe noise with five evaluations, mean mismatch loss is 2.000 dB on the 3 × 3 exponent grid and 1.256 dB on the 5 × 5 grid; worst-case PSNR is the minimum across tested data exponents.Among fixed-exponent models, the minimax model maximizes that minimum.Source: B. Evaluation across noise exponents · p. 22

B. Sensitivity to estimator settings

  • Across tested estimator settings, maximum absolute exponent error ranges from 0.067 to 0.089; on constant-intensity tiles, |p̂ − p| ≤ 0.0066 and ν is recovered to three significant figures.The tested settings vary local window size, intensity-bin count, and retained fraction of flattest samples.Source: B. Sensitivity to estimator settings · p. 22

A. Conditioning and training-step sampling

  • Training samples bridge steps uniformly from {t_min, …, T − 1}; conditional training caches 21 bridge configurations and accumulates gradients over 12 micro-batches with separately sampled exponents.At inference, the exponent embedding and bridge configuration are updated together, including the corresponding measurement index.Source: A. Conditioning and training-step… · p. 22, B. The conditional model · p. 22

C. Noise-severity transfer

  • For noise-severity transfer, only the conditioning input is thinned to the training measurement index; the observed image initializing sampling remains unchanged.For 1 < p < 2, synthetic evaluations use simulator-retained latent counts for thinning.Source: C. Noise-severity transfer · p. 22

D. Noise-exponent estimation

  • The single-image estimator uses cross-difference residuals, local-mean intensity bins, and a Theil–Sen fit to estimate the exponent and measurement index.It retains the flattest 30% of samples within each of 10 bins using a 15 × 15 box filter; the pseudocode returns estimates before range clipping.Source: D. Noise-exponent estimation · p. 22, B. The conditional model · p. 22

A. Additional restoration examples

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