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Consecutive Posterior Fusion for Diffusive Recovery of Unobservable Image Structures

Elena Morotti, Davide Evangelista, Elena Loli Piccolomini

arXiv:2610.03261v1cs.CVcs.AI

TL;DR

Ill-posed imaging measurements can leave image structures unobservable, while standard diffusion posterior samplers do not explicitly exploit information in previous corrections. CPF-DDNM fuses consecutive measurement-aware estimates within DDNM, preserving the measurement-determined component while modifying the prior-driven null-space completion. Experiments report improvements over DDNM, with performance varying by task and fusion schedule.

  • Problem

    Severely ill-posed imaging problems leave image components unobservable or weakly constrained, and standard diffusion posterior samplers do not explicitly exploit information carried by previous posterior corrections.

    Source: §1 · pp. 1–2

  • Method

    CPF-DDNM fuses consecutive measurement-aware estimates in DDNM, preserving the measurement-determined component while modifying only the prior-driven null-space completion.

    Source: §1 · p. 2

  • Results

    CPF-DDNM consistently improved on DDNM and outperformed considered diffusion-based inverse solvers in sparse-view CT; in super-resolution it improved structural similarity, though DiffPIR had better average performance for some metrics in the noisy setting.

    Source: §6 · p. 18

  • Takeaways & Limitations

    With an exact Moore–Penrose pseudo-inverse, CPF-DDNM preserves exact data consistency for noiseless data, and local error analysis gives conditions for small null-space recovery error with extrapolative fusion.

    Source: §6 · p. 18

    The fusion schedule is selected on validation data and depends on the forward operator.

    Source: §6 · p. 18

Abstract

from arXiv · show

Solving severely ill-posed imaging inverse problems requires recovering image structures that are unobservable or weakly constrained by the measurements. Diffusion models provide expressive learned priors for inferring such missing information, while posterior sampling incorporates measurement consistency along the reverse process. Standard diffusion posterior samplers, however, rely on instantaneous measurement-aware estimates, without explicitly exploiting information carried by previous posterior corrections. We introduce Consecutive Posterior Fusion Denoising Diffusion Null-Space Models (CPF-DDNM), an inference-time strategy that fuses consecutive measurement-aware estimates to improve the diffusive recovery of unobservable image structures, without requiring retraining or additional denoiser evaluations. We instantiate this principle within DDNM, whose range/null-space decomposition reveals that consecutive fusion preserves the measurement-determined component while acting exclusively on the prior-driven null-space estimate. We thus provide a geometric interpretation of CPF-DDNM and a local error analysis that characterizes the optimal time-dependent fusion coefficient, including the extrapolative regime. Experiments on sparse-view and simulated low-dose computed tomography, as well as medical image super-resolution, show consistent improvements over DDNM and competitive performance against diffusion-based inverse solvers.

1 Introduction

Ill-posed imaging measurements can leave image components unobservable, making reconstruction depend on priors to infer missing or weakly constrained information. CPF-DDNM fuses consecutive measurement-aware estimates within DDNM, preserving the measurement-determined component while modifying only the prior-driven completion; experiments report improvements over DDNM across evaluated tasks.

1 Introduction

  • Undersampling and noise can make imaging inverse problems severely ill-posed, leaving image components unobservable or weakly constrained by measurements.Reconstruction quality therefore depends critically on the prior used to infer missing or poorly constrained information.Source: §1 · pp. 1–2
  • Standard diffusion posterior samplers use measurement-aware corrections, but consecutive corrections can be volatile early in reverse diffusion and their problem-specific interaction remains underexplored.LAMP [8] combines consecutive estimates, but does not investigate their interaction with the inverse problem's structure.Source: §1 · p. 2
  • CPF-DDNM combines consecutive DDNM corrections that share the same measurement-determined component, so fusion preserves that component and modifies only the prior-driven recovery of unobservable image content.The fusion coefficient is time-dependent.Source: §1 · p. 2
  • The DDNM range/null-space decomposition gives fusion a geometric interpretation, including extrapolative coefficients that move beyond the segment between consecutive null-space estimates without changing the measurement-determined component.The paper derives a local optimality condition for when extrapolation can minimize expected completion error.Source: §1 · pp. 2–3
  • CPF-DDNM improves the original DDNM baseline across sparse-view CT, low-dose CT, and medical image super-resolution, with competitive or superior performance against other diffusion-based inverse solvers.The strategy requires neither retraining the diffusion prior nor additional denoiser evaluations.Source: §1 · p. 3

2 Related Works

Diffusion inverse solvers differ in how they incorporate measurement information, while multistep methods use temporal information from the reverse trajectory. CPF-DDNM is complementary to these approaches, focusing on fusion of consecutive estimates within DDNM and its extrapolative regime.

2 Related Works

  • Diffusion inverse solvers use different measurement-integration strategies: DDRM [16] uses spectral posterior updates, DPS [5] approximate likelihood gradients, DDNM [27] range/null-space consistency, and DiffPIR [35] plug-and-play splitting.All generate measurement-aware estimates along the reverse trajectory.Source: §2 · p. 3
  • Multistep discretizations use information across diffusion steps, but temporal relations between consecutive measurement-aware estimates have been comparatively underexplored.LAMP [8] provides initial evidence that consecutive estimates contain useful information discarded by standard posterior updates.Source: §2 · p. 3
  • LAMP [8] uses a backbone-independent convex combination with β_t ∈[0, 1], whereas CPF-DDNM studies extrapolative over-relaxation within the DDNM null-space.The methods use consecutive estimates in different regimes and for different purposes.Source: §2 · p. 3
  • Medical diffusion reconstruction spans CT, MRI, PET, and multimodal imaging; CPF-DDNM complements modality-specific methods by changing how DDNM uses consecutive estimates without retraining or changing its diffusion prior.Source: §2 · p. 3

3 Method

The method conditions reverse diffusion with measurement-aware corrections, then fuses consecutive DDNM corrections so that fusion changes only the prior-driven null-space completion. Its coefficient controls interpolation or extrapolation within that null space.

3.1 Diffusion models for image generation

  • DDIM reverses a noisy-image process to generate samples, with each clean-image estimate becoming more detailed and coherent as denoising proceeds.The reverse-step size determines the number of denoising iterations.Source: §3.1 · p. 4

3.2 Diffusion posterior samplers

  • Posterior samplers insert measurement-aware corrections into the reverse process; DDNM replaces the range-space estimate with K†y and retains the prior-estimated null-space component.For noiseless, range-consistent measurements and an exact pseudo-inverse, each corrected estimate is exactly data-consistent; applying the correction to noisy measurements may amplify noise.Source: §3.2 · pp. 4–5

3.3 The CPF-DDNM method

  • CPF-DDNM fuses DDNM corrections from consecutive reverse iterations using βt, preserving K†y while modifying only the prior-driven null-space completion.It requires storing the preceding correction, but adds no denoiser evaluations and does not require retraining the diffusion model.Source: §3.3 · p. 5
  • For any βt, the fused correction remains data-consistent when K† is the Moore-Penrose pseudo-inverse and y lies in Range(K).In CT experiments, finite CGLS iterations approximate the required pseudo-inverse operations, so these properties hold up to the resulting numerical error.Source: §3.3 · pp. 5–6

3.4 Insights on the fusion parameter

  • βt places the fused null-space estimate along the line through consecutive estimates: values above 1 extrapolate beyond the preceding estimate as an over-relaxation.The added displacement acts only within Ker(K), leaving the measurement-determined component unchanged.Source: §3.4 · p. 6
  • Under the local error model, the unique expected-error-minimizing coefficient depends on consecutive-estimate error magnitudes and correlation, and may exceed 1 when Ct > Bt.Because these error statistics are unavailable during inference, the method uses task-dependent time schedules that can vary fusion strength across the reverse trajectory.Source: §3.4 · p. 7

4 Experimental setup

The experiments use patient-separated Mayo Clinic chest CT data, a diffusion prior trained on those images, and simulated CT and ×4 super-resolution tasks. Comparisons evaluate reconstruction quality across established diffusion and model-based methods, with task-specific pseudo-inverse implementations and fusion schedules.

4 Experimental setup

  • Experiments used Python, PyTorch, MONAI, and ASTRA on an NVIDIA A4000 GPU, with random seed 42 unless otherwise specified.Source: §4 · p. 8

4.1 Dataset and preprocessing

  • The Mayo Clinic dataset contains 3,631 chest CT images from 11 patients; 3,305 images from 10 patients train the model, and 326 images from one held-out patient test it.Source: §4.1 · p. 8

4.2 Diffusion model and training

  • A two-dimensional MONAI DiffusionModelUNet was trained from scratch on 256 × 256 grayscale images using a 1,000-step DDPM noise-prediction objective.The training set comprised the downscaled Mayo Clinic images.Source: §4.2 · p. 8, §4.1 · p. 8

4.3 Medical applications and imaging tasks

  • CPF-DDNM was evaluated on sparse-view and low-dose CT reconstruction and single-image super-resolution, tasks with image components unobservable or weakly constrained by measurements.Source: §4.3 · p. 9

4.3.1 CT image reconstruction

  • Sparse-view CT used 60, 45, or 30 projection angles; low-dose CT used 180 angles with simulated post-log Gaussian noise at relative levels 0.1, 0.3, or 0.5.The low-dose noise model is described as a simplified approximation, not a complete photon-counting simulation; the operator is expected to have a non-trivial kernel.Source: §4.3.1 · p. 9

4.3.2 Medical image super-resolution

  • ×4 super-resolution uses block-averaged downsampling, which removes fine anatomical detail and leaves a non-trivial null-space to be inferred from prior information.Experiments include noiseless and mildly noisy observations.Source: §4.3.2 · pp. 9–10

4.4 Pseudo-inverse computation and algorithmic implementation

  • Super-resolution uses an exact blockwise-SVD pseudo-inverse, whereas CT uses 16-iteration CGLS because its operator is too large for explicit factorization.Finite-iteration CGLS makes data consistency and the range/null-space decomposition approximate, with errors from both the prior prediction and pseudo-inverse action.Source: §4.4 · p. 10

4.5 Comparison methods, parameter settings, and evaluation metrics

  • Comparisons include DDNM, DiffPIR, DPS, PS+, and, for CT, FBP, SIRT, and ℓ2-TV; quality is measured by SSIM, PSNR, and LPIPS across five initializations.CT uses an extrapolative fusion schedule with β_t in [1.2, 1.8], while super-resolution uses a schedule that decreases toward zero.Source: §4.5 · pp. 10–11

5 Results and discussion

Across sparse-view CT and low-dose CT, CPF-DDNM improves on DDNM, while ×4 super-resolution results vary by noise setting. The coefficient ablation favors a time-dependent extrapolative schedule over constant extrapolation.

5.1 Results on sparse-view CT reconstruction

  • At 30 views, CPF-DDNM raises DDNM SSIM from 0.7837 to 0.8511 and PSNR from 31.06 to 32.40 dB, while reducing LPIPS from 0.1322 to 0.0829.The table covers 30-, 45-, and 60-view geometries; these values are for the most challenging 30-view setting.Source: Table 1 · p. 12, §5.1 · p. 12
  • At 45 views, CPF-DDNM improves PSNR over DiffPIR by approximately 0.77 dB, lowers LPIPS from 0.0601 to 0.0356, and raises SSIM from 0.8877 to 0.9125.At 60 views, CPF-DDNM reaches 37.27 dB PSNR, approximately 1.66 dB higher than DiffPIR, and reduces LPIPS from 0.0433 to 0.0208.Source: §5.1 · p. 12
  • In qualitative sparse-view CT comparisons, CPF-DDNM gives slightly sharper boundaries and better fine-detail preservation, with differences clearer at 45 and 60 views.The comparisons include 30-, 45-, and 60-view reconstructions, with magnified details shown for each setting.Source: §5.1 · p. 12, Figure 3 · p. 13

5.2 Ablation Study on the Fusion Coefficient

  • Schedule C, which progressively reduces extrapolation from 1.8 to 1.2, achieves the best final image-quality metrics; constant βt = 1.5 can end in an unfavorable oscillation phase.The constant βt = 1.5 and Schedule C both remain extrapolative; their trajectories show late-stage oscillations, while constant βt ≤ 1 generally yields smoother trajectories closer to standard DDNM.Source: §5.2 · pp. 15–16

5.3 Low-dose CT reconstruction

  • In low-dose CT, CPF-DDNM achieves the highest SSIM at all tested noise levels and the lowest LPIPS at δ = 0.3 and 0.5, while DiffPIR has the best LPIPS at δ = 0.1.The reported noise levels are δ = 0.1, 0.3, and 0.5; qualitative results at δ = 0.3 show noise suppression with sharp, well-defined object boundaries.Source: §5.3 · pp. 16–17, Figure 6 · p. 17

5.4 Super-resolution of CT images

  • For ×4 super-resolution, CPF-DDNM has the highest SSIM in the noiseless setting, but DiffPIR leads all three metrics when noise δ = 0.01 is added.For noisy examples, CPF-DDNM shows sharper tissue boundaries and localized details than DiffPIR despite its lower average metric scores.Source: §5.4 · p. 18, Table 2 · p. 18

6 Conclusions

CPF-DDNM improves DDNM across imaging settings, with the largest gains in sparse-view CT, while preserving the measurement-determined component and modifying only prior-driven recovery. Its fusion schedule remains validation-selected and forward-operator-dependent.

6 Conclusions

  • CPF-DDNM combines consecutive measurement-aware estimates at inference time to improve prior-driven recovery of unobservable image components without modifying or retraining the diffusion prior.Source: §6 · p. 18
  • With an exact Moore–Penrose pseudo-inverse, fusion preserves exact data consistency for noiseless data by modifying only the prior-driven null-space component.Local error analysis identifies conditions under which an extrapolative optimal fusion coefficient yields a small null-space recovery error.Source: §6 · p. 18
  • CPF-DDNM consistently improves upon DDNM across sparse-view CT acquisition geometries and noise levels, and outperforms the considered diffusion-based inverse solvers.In super-resolution it improves structural similarity over DDNM, although DiffPIR achieves better average performance for some metrics in the noisy setting.Source: §6 · p. 18
  • The method adds no denoiser evaluations or pseudo-inverse operations compared with DDNM, but its fusion schedule is validation-selected and depends on the forward operator.The added computation is limited to storing the preceding correction and taking a linear combination.Source: §6 · p. 18
  • Figure 7 compares methods for ×4 super-resolution under measurement noise δ = 0.01 using two test images and enlarged regions of interest.Source: Figure 7 · p. 19
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