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Diffusion Posterior Sampling with Uncertain Forward Models

Wentao Shangguan, Vivek K Goyal

arXiv:2610.29102v1cs.CV

TL;DR

Fixed forward-model estimates leave parameter uncertainty out of diffusion reconstruction likelihoods, even when calibration data characterize that uncertainty. UAL approximates the nuisance-marginal likelihood with a structured Gaussian covariance and can be used in existing sampler updates. Across the reported applications, it improves reconstruction and ensemble prediction, with motion-deblurring comparisons also showing higher ensemble-mean PSNR than published blind samplers.

  • Problem

    Fixing uncertain forward-model parameters at estimated values omits their uncertainty from the likelihood, allowing parameter error to be absorbed into the reconstruction.

    Source: §1 · p. 1

  • Method

    UAL approximates the likelihood marginalized over nuisance parameters with a Gaussian whose mean and structured covariance are computed by linearizing the forward model.

    Source: §1 · p. 2, §3.2 · p. 5

  • Results

    UAL improves reconstruction across CT, MRI, and fluorescence microscopy and lowers ensemble prediction scores in all 12 reported comparisons; in motion deblurring, its DAPS and PnP-DM ensemble-mean PSNR exceeds published blind samplers by 6.2 to 6.4 dB.

    Source: §4.2 · p. 6, §5 · p. 9

  • Takeaways & Limitations

    UAL uses calibration and reference statistics in existing sampler data updates while retaining pretrained priors and sampling schedules, without additional diffusion-model training.

    Source: §1 · p. 3

    The nuisance uncertainty distribution is held fixed, and for nonlinear nuisance dependence UAL uses a Gaussian approximation based on linearization.

    Source: §3.1 · pp. 4–5

Abstract

from arXiv · show

Diffusion posterior samplers reconstruct signals by combining a pretrained diffusion prior with a measurement likelihood specified by a forward model and a noise model. Calibration and reference data can characterize uncertainty in the forward model's parameters, such as blur kernels or detector responses. Fixing these parameters at estimated values leaves their uncertainty out of the likelihood, and parameter errors are thus absorbed into the reconstruction. We introduce an uncertainty-aware likelihood (UAL), a Gaussian approximation to the marginal likelihood obtained by integrating over the uncertain parameters. For each candidate signal, UAL propagates parameter uncertainty into a structured measurement covariance, reducing the quadratic penalty along residual directions that parameter variation can explain. Existing samplers can use UAL in their data updates while retaining their pretrained diffusion priors and sampling schedules. Under a fixed Gaussian nuisance model, we bound likelihood error from linearization and average divergence between the exact marginal and UAL posteriors. We compare samplers with and without UAL across six inverse problems spanning Gaussian and motion deblurring, CT, MRI, ultrasound, and fluorescence microscopy, and find improvements in reconstruction and ensemble prediction. UAL reduces geometric mean MSE by 9% to 20% across samplers in the main Gaussian-deblurring setting and by 51% to 92% in motion deblurring when non-blind samplers use the population-mean kernel. In motion deblurring, DAPS and PnP-DM with UAL also give 6.2 to 6.4 dB higher ensemble-mean PSNR than published BlindDPS and Fast Diffusion EM. On the ablation set, UAL outperforms trace-matched isotropic inflation for PnP-DM in both deblurring tasks; in motion deblurring, it also exceeds the best-tested fixed scalar data weighting by 3.3 dB for PnP-DM and 4.2 dB for DAPS in average PSNR.

1 INTRODUCTION

Fixed-parameter likelihoods omit forward-model uncertainty, which can make reconstructions absorb parameter error. UAL marginalizes over that uncertainty and incorporates its effects into existing diffusion samplers.

1 INTRODUCTION

  • Fixed forward-model estimates leave parameter uncertainty out of the likelihood, so reconstruction can distort the signal to fit residuals caused by parameter error.Variation in nuisance parameters predicts correlated measurement changes, which should receive smaller quadratic penalties than unexplained residual directions.Source: §1 · pp. 1–2
  • UAL approximates the nuisance-marginal likelihood with a Gaussian and uses parameter means and covariances to weight residual directions in existing samplers.It retains pretrained priors and sampling schedules; no additional diffusion-model training is needed.Source: §1 · pp. 2–3
  • For a fixed Gaussian nuisance model, the paper bounds linearization likelihood error and average posterior divergence; UAL’s linear estimator minimizes average MSE among linear estimators under specified matched-moment conditions.At a fixed nuisance realization, the point-likelihood estimator can instead have lower error.Source: §1 · p. 2
  • 27.48 dB versus 27.03 dB in Gaussian deblurring, and 26.64 dB versus 23.37 dB in motion deblurring, are PnP-DM’s mean per-trajectory PSNRs with UAL versus its best tested scalar inflation.UAL also lowers mean CRPS, energy, and local variogram scores for every sampler and application in Table 1.Source: §1 · p. 3

2 BACKGROUND AND RELATED WORK

The background distinguishes fixed-operator diffusion methods and blind joint estimation from UAL’s marginal treatment of forward-model uncertainty. It also relates UAL to earlier Gaussian uncertainty models and describes the calibration information used to specify uncertainty.

2 BACKGROUND AND RELATED WORK

  • Fixed-operator diffusion methods condition on one nuisance value, while blind methods sample or estimate nuisance parameters jointly with the signal; UAL instead approximates a marginal likelihood from a specified uncertainty model.BlindDPS, GibbsDDRM, and PRISM are joint samplers; Fast Diffusion EM and DMPlug estimate parameters, while Kernel Diffusion samples a conditional kernel distribution.Source: §2 · p. 3
  • Calibration and reference data specify nuisance-parameter means and covariances, including variation in CT flat and dark fields, motion-kernel banks, MRI coil maps, and ultrasound sound speeds.These statistics define the distribution used to construct UAL.Source: §2 · pp. 3–4
  • Earlier work uses Gaussian uncertainty treatments for regression, sampling-location errors, optical coefficients, and random blurs; UAL adapts this treatment to diffusion samplers’ existing data updates.The paper tests UAL against numerical marginalization and scalar variance inflation.Source: §2 · p. 4

3 METHOD

The method marginalizes a fixed nuisance-parameter uncertainty model and approximates its measurement likelihood by a Gaussian whose covariance reflects parameter variation. It evaluates this likelihood through existing sampler updates and analyzes approximation error and reconstruction risk.

3.1 FORWARD MODEL AND UAL CONSTRUCTION

  • For each candidate signal, UAL linearizes the forward model in nuisance parameters and forms a Gaussian likelihood using the resulting measurement mean and covariance.The nuisance model is fixed; the signal and nuisance parameters are modeled as independent, and measurement noise is independent of both. The Gaussian marginal is exact for affine nuisance dependence with Gaussian nuisance parameters and independent Gaussian noise.Source: §3.1 · p. 4

3.2 UAL EVALUATION AND SAMPLER INTEGRATION

  • UAL weights residual directions according to predicted nuisance variation, while scalar covariance inflation weights all noise-whitened directions equally.The nuisance covariance can be factored so evaluation uses an r × r system rather than explicitly inverting the measurement covariance; UAL enters samplers through their published data updates.Source: §3.2 · p. 5

3.3 APPROXIMATION ERROR AND RECONSTRUCTION RISK

  • For nonlinear nuisance dependence, likelihood error is controlled by the forward-model linearization remainder, and the paper bounds the resulting average divergence between exact marginal and UAL posteriors.In a linear model with fixed nuisance sensitivity and matched nuisance moments, the UAL-based linear estimator minimizes MSE among linear estimators on average, though the point estimator can do better at a fixed nuisance realization near the mean.Source: §3.3 · pp. 5–6

4 EXPERIMENTS

Across the evaluated inverse problems, UAL improves reconstruction and ensemble-prediction results, while likelihood and ablation controls test whether its gains reflect accurate, structured uncertainty modeling. The experiments use paired sampler comparisons and examine likelihood accuracy, covariance choices, uncertainty levels, and covariance rank.

4.1 EXPERIMENTAL PROTOCOL

  • Paired comparisons hold measurement data, diffusion checkpoint, sampling schedule, step sizes, and reconstruction seeds fixed; deblurring, CT, MRI, and microscopy use 50 cases, while ultrasound uses acquired phantoms.Each case has 12 reconstruction seeds; the fixed deblurring ablation set contains 16 images with 12 paired seeds per image.Source: §4.1 · p. 6, §4.2 · p. 6

4.2 RECONSTRUCTION ACROSS SAMPLERS AND APPLICATIONS

  • UAL also improves reconstruction and ensemble prediction in CT, MRI, and fluorescence microscopy; on the PICMUS ultrasound contrast phantom, CNR rises from 9.929 to 12.363 dB and gCNR from 0.9141 to 0.9743.MRI gains are small but consistent across seven metrics; under the Gaussian coil-sensitivity model, UAL gives the exact marginal measurement likelihood.Source: §4.2 · pp. 6–7
  • In motion deblurring, DAPS and PnP-DM with UAL exceed published BlindDPS and Fast Diffusion EM by 6.2 to 6.4 dB in ensemble-mean PSNR; UAL improves all reported deblurring ensemble metrics.Table 1 compares reconstruction and ensemble prediction; paired 95% intervals for the motion-deblurring comparison exclude zero.Source: §4.2 · p. 6, Table 1 · p. 7

4.3 LIKELIHOOD VALIDATION AND CONTROLS

  • At final reconstructions, UAL's mean relative image-score error stays below 1% at every tested uncertainty level, versus 6% to 15% for the point likelihood; UAL also has the lowest score error at five saved states.Against numerical nuisance marginalization, UAL also has the lowest mean absolute negative-log-likelihood error; later-state score error is at most 2%, versus approximately 6% to 19% for the point likelihood.Source: §4.3 · p. 7, Figure 2 · p. 8
  • UAL's structured covariance raises per-trajectory PSNR and reduces CRPS versus trace-matched isotropic inflation in both deblurring tasks; it also beats the best tested scalar weighting in Gaussian and motion deblurring.For PnP-DM, UAL gives 27.475 dB versus 27.029 dB in Gaussian deblurring, and 26.641 dB versus 23.369 dB in motion deblurring.Source: §4.3 · pp. 7–8, Table 2 · p. 8
  • Recomputing covariance at each candidate preserves PnP-DM's Gaussian-deblurring gain whether or not derivatives are propagated, while fixing it at the ensemble mean removes nearly all the PSNR gain.For Fast Diffusion EM with a fixed population-mean kernel, differentiating covariance raises motion-deblurring PSNR from 23.80 to 28.49 dB; including the log determinant gives 29.10 dB.Source: Table 3 · p. 8, §4.3 · p. 8
  • At τ = 1.5 pixels, UAL exceeds Published by 1.75 dB and Isotropic by 0.47 dB in mean per-trajectory PSNR; its advantage grows as matched Gaussian-width uncertainty increases.Across tested changes of up to approximately 20% to parameter mean or covariance scale, UAL's Gaussian-deblurring PSNR remains 27.408–27.475 dB; motion deblurring is similarly insensitive to 20% covariance-scale changes.Source: Figure 3 · p. 9, §4.3 · p. 9
  • In motion deblurring, retaining 256 kernel-covariance modes preserves 96% of kernel variance and gives 26.641 dB mean per-trajectory PSNR, versus 26.251 dB with 2048 modes.Mean wall time is 89.5 seconds per trajectory at 256 modes versus 2171.3 seconds at 2048 modes.Source: Figure 4 · p. 9, §4.3 · p. 9

5 CONCLUSION AND LIMITATIONS

UAL improves reconstruction and ensemble prediction across the reported inverse problems while retaining the existing samplers’ priors and schedules. The appendix characterizes its Gaussian marginal approximation, likelihood score, and limitations under nonlinear nuisance dependence.

5 CONCLUSION AND LIMITATIONS

  • UAL improves reconstruction in CT, MRI, and fluorescence microscopy, contrast in ultrasound, and ensemble prediction across all 12 reported score comparisons.Its benefits depend on the nuisance model and, for nonlinear nuisance dependence, the local forward-model approximation; computation grows with covariance rank and depends on the sampler.Source: §5 · p. 9

A UAL LIKELIHOOD: DERIVATIONS AND APPROXIMATION THEORY

  • Under fixed Gaussian nuisance models, the paper bounds linearization likelihood error and measurement-averaged posterior divergence; affine-Gaussian nuisance dependence yields an exact marginal.For nonlinear dependence, the moments are exact for the linearized forward model, which drops the remainder term.Source: A UAL LIKELIHOOD: DERIVATIONS AND… · p. 14, §A.1 · p. 14

A.1 AFFINE GAUSSIAN MARGINAL AND LIKELIHOOD SCORE

  • For nonlinear nuisance dependence, dropping the forward-model remainder gives the linearized response, with omitted mean and covariance bounded at the nuisance mean.The conclusion notes that a different motion-kernel population was not tested.Source: §5 · p. 9, §A.1 · p. 14
  • When covariance depends on the signal, the likelihood score accounts for changes in both residual weighting and Gaussian normalization; fixing covariance leaves the mean-response term.The score is obtained by differentiating the normalized UAL energy while holding the nuisance model and reference fixed.Source: §A.1 · p. 14

A.2 FACTORIZED EVALUATION OF UAL

The factorized evaluation reduces UAL computations to a Cholesky factorization in nuisance-rank space. Its energy can be interpreted through a penalized nuisance profile, while scalar covariance inflation cannot reproduce directional weighting in general.

A.2 FACTORIZED EVALUATION OF UAL

  • A Cholesky factorization of the r × r matrix G provides both S^-1d and the determinant contribution, without requiring explicit storage of U in motion-deblurring Fourier coordinates.Explicit columnwise construction is an alternative that requires r applications of the nuisance Jacobian.Source: §A.2 · pp. 14–15
  • The factorized quadratic can be evaluated as nonnegative terms, avoiding cancellation between large quadratic terms.Substitution into the energy also includes the log-determinant contributions.Source: §A.2 · p. 15
  • For fixed x, the profiled nuisance solution is its conditional mean, and the UAL energy adds a log-determinant term that distinguishes marginalization from optimization.The penalized profile energy combines observation-noise and standard Gaussian prior penalties.Source: §A.2 · p. 15
  • Covariance truncation is a separate approximation, with a likelihood-error bound when the mean is unchanged and discarded covariance is positive semidefinite.Source: §A.2 · p. 15
  • The KL-optimal scalar inflation matches average whitened variance, but anisotropic covariance leaves positive divergence unless UU^T is proportional to Σe.Under affine-Gaussian assumptions, UAL is the exact marginalized likelihood in this comparison.Source: §A.2 · p. 16

A.3 LIKELIHOOD AND POSTERIOR APPROXIMATION ERROR

The appendix bounds likelihood and posterior error from nuisance linearization, then analyzes how covariance weighting affects reconstruction risk in a fixed-sensitivity model. It also characterizes moment errors and covariance truncation.

A.3 LIKELIHOOD AND POSTERIOR APPROXIMATION ERROR

  • Small expected squared linearization remainder in observation-noise units implies small likelihood divergence, and the prior-averaged likelihood bound also bounds posterior divergence averaged over measurements.For Gaussian nuisance parameters, the affine-mixture likelihood is UAL; for non-Gaussian parameters, the argument stops before Gaussian moment matching.Source: §A.3 · p. 16

A.4 MOMENT ERRORS FROM NUISANCE LINEARIZATION

  • Nuisance curvature can shift the omitted mean and alter covariance, with scalar Gaussian nuisance variance c yielding an O(c) mean error and an O(c^3/2 + c^2) covariance-error bound.The covariance change can increase or decrease variance along a measurement direction through cross terms with the linear response.Source: §A.4 · p. 17

A.5 LIKELIHOOD ERROR FROM COVARIANCE TRUNCATION

  • With the UAL mean unchanged and positive-semidefinite covariance discarded, likelihood divergence depends on the discarded covariance in coordinates whitened by the retained covariance.The bound applies when the retained covariance is no greater than the full UAL covariance.Source: §A.5 · p. 18

B.1 FIXED-SENSITIVITY MODEL AND ESTIMATOR DEFINITIONS

  • The fixed-sensitivity analysis assumes a centered signal independent of nuisance parameters and noise, with effective measurement covariance S = Σe + JΣψJ^T.It isolates reconstruction effects of changing the measurement-error covariance while holding the forward sensitivity and signal moments fixed.Source: §B.1 · p. 18

B.2 AVERAGE RISK AND THE OPTIMAL LINEAR ESTIMATOR

  • Under matched nuisance moments, the UAL linear filter uniquely minimizes average squared error among linear estimators, and is the posterior mean when signal, nuisance, and noise are Gaussian.The same signal covariance is used for both filters.Source: §B.2 · p. 19

B.3 AVERAGE GAIN AND ITS NORMALIZATION

  • The average gain recovers a nonnegative weighted fraction of the point estimator’s nuisance-error contribution, approaching all of it as λ grows; the absolute gain is H0κ0.For multiple parameters, the weights are associated with eigenvalues of Λ.Source: §B.3 · pp. 19–20

B.4 CONDITIONAL RECONSTRUCTION ERROR AND THE CROSSOVER

The analysis distinguishes average reconstruction gains from conditional gains at a fixed nuisance value, and bounds the cost of discarding covariance modes.

B.4 CONDITIONAL RECONSTRUCTION ERROR AND THE CROSSOVER

  • For one nuisance parameter and distinct estimators, the point-likelihood estimator has lower conditional error below |δ⋆|, while UAL has lower error above it; the crossover relative to nuisance standard deviation is (λ + 2)^−1/2.That relative crossover distance decreases as λ increases.Source: §B.4 · p. 21
  • For a fixed nuisance-error direction, increasing its magnitude favors UAL only when covariance weighting reduces nuisance error along that direction.If av ≤ 0, increasing the magnitude does not produce a positive gain.Source: §B.4 · p. 21
  • In the scalar fixed-sensitivity model, an assumed variance 0 < c ≤ 2E[(ψ⋆−µ)^2] is sufficient for positive average gain.Below the stated threshold, filter-change cost exceeds the reduction in nuisance-induced error; risks agree at the threshold.Source: §B.4 · pp. 21–22

B.5 RECONSTRUCTION ERROR FROM COVARIANCE TRUNCATION

  • When covariance is truncated, the reconstruction-risk bound is zero if the retained covariance equals the full covariance; it also applies to truncating nuisance covariance.The bound weights discarded modes by their variances and the reconstruction filter’s amplification.Source: §B.5 · p. 22

C TASK MODELS AND UAL IMPLEMENTATIONS

This section describes task-specific implementations of UAL across diffusion samplers and inverse problems, adapting their data guidance or weighting to account for nuisance-parameter uncertainty. These updates retain each sampler’s core sampling procedure, including its pretrained prior and schedule where applicable.

C TASK MODELS AND UAL IMPLEMENTATIONS

  • Each sampler uses UAL’s measurement mean and covariance in its existing data update while retaining its pretrained prior and sampling schedule.Implementations preserve published data-fitting scales and step sizes, with sampler-specific rescaling or covariance handling where needed.Source: C TASK MODELS AND UAL IMPLEMENTATIONS · p. 23

C.1 DEBLURRING FORWARD MODELS AND NUISANCE STATISTICS

  • Gaussian deblurring propagates two kernel-width uncertainties, whereas motion deblurring models kernel variation with retained covariance modes.The motion-blur model retains 256 modes, which capture 96% of kernel variance; Gaussian widths use specified means and covariance.Source: §C.1 · pp. 23–25

C.2 UAL IN DEBLURRING SAMPLERS

  • Two PnP-DM controls isolate covariance recomputation from differentiation and test fixed covariance, while scalar-noise controls vary data weighting on ablation sets.The fixed covariance is evaluated at the published sampler’s 12-reconstruction ensemble mean; Langevin step size is unchanged in scalar-weight comparisons.Source: §C.2 · p. 26

C.2.1 DAPS

  • DAPS replaces its inner measurement update with the UAL energy gradient while preserving its outer loop, inner step sizes, prior, and re-noising schedule.The data drift uses the published data-fitting scale, with two width-sensitivity columns for Gaussian blur and retained kernel-mode responses for motion blur.Source: §C.2.1 · p. 26

C.2.2 PNP-DM

  • PnP-DM substitutes the UAL negative log likelihood in its likelihood conditional and evaluates its gradient at each inner Langevin candidate.The gradient accounts for the predicted measurement mean, log determinant, and covariance-weighted residual penalty; the prior update remains unchanged.Source: §C.2.2 · pp. 26–27

C.2.3 DIFFPIR

C.2.4 BLINDDPS

  • BlindDPS replaces point-likelihood residual fidelity with UAL energy for image guidance and rescales guidance to the published norm before the denoiser Jacobian.Its kernel weight and guidance step size remain unchanged; with zero nuisance covariance, the UAL fidelity reduces to the residual norm at the UAL mean.Source: §C.2.4 · p. 29
  • For blind deblurring, BlindDPS retains its kernel diffusion, projection, and update while replacing image guidance with UAL; in motion blur, the effective kernel shrinks retained deviations toward the population mean.For Gaussian blur, the fidelity depends on the fitted kernel through reference widths.Source: §C.2.4 · p. 29

C.2.5 FAST DIFFUSION EM

  • Fast Diffusion EM replaces image guidance with the gradient of Gaussian UAL energy at the denoised candidate, using the mean-kernel response and covariance for clean-image and kernel uncertainty while leaving kernel maximization unchanged.The covariance includes the interaction between image and kernel uncertainty; the mapping to the diffusion state and step size remain unchanged.Source: §C.2.5 · p. 30

C.3 CT WITH FLAT AND DARK REFERENCE UNCERTAINTY

  • For CT, UAL models shared flat- and dark-field fluctuations as correlated measurement covariance and replaces DAPS’s likelihood-score drift with a covariance-weighted UAL score while preserving published step sizes and data-fitting scale.The application uses 60 projection views and mean reference fields, whereas the published version uses one flat and one dark frame.Source: §C.3 · pp. 31–32

C.4 MRI WITH COIL-MAP UNCERTAINTY

  • For MRI, 32 recalibration replicates estimate coil-map covariance; because the response is affine in map coefficients, its Gaussian marginal needs no nuisance linearization, and Ambient DPS substitutes the resulting energy and derivative into released guidance.The guidance descends the square-root fidelity, with the candidate-dependent mean and covariance included in the derivative.Source: §C.4 · p. 33

C.5 ULTRASOUND WITH SOUND-SPEED UNCERTAINTY

  • For ultrasound, DRUS retains the nominal mean response but replaces scalar noise with per-mode variances estimated from nonlinear sound-speed perturbations, using the same spectral variances in each channel.The sound-speed model is centered at 1540 m/s with a 26.9 m/s scale, and the diffusion prior and remaining sampling operations are unchanged.Source: §C.5 · p. 35
  • DRUS’s UAL spectral update partitions measured coordinates by singular-value thresholds, applying residual/noise mixing on one subset and measurement blending on the other (Zhang et al., 2024).Unmeasured coordinates retain the published update, with independent Gaussian draws in the published ordering.Source: §C.5 · p. 36

C.6 FLUORESCENCE MICROSCOPY WITH PUPIL-PHASE UNCERTAINTY

  • The fluorescence evaluation uses 50 cropped sections selected from four CARE Planaria stacks without reconstruction-score-based selection.A realized aberration and noisy measurement are fixed per section and shared across both sampler versions and all 12 trajectories.Source: §C.6 · pp. 36–38
  • For fluorescence microscopy, UAL propagates five Gaussian pupil-phase uncertainties through linearized PSF changes into a structured measurement covariance.The coefficient distribution is centered at the linearization point, so the first-order mean correction vanishes.Source: §C.6 · p. 37
  • Both fluorescence sampler versions use PnP-DM (Wu et al., 2024), the ED-Diff CARE checkpoint, and the same mean kernel, while only UAL changes the likelihood step.The diffusion-prior update remains unchanged; no retraining is performed. The examined checkpoint release does not specify a separate redistribution license.Source: §C.6 · p. 38

C.7 EVALUATION METRICS AND REPORTING

The section defines reconstruction and ensemble metrics, explains how scores and uncertainty are reported, and compares UAL likelihood approximations against numerical marginalization during sampling.

C.7 EVALUATION METRICS AND REPORTING

  • Reported 95% intervals use paired bootstrap resampling, with resampling units varying by task; microscopy and ultrasound whiskers use task-specific procedures.Ablation error bars are standard errors across the ablation set.Source: §C.7 · p. 38, Table 6 · p. 39
  • PSNR and SSIM are reported both as average trajectory scores and as scores of the ensemble-mean reconstruction, with MRI ensemble means formed before taking magnitude.Case-level scores are averaged over reported cases; PSNR uses the task-specific peak D.Source: §C.7 · p. 38
  • The section reports CRPS, energy score, and local variogram score using fair finite-ensemble estimators; lower values of all three ensemble scores are better.Energy score uses the full image-vector Euclidean norm, while local variogram scores group pixel pairs across 20 spatial offsets.Source: §C.7 · p. 39

D.1 NUMERICAL LIKELIHOOD REFERENCE

  • UAL has the lowest mean energy error at every saved sampling state and uncertainty level in Gaussian deblurring.At the first saved state, its mean relative image-score error ranges from 17.5% to 19.9% across uncertainty levels.Source: §D.1 · p. 40, Table 7 · p. 40

D.2 DEBLURRING COMPARISONS

UAL improves deblurring reconstructions across samplers, with especially large gains in motion deblurring when non-blind samplers use the population-mean kernel. In motion deblurring, DAPS and PnP-DM with UAL also outperform published blind samplers in ensemble-mean PSNR.

D.2 DEBLURRING COMPARISONS

  • For Gaussian deblurring, UAL increases per-trajectory PSNR for all four samplers, with gains from 0.39 dB for DAPS to 0.99 dB for DiffPIR.Mean CRPS also decreases for all four samplers.Source: §D.2 · p. 40
  • UAL raises mean per-trajectory PSNR in motion deblurring for all four samplers, with gains from 3.11 dB for DiffPIR to 11.16 dB for DAPS.DAPS rises from 17.68 to 28.84 dB; PnP-DM from 17.04 to 26.54 dB; DiffPIR from 11.44 to 14.55 dB; and Fast Diffusion EM from 22.76 to 29.10 dB.Source: §D.2 · p. 40
  • For DAPS and PnP-DM in motion deblurring, UAL closes 86.0% and 89.5% of the respective Published-to-Oracle per-trajectory PSNR gaps.Oracle uses the true measurement kernel; UAL retains 1.81 dB of DAPS’s gap and 1.11 dB of PnP-DM’s gap.Source: §D.2 · p. 41, Table 8 · p. 41
  • For motion-deblurring BlindDPS, UAL improves all three ensemble scores, although mean per-trajectory PSNR is 21.95 dB versus 22.12 dB for Published.Ensemble-mean PSNR rises from 23.21 to 23.33 dB.Source: §D.2 · p. 41
  • DAPS and PnP-DM with UAL exceed published blind samplers by 6.2 to 6.4 dB in motion-deblurring ensemble-mean PSNR.The comparisons use the same 50 observations; blind samplers sample or estimate an image-specific kernel, while UAL uses the population-mean kernel and propagated covariance.Source: §D.2 · p. 41, Table 9 · p. 41

D.3 APPLICATION DETAILS

Across CT, MRI, fluorescence microscopy, and ultrasound applications, UAL improves reported reconstruction or ensemble metrics. The supplied comparisons include acquired-data applications and paired reconstructions across the evaluated modalities.

D.3 APPLICATION DETAILS

  • In CT, UAL gains 0.204 dB in per-trajectory PSNR and improves SSIM and all three ensemble scores.With the same mean flat and dark reference frames, PSNR rises from 25.942 to 26.145 dB and CRPS falls from 0.021955 to 0.021352.Source: §D.3 · pp. 41, 44, Figure 7 · p. 44
  • In MRI, mean per-trajectory PSNR improves on 47 of 50 scans, with a mean paired gain of 0.0127 dB.All seven reported metrics favor UAL, with pointwise 95% paired-bootstrap intervals excluding zero.Source: §D.3 · p. 44, Table 10 · p. 44, Figure 8 · p. 45
  • In fluorescence microscopy, UAL increases per-trajectory PSNR by 0.425 dB and ensemble-mean PSNR from 33.67 to 34.22 dB.The ensemble mean's root-mean-square error falls from 0.085 to 0.052 for the Figure 10 section; its root-mean-square sample standard deviation changes from 0.063 to 0.059.Source: §D.3 · p. 44, Figure 9 · p. 46, Figure 10 · p. 47
  • On the acquired PICMUS contrast phantom, UAL increases CNR and gCNR; on the resolution phantom, both mean FWHM changes are below 0.01 mm.Figure 11 presents all 12 paired contrast-phantom reconstructions, with CNR and gCNR shown for displayed draws.Source: §D.3 · p. 46, Table 11 · p. 46, Figure 11 · p. 47

D.4 COVARIANCE-STRUCTURE CONTROL AND SECONDARY METRICS

Covariance-structure controls show that UAL's directional weighting and likelihood normalization matter beyond simply increasing observation-noise variance. Final-state diagnostics show that added covariance can concentrate variance in a few whitened directions.

D.4 COVARIANCE-STRUCTURE CONTROL AND SECONDARY METRICS

  • Structured UAL gains 0.455 ± 0.197 dB in Gaussian-deblurring per-trajectory PSNR over trace-matched isotropic inflation and improves PSNR on 13 of 16 images.It also reduces CRPS by 0.00260 ± 0.00056; the isotropic control changes little from the published sampler.Source: §D.4 · p. 48, Table 13 · p. 48
  • In motion deblurring, structured UAL adds 2.557 ± 0.317 dB in per-trajectory PSNR over isotropic inflation and improves PSNR on 15 of 16 images.The ensemble-mean PSNR margin is 4.431 ± 0.325 dB; isotropic inflation alone improves PSNR by 6.212 ± 0.398 dB over Published.Source: §D.4 · p. 48, Table 12 · p. 48
  • For motion-deblurring PnP-DM, including the log determinant improves all seven metric means over the quadratic-only control.With the log determinant, UAL's mean per-trajectory PSNR advantage is 0.0257 ± 0.0066 dB, with higher PSNR on 15 of 16 images.Source: §D.4 · p. 49, Table 14 · p. 49
  • For motion-deblurring Fast Diffusion EM, UAL improves every reported metric on all 50 images relative to covariance weighting alone.The table's adapted arms use the population-mean kernel and propagated covariance; covariance weighting holds covariance fixed during differentiation and omits the log determinant.Source: §D.4 · p. 49, Table 15 · p. 49

D.5 COVARIANCE SPECTRA AT FINAL RECONSTRUCTIONS

  • The Gaussian covariance concentrates added variance in two directions, so a small average variance increase can coexist with large directional variances.In motion deblurring, the added covariance trace differs across final candidate reconstructions; these diagnostics use measurement-noise whitening.Source: §D.5 · p. 49

D.6 COVARIANCE SCALE AND MATCHED UNCERTAINTY

In the tested fixed-observation settings, Gaussian-deblurring performance is little changed by nuisance-moment mismatch, and motion-deblurring PSNR is similar under reference and doubled covariance. When generating and supplied width uncertainty rise together, UAL's gains over Published grow.

D.6 COVARIANCE SCALE AND MATCHED UNCERTAINTY

  • In motion deblurring, removing nuisance covariance returns mean per-trajectory PSNR to the Published sampler's 17.872 dB; doubling covariance gives 26.644 dB versus 26.641 dB at the reference covariance.For Gaussian deblurring, mean per-trajectory PSNR remains 27.475 dB to three decimals across the tested nonzero covariance settings.Source: §D.6 · p. 50
  • Across approximately ±20% covariance error, mean per-trajectory PSNR changes by less than 0.001 dB in Gaussian deblurring and at most 0.0641 dB in motion deblurring.The largest absolute relative CRPS changes are 0.0011% and 0.0572%, respectively.Source: §D.6 · p. 50, Table 17 · p. 51
  • As matched generating and UAL width uncertainty rises, UAL's mean per-trajectory PSNR gain over Published grows from 0.043 ± 0.067 dB at τ = 0.25 to 1.746 ± 0.397 dB at τ = 1.5.At the largest tested uncertainty, UAL reduces average per-trajectory MSE by approximately 38% and outperforms trace-matched isotropic inflation on mean MSE, PSNR, and CRPS at every tested level.Source: §D.6 · p. 50, Table 18 · p. 51

D.7 FULL GAUSSIAN-WIDTH RESULTS

Across Gaussian-width settings, UAL improves reconstruction and ensemble prediction for all four samplers; rank ablations also report quality comparisons and per-trajectory runtime conditions.

D.7 FULL GAUSSIAN-WIDTH RESULTS

  • Across three Gaussian-width settings, UAL gives all four samplers higher mean per-trajectory PSNR and lower CRPS than Published.The comparisons use 50 images and 12 paired trajectories per image.Source: Table 19 · p. 51, §D.7 · p. 51, Table 20 · p. 52

D.8 COVARIANCE RANK AND COMPUTATIONAL COST

  • At covariance ranks P = 256 and P = 2048, UAL improves mean per-trajectory PSNR for three non-blind samplers and lowers ensemble scores for all four.The lowered scores are mean CRPS, energy, and local variogram scores.Source: §D.8 · p. 52, Table 21 · p. 52
  • Mean wall time per trajectory is reported on NVIDIA L40S GPUs with one reconstruction job per GPU.DAPS deblurring times exclude setup; DiffPIR includes initialization and output writing; DRUS excludes model loading, singular value decomposition, and covariance preparation.Source: §D.8 · p. 52, Table 22 · p. 53

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