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ReBridge-Flow: Re-Coupling Posterior Bridges in Flow Matching for Image Restoration

Jiaqi Zhang, Yiqi Wang, Hongjie Wu, Bohan Guo, Xinan Wang, Zichen Luo, Taotao Cai, Zhi Chen, Mingkai Zheng

arXiv:2609.00811v1cs.CV

TL;DR

Existing image-restoration methods can impose measurement corrections that disrupt the source–clean endpoint coupling encoded by Flow Matching. ReBridge-Flow re-couples measurement-aware endpoints through clean-side anchoring and source-side updates, and experiments across natural and medical tasks show improved structural consistency and stable restoration performance.

  • Problem

    Existing Flow Matching restoration methods apply local measurement corrections without explicitly preserving source–clean endpoint coupling, which can create bridge mismatch.

  • Method

    ReBridge-Flow decodes local endpoints, anchors the clean endpoint, synchronously re-couples the source endpoint, and uses the pair to define posterior-informed transport.

  • Results

    Experiments across diverse natural and medical restoration tasks demonstrate reduced bridge mismatch and stable performance with clearer, more structurally consistent restored images.

  • Takeaways & Limitations

    Posterior bridge re-coupling coordinates measurement constraints with endpoint priors while preserving local bridge compatibility during sampling.

  • Takeaways & Limitations

    The method primarily targets known linear degradations, assumes reasonably estimated measurement noise, and may incur additional cost for general large-scale operators.

Abstract

from arXiv · show

Flow Matching provides an efficient generative prior for image restoration by learning continuous transport between source and data distributions. However, existing methods typically incorporate measurement constraints through local corrections. Such corrections may disrupt the source-clean endpoint coupling implicitly encoded by the pretrained flow, making the corrected endpoint pair incompatible with the current state. To address this issue, we propose ReBridge-Flow, a posterior bridge re-coupling method. Specifically, given the current state, ReBridge-Flow first decodes the corresponding local source and clean endpoints. It then incorporates measurement information through clean-side anchoring and synchronously re-couples the source endpoint, yielding a measurement-aware endpoint pair with improved local bridge compatibility. The re-coupled endpoints further define a posterior-informed transport direction for advancing the sampling process. We also introduce the Posterior Bridge Defect, which jointly characterizes measurement error, deviation from the flow prior, and bridge mismatch, and leads to explicit updates for clean-side anchoring and source-side re-coupling. Extensive experiments on multiple natural and medical image restoration tasks demonstrate that ReBridge-Flow effectively alleviates bridge mismatch and improves the structural consistency of restored images.

1 Introduction

Flow Matching enables efficient image restoration, but local measurement corrections can disrupt the source–clean endpoint coupling that determines intermediate states. ReBridge-Flow addresses this mismatch by re-coupling both endpoints with measurement-aware updates and reports clearer, more structurally consistent restorations.

  • Flow Matching connects source and data distributions through continuous transport, enabling efficient sampling for image restoration.
  • ReBridge-Flow decodes local endpoints, anchors the clean endpoint with measurements, and synchronously re-couples the source endpoint.The resulting measurement-aware pair improves local bridge compatibility and defines a posterior-informed transport direction.
  • The Posterior Bridge Defect jointly accounts for measurement error, flow-prior preservation, and bridge residual.These terms provide a unified characterization of the re-coupling process and support explicit endpoint updates.
  • Local measurement interventions can disrupt source–clean endpoint coupling and affect subsequent local transport.Intermediate states are jointly determined by endpoint pairs, so correcting one variable can make the corrected pair incompatible with the current state.
  • Experiments across diverse natural and medical image restoration tasks show reduced structural drift and artifacts with clearer image details.

2 Related Work

Generative priors, especially diffusion models and Flow Matching, support image restoration by modeling complex image distributions. Related methods inject measurement constraints during sampling, but Flow Matching approaches use varied local interventions without explicitly addressing endpoint coupling.

  • Generative priors have become an important paradigm for image restoration inverse problems because they model complex image distributions.
  • Diffusion-based restoration methods introduce measurement constraints through gradient guidance or related reverse-sampling procedures.
  • Flow Matching-based restoration methods modify pretrained-flow sampling through measurement gradients, trajectory optimization, data-consistency updates, or guidance.

3 ReBridge-Flow

ReBridge-Flow treats measurement-conditioned restoration as posterior re-coupling of source and clean endpoints rather than independent local correction. It jointly anchors the clean endpoint, re-couples the source endpoint, and uses the resulting pair to define a posterior-informed transport direction.

  • 3 ReBridge-Flow: ReBridge-Flow decodes local endpoints from the current state, applies clean-side anchoring, synchronously re-couples the source endpoint, and advances sampling with the resulting transport direction.Measurement information is encoded through the new endpoint pairing rather than added as an independent external guidance term.
  • Motivation: Why Local Correction Breaks the Bridge: Local measurement correction can reduce observation error while breaking compatibility between the corrected endpoint pair and the current state.Clean-side-only correction leaves the state unchanged but causes the corrected pair to no longer interpolate to it.
  • 3.1 Posterior Bridge Defect: The observation posterior changes the pairing between source and clean endpoints, so correcting only the clean side is generally insufficient.This motivates explicitly modeling a measurement-conditioned endpoint coupling.
  • 3.1 Posterior Bridge Defect: The Posterior Bridge Defect jointly measures measurement error, deviation from local flow predictions, and mismatch between the corrected pair and the current state.Its weights control clean-side prior preservation, source-side prior preservation, and bridge re-coupling strength.
  • 3.2 Closed-Form Posterior Bridge Re-Coupling: For linear degradations, clean-side anchoring and source-side re-coupling are closed-form components of one joint quadratic optimization problem.The endpoint updates jointly produce the unique global minimizer under the stated positive-weight assumptions.
  • 3.4 Theoretical Analysis: When κ > 0 and t ∈ (0, 1), re-coupling strictly contracts the bridge residual, with stronger relative re-coupling strength producing stronger local repair.The analysis also compares cumulative mismatch across early, middle, late, and full-trajectory activation schedules.

4 Experiment

Experiments across six natural and medical datasets evaluate ReBridge-Flow under diverse restoration tasks, endpoint-handling variants, parameter settings, and computational constraints. The method provides competitive restoration quality, preserves structural details, and benefits from jointly maintaining priors at both endpoints.

  • Experimental setup: ReBridge-Flow is evaluated on six natural and medical datasets spanning denoising, deblurring, super-resolution, and inpainting tasks.The natural datasets are CelebA, AFHQ-Cat, and COCO; the medical datasets are IXI-Brain, PMUB, and X-Ray Hand.
  • Comparison with state-of-the-art methods: Across tasks and domains, ReBridge-Flow shows competitive performance, generally improves PSNR and SSIM over several flow-based baselines, and more reliably preserves medical-image structures.Qualitative comparisons also indicate sharper textures and edges, with fewer residual artifacts, oversmoothing effects, and local structural shifts.
  • Ablation study: The full endpoint strategy performs best on CelebA random inpainting and AFHQ-Cat 4× super-resolution, whereas removing the clean-side prior causes a substantial performance drop.Clean-side anchoring improves measurement consistency, but fixing the source endpoint breaks local bridge compatibility; hard re-coupling reduces residuals but is less stable.
  • Parameter sensitivity: At ρ = λ = 1, ReBridge-Flow reaches PSNR 28.16 and LPIPS 0.119 while remaining stable across a moderate parameter range.Large λ restricts source-endpoint adaptation, small λ weakens the source-side prior, and extreme ρ values disrupt the balance between priors and measurement correction.
  • Computational efficiency: ReBridge-Flow achieves the best results across all three quantitative metrics on CelebA deblurring while using 6.75s average inference time and 0.79 GB GPU memory.Restora-Flow is faster and OT-ODE uses slightly less memory, but both provide substantially lower restoration performance.

5 Discussion

The discussion distinguishes endpoint re-coupling from stronger measurement guidance and identifies the method’s supported operating conditions and limitations. ReBridge-Flow allocates observation corrections across both endpoints while retaining local priors, but its guarantees and scope remain limited.

  • Endpoint re-coupling: ReBridge-Flow synchronously updates source and clean endpoints, unlike clean-side-only correction, and propagates with the direction ¯bt − ¯at.This allocation preserves local flow priors on both sides rather than simply increasing the measurement residual correction.
  • Bridge residual: The bridge residual measures whether the corrected endpoint pair can explain the current state, but reducing it does not guarantee monotonic PSNR or SSIM improvement at every step.It can nevertheless help prevent later transport directions from repeatedly using an inconsistent endpoint pair.
  • Limitations: The current method is primarily limited to known linear degradations, requires reasonably estimated measurement-noise levels, and may incur extra cost for large-scale linear systems.Local endpoint quality also depends on how well the pretrained velocity field matches the target domain.
  • Limitations: Theoretical guarantees cover local PBD optimality and bridge-residual contraction, not monotonic global reconstruction-error decrease.Medical experiments use known synthetic degradations and therefore do not replace evaluation under real clinical degradations.
  • Future work: Future work targets nonlinear and unknown degradations and adaptive adjustment of ρ, λ, and κ across sampling conditions.The proposed extensions aim to reduce manual parameter selection across domains and degradation levels.

6 Conclusion

ReBridge-Flow reformulates Flow Matching-based image restoration as measurement-conditioned posterior bridge re-coupling. It uses clean-side anchoring and source-side re-coupling to construct compatible endpoint pairs and a posterior-informed transport direction.

  • ReBridge-Flow reformulates Flow Matching-based image restoration as a measurement-conditioned posterior bridge re-coupling problem.
  • The re-coupled endpoints define a posterior-informed transport direction for subsequent sampling, and experiments report stable performance across diverse restoration tasks.
  • The method decodes local source and clean endpoints from the current state using the pretrained velocity field.
  • Clean-side anchoring incorporates measurement information, while source-side re-coupling synchronously adjusts the source endpoint to improve bridge compatibility.
  • The Posterior Bridge Defect jointly accounts for measurement error, deviation from the flow prior, and endpoint–state bridge mismatch, yielding closed-form updates.

D Complete Derivation of Closed-Form Posterior Bridge Re-Coupling

The derivation minimizes a joint Posterior Bridge Defect over clean and source endpoints. Eliminating the source endpoint yields closed-form clean-side anchoring, followed by analytically coupled source recovery.

  • D.1 Restatement of the PBD Objective: The Posterior Bridge Defect combines measurement consistency, clean- and source-side proximity to local pseudo-endpoints, and current-state bridge compatibility.
  • D.2 First-Order Optimality Condition for a: For fixed clean endpoint b, the unique source minimizer analytically balances the source-side flow prior with current bridge compatibility.
  • D.3 Elimination of the Source Endpoint: Eliminating the source endpoint produces a reduced objective in b whose effective clean-side weight includes a contribution from re-coupling and bridge residual.
  • D.4 Normal Equation for the Clean Endpoint: Because H⊤H + γtI is positive definite, the clean endpoint has a unique solution, and the final update can be computed by solving a regularized measurement-space linear system.
  • D.5 Push-Through Identity: The push-through identity converts the clean-side solution into the measurement-space expression used by the main method.
  • D.6 Recovery of the Source Endpoint: After clean-side anchoring, the source endpoint moves in the opposite direction, with λ preserving proximity to the prior and κ increasing compensation for bridge mismatch.
  • D.7 Summary of the Joint Minimization: Clean-side anchoring and source-side re-coupling jointly form the global minimizer of the original PBD objective rather than independent correction steps.

E Proof of Proposition 1: Unique Minimizer of the PBD Objective

The PBD objective is shown to have a unique global minimizer. Strong convexity and exact quadratic expansion establish both uniqueness and a quadratic objective-gap guarantee.

  • The proof first establishes strong convexity of the PBD objective in the joint endpoint variable (a, b).
  • The Hessian quadratic form is positive because the measurement, prior, and bridge-residual contributions are nonnegative.
  • The resulting global minimizer is unique because the objective is strictly convex and has at most one stationary point.
  • Conditional source minimization followed by reduced clean-side minimization yields the endpoint pair specified by the closed-form updates.
  • The objective value away from the minimizer increases by at least a quadratic amount in the joint endpoint distance.

F Proof of Proposition 2: Exact Contraction of the Bridge Residual

The proof shows that source-side re-coupling contracts the local bridge residual introduced by clean-side-only correction. This result is local to the current sampling time and does not imply contraction of global reconstruction error.

  • The bridge residual is defined before and after source-side re-coupling to compare endpoint–state consistency.
  • Using the local pseudo-endpoint identity and the source update, the two residuals satisfy an exact vector relation.
  • When κ = 0 or t = 1, re-coupling provides no contraction; when t = 0, both residuals are zero.
  • For κ > 0, t ∈(0, 1), and changed clean endpoints, the contraction factor is strictly below one.
  • The exact multiplicative contraction applies to the local endpoint-state consistency residual, not to global reconstruction error.

G.3 Restoration Task Settings

The evaluation covers diverse natural and medical restoration settings and compares ReBridge-Flow with Flow Matching baselines using consistent task protocols. Sampling-step and bridge-strength studies show that performance depends on both inference length and the recoupling coefficient.

  • Restoration Task Settings: Natural-image evaluation includes denoising, deblurring, 2× or 4× super-resolution, random inpainting, and box inpainting across CelebA, COCO, and AFHQ-Cat.The protocols specify task-dependent noise levels and mask sizes.
  • Restoration Task Settings: Medical-image evaluation covers denoising, 2× super-resolution, random inpainting, and box inpainting on IXI-Brain, PMUB, and X-Ray Hand.Medical settings use task-specific noise levels and centered masks.
  • Compared Methods: Six Flow Matching baselines span velocity correction, trajectory optimization, source optimization, intermediate-state correction, and clean-endpoint updating.The compared methods include OT-ODE, Flow-Priors, D-Flow, PnP-Flow, Restora-Flow, and Flower.
  • Sampling-Step Study: K = 100 achieves the highest PSNR of 28.16 dB and SSIM of 0.820, while K = 200 yields the lowest LPIPS of 0.116 on AFHQ-Cat 4× super-resolution.All three metrics improve consistently as sampling steps increase from K = 10 to K = 100.
  • Bridge-Strength Study: At κ = 5, CelebA 2× super-resolution reaches 34.51 dB PSNR, 0.962 SSIM, and 0.014 LPIPS, whereas stronger constraints reduce PSNR.Relative to κ = 0, this setting improves PSNR by 0.79 dB and SSIM by 0.012 while reducing LPIPS by 0.007.

H.3 Trajectory-Level Analysis of PBD Components

The trajectory analysis tracks Posterior Bridge Defect components across four CelebA tasks and separates clean-side anchoring from source-side re-coupling. ReBridge-Flow concentrates source-side adjustment in the middle of sampling to balance consistency, prior preservation, and bridge compatibility.

  • PBD Components: The analysis tracks Measurement Defect, clean-side Flow-Prior Deviation, source-side Flow-Prior Deviation, and Bridge Residual across early, middle, and late sampling stages.Values are normalized by image dimensionality and averaged with one standard deviation across samples.
  • Comparative Analysis: Same-¯bt, Frozen Source and ReBridge-Flow have nearly identical Measurement Defect and clean-side Flow-Prior Deviation, isolating source-endpoint updating as their main difference.Frozen Source applies the same clean-side anchoring but keeps the source endpoint fixed.
  • Temporal Behavior: ReBridge-Flow’s source-side Flow-Prior Deviation concentrates in the middle stage and approaches zero at both ends, consistent with the temporal factor t(1−t).The pattern represents controlled source-side mismatch during middle-stage transport.
  • Interpretation: ReBridge-Flow balances observation consistency, flow-prior preservation, and local bridge compatibility rather than minimizing Measurement Defect alone.The balance is achieved through controlled source-side adjustment.

H.4 Challenging Tasks

High-factor CelebA super-resolution becomes harder as observed information decreases, but ReBridge-Flow maintains the strongest overall reconstruction quality at both evaluated scales. Its qualitative advantage is clearest in identity, facial structure, and local details.

  • Experimental Setting: All methods degrade from 4× to 8× super-resolution because the observed information is further reduced.The challenging-task experiment uses ten fixed CelebA test images with measurement noise σy = 0.01.
  • Quantitative Results: ReBridge-Flow achieves the best overall reconstruction quality at both 4× and 8× super-resolution.It also uses less GPU memory and infers faster than Restora-Flow and Flower.
  • Qualitative Results: At large upscaling factors, ReBridge-Flow more consistently preserves identity, facial structure, hair, and glasses than the compared methods.PnP-Flow shows over-smoothing, Flower blurs local details, and OT-ODE and Restora-Flow deviate in contours and texture recovery.
  • Interpretation: The results indicate that source–clean endpoint re-coupling helps maintain local transport consistency under severely limited observations.The reported consequence is reduced structural shifts and detail loss.

I More Quantitative and Qualitative Results

Additional results extend the evaluation to medical datasets and restoration tasks, with quantitative comparisons organized by dataset, task, and PSNR, SSIM, and LPIPS. The reported tables cover IXI-Brain, PMUB, and X-Ray Hand under matched comparison protocols.

  • Medical Evaluation: Medical-image results are reported for IXI-Brain, PMUB, and X-Ray Hand across denoising, 2× super-resolution, random inpainting, and box inpainting.Table 15 presents quantitative comparisons, while additional qualitative results are provided in Figures 10–36.
  • Metrics and Layout: The medical comparison reports PSNR, SSIM, and LPIPS for each restoration setting, with best and suboptimal results highlighted.The table covers four task columns for each medical dataset.
  • Compared Methods: The reported comparison includes Flow-Priors, D-Flow, PnP-Flow, OT-ODE, and other restoration baselines alongside ReBridge-Flow.The displayed rows include representative baseline results for medical restoration tasks.
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