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

• Because degradation operators are typically non-invertible, image restoration requires both data consistency and an effective image prior.Source: §1 · p. 1
• Local corrections can disrupt source–clean endpoint coupling, leaving endpoint pairs unable to explain the current state and causing accumulated directional errors.These errors may manifest as structural drift, artifacts, or over-smoothing.Source: §1 · pp. 2–3
• ReBridge-Flow decodes local endpoints, anchors the clean endpoint with measurement information, and synchronously re-couples the source endpoint to define a posterior-informed transport direction.The resulting measurement-aware endpoint pair is intended to improve local bridge compatibility.Source: §1 · p. 3
• The Posterior Bridge Defect jointly characterizes measurement error, flow-prior preservation, and bridge residual to guide endpoint anchoring and re-coupling.Source: §1 · p. 3
• ReBridge-Flow suppresses error propagation, reduces structural drift and artifacts, and preserves clearer details across diverse natural and medical restoration tasks.Source: §1 · p. 3

Diffusion-Based Image Restoration (DBIR)

• Diffusion-based restoration methods either guide reverse sampling with measurement gradients or enforce consistency through projections and structured constraints.Examples include DPS Chung et al. (2023), ΠGDM Song et al. (2023), DDRM Kawar et al. (2022), and DDNM Wang et al. (2023).Source: §2 · p. 3

Flow Matching-Based Image Restoration (FMBIR)

• Flow Matching restoration methods modify sampling through measurement gradients, local trajectory optimization, alternating consistency updates, or mask guidance and trajectory correction.Examples include OT-ODE Pokle et al. (2024), Flow-Priors Zhang et al. (2024), PnP-Flow Martin et al. (2025), and Restora-Flow Hadzic et al. (2026).Source: §2 · p. 4
• Endpoint-based methods optimize the initial source point or estimate a flow-consistent clean endpoint before measurement-informed refinement.D-Flow Ben-Hamu et al. (2024) backpropagates through the complete Flow ODE; Flower Pourya et al. (2026) refines an estimated clean endpoint using measurement information.Source: §2 · p. 4

3 ReBridge-Flow

• Clean-side-only correction can break compatibility between the endpoint pair and current state, motivating ReBridge-Flow’s clean anchoring and synchronous source re-coupling.The induced residual grows with the correction magnitude and time t; repeatedly using incompatible pairs may accumulate trajectory error, structural drift, artifacts, or over-smoothing.Source: §3 · pp. 4–5

3.1 Posterior Bridge Defect

• When observations depend on the clean endpoint, conditioning changes the posterior pairing of source and clean endpoints, so clean-side correction alone is insufficient.The measurement restricts feasible clean endpoints and changes how they pair with source endpoints.Source: §3.1 · p. 5
• The Posterior Bridge Defect jointly balances measurement error, deviation from local flow predictions, and bridge residual, with ρ, λ, and κ controlling clean-side prior, source-side prior, and re-coupling strength.The measurement term assesses how well the clean endpoint explains the observation; the prior terms limit deviation from pretrained local predictions, while the bridge term measures endpoint-pair mismatch with the current state.Source: §3.1 · pp. 5–6

3.2 Closed-Form Posterior Bridge Re-Coupling

• For linear degradation, clean-side anchoring is a noise-aware proximal correction, and the closed-form source update balances its flow prior against bridge-residual reduction as the joint objective’s optimum.A smaller γt enforces a stronger observation constraint, while a larger γt preserves more flow-predicted clean structure; source re-coupling is not an independent heuristic adjustment.Source: §3.2 · p. 6

3.3 ReBridge-Flow Sampling Algorithm

• At each sampling step, ReBridge-Flow decodes a local endpoint pair, performs clean-side anchoring and source-side re-coupling, then advances the state along their posterior-informed direction.Measurement information enters through the re-coupled pairing and its induced local velocity, rather than as an independent guidance term added to the pretrained velocity.Source: §3.2 · pp. 6–7, §3.3 · p. 7

3.4 Theoretical Analysis

• Under the stated linear-operator and positive-weight assumptions, the clean- and source-side updates jointly yield the unique global minimizer of the PBD objective.The endpoint-prior terms make the objective strongly convex, so the two updates are components of one joint optimization rather than independent corrections.Source: §3.4 · p. 7
• When κ > 0, t is between 0 and 1, and clean-side correction is nonzero, source re-coupling strictly contracts the bridge residual; full-trajectory re-coupling suppresses cumulative mismatch, while middle-stage partial re-coupling reduces it most among partial schedules.Increasing κ/λ strengthens local bridge repair by decreasing the residual contraction factor.Source: Figure 4 · p. 7, §3.4 · p. 8

4 Experiment

• Evaluation uses 100 shared test images from each of six natural and medical datasets, measuring PSNR, SSIM, and LPIPS.The datasets are CelebA, AFHQ-Cat, COCO, IXI-Brain, PMUB, and X-Ray Hand.Source: §4 · p. 8
• Natural-image evaluation covers five restoration tasks, while medical-image evaluation covers four, with task-specific noise levels, sampling scales, and inpainting masks.Natural tasks are denoising, deblurring, super-resolution, random inpainting, and box inpainting; medical tasks omit deblurring. Denoising uses σy = 0.2 for natural images and σy = 0.08 for medical images; random masks remove 70% and 30% of pixels, respectively.Source: §4 · p. 9

4.1 Comparison with State-of-the-Art Methods

• Across six datasets, ReBridge-Flow generally achieves higher PSNR and SSIM than PnP-Flow, Restora-Flow, and Flower, with more reliable structural preservation on medical images.Qualitative comparisons also report sharper textures, edges, contours, and tissue boundaries, with fewer artifacts and local structural shifts than baselines.Source: §4.1 · p. 9, Table 1 · p. 10, Table 2 · p. 10

4.2 Ablation Study

• The full ReBridge-Flow performs best on both ablation tasks, while clean-side anchoring alone leaves bridge mismatch and removing either endpoint prior harms restoration.Hard Re-Coupling reduces bridge residual but is less stable without the source-side prior; omitting the clean-side prior substantially reduces performance.Source: §4.2 · pp. 10–12, Table 3 · p. 11

4.3 Hyperparameter Sensitivity Analysis

• With κ = 5, the best reported performance occurs at ρ = λ = 1, reaching PSNR 28.16 and LPIPS 0.119.Large λ restricts source-endpoint adaptation, small λ weakens the source prior, and extreme ρ values disrupt measurement-consistency and clean-prior balance.Source: §4.3 · p. 12, Figure 6 · p. 12

4.4 Computational Efficiency

• On CelebA deblurring, ReBridge-Flow leads all three metrics with 6.75s average inference time and 0.79 GB GPU memory, without iterative optimization.Restora-Flow is faster and OT-ODE uses slightly less memory, but both have substantially lower restoration performance.Source: §4.4 · p. 12

5 Discussion

• OT-ODE Pokle et al. (2024) adds measurement gradients to velocity, Flower Pourya et al. (2026) corrects only the clean endpoint, while ReBridge-Flow synchronously re-couples both endpoints.Given the same corrected clean endpoint, ReBridge-Flow allocates the observation correction across both endpoints rather than merely amplifying measurement guidance; stronger guidance alone does not resolve endpoint-pair inconsistency with the current state.Source: §5 · pp. 12–13
• The Measurement Defect, Flow-Prior Deviation, and Bridge Residual respectively constrain measurement consistency, prior preservation, and local propagation; reducing the residual alone guarantees neither lower reconstruction error nor monotonic PSNR or SSIM.Re-coupling typically reduces the residual more at intermediate sampling times, when both endpoint interpolation weights are non-negligible; the residual is not a replacement for final data consistency.Source: §5 · p. 13

Limitations

• ReBridge-Flow assumes known linear degradations, estimable noise, and a reasonably matched pretrained flow; large operators can add solve costs, while domain shift or information loss can preserve prior bias.The theory establishes local PBD optimality and bridge-residual contraction, not monotonic global reconstruction-error reduction; medical experiments use known synthetic degradations and do not establish performance under real clinical degradations.Source: §5 · pp. 13–14

Future Work

• Future work will address nonlinear or unknown degradations through local linearization or joint operator estimation, and adapt ρ, λ, and κ to sampling stage and residuals.Adaptive parameter adjustment is proposed to reduce manual selection across data domains and degradation levels.Source: §5 · p. 14

6 Conclusion

• ReBridge-Flow reformulates FMBIR as posterior bridge re-coupling, using clean-side anchoring and source-side re-coupling to form measurement-consistent, locally bridge-compatible endpoints.The Posterior Bridge Defect yields closed-form updates; experiments show stable performance across diverse restoration tasks, and future work will extend the method to nonlinear and unknown degradations.Source: §6 · p. 14

A More Related Work

• Prior methods inject measurements through score, state, or one-sided endpoint corrections, but leave joint source–clean endpoint compatibility implicit; ReBridge-Flow explicitly constrains both endpoints.The described approaches include diffusion posterior sampling, optimization or proximal updates, and Flow Matching velocity, trajectory, or endpoint interventions.Source: A More Related Work · p. 18

C.1 Linear Flow Matching Bridge

• For a linear Flow Matching bridge, endpoint interpolation yields a constant conditional velocity equal to the clean-minus-source endpoint difference, enabling local endpoint decoding from the pretrained velocity.The bridge has source and clean endpoints at its time-zero and time-one conditions.Source: §C.1 · p. 20

B Symbols Table; C.2 Local Pseudo-Endpoint Identities

• At step k, x_k, v_k, and decoded endpoints â_k and b̂_k denote the sampling state, pretrained velocity, and local pseudo-endpoints, which exactly reconstruct the state and reproduce its velocity.The endpoints are decoded from the current state and velocity as â_t = x_t − t vθ(t, x_t) and b̂_t = x_t + (1 − t)vθ(t, x_t).Source: §C.2 · pp. 20–21, B Symbols Table · p. 20

C.3 Clean-Side-Only Bridge Residual

• Clean-side-only correction leaves the current state unchanged but creates a nonzero bridge residual when the clean endpoint changes at t>0; for equal correction magnitudes, the residual grows linearly with t.The residual reflects inconsistency in the corrected endpoint representation, not a change to the current state.Source: §C.3 · p. 21

C.4 Measurement-Conditioned Endpoint Coupling

• Posterior reweighting changes the joint source–clean endpoint pairing even when the linear-observation likelihood depends only on the clean endpoint; the PBD accounts for observation consistency, flow-prior deviation, and bridge compatibility.ReBridge-Flow uses the joint PBD objective to construct measurement-aware endpoints around the current state.Source: §C.4 · p. 22

D.1 Restatement of the PBD Objective

• The joint PBD objective combines measurement consistency, endpoint-prior constraints, and compatibility of the endpoint pair with the current state through the linear bridge.Its terms measure clean-endpoint consistency with the observation, deviation from local pseudo-endpoints, and the bridge residual.Source: §D.1 · p. 22

D.2 First-Order Optimality Condition for a

• For any candidate clean endpoint, the source endpoint has a unique analytic minimizer balancing the source-side flow prior against current bridge compatibility.Setting the clean endpoint to its anchored value recovers the source-side re-coupling update used in the main paper.Source: §D.2 · p. 23

D.3 Elimination of the Source Endpoint

• Eliminating the source endpoint yields a clean-only objective whose effective clean-side prior weight includes an additional contribution induced jointly by source re-coupling and bridge residual.This reduced objective is then minimized with respect to the clean endpoint.Source: §D.3 · p. 24

D.4 Normal Equation for the Clean Endpoint

• Because γt > 0, H⊤H + γtI is positive definite, so the clean-endpoint normal-equation matrix is invertible.The derivation uses this invertibility to obtain the clean-endpoint solution.Source: §D.4 · p. 24

D.5 Push-Through Identity

• The clean-side anchoring update is identical to the main-paper expression and requires solving only a regularized linear system in measurement space.The derivation obtains it using a push-through identity.Source: §D.5 · p. 25

D.6 Recovery of the Source Endpoint

• When clean-side anchoring shifts the clean endpoint, the source endpoint adjusts synchronously in the opposite direction, with the magnitude governed by t, λ, and κ.Larger λ keeps the source closer to its local pseudo-endpoint, while larger κ increases compensation for bridge mismatch.Source: §D.6 · p. 25

D.7 Summary of the Joint Minimization

• The anchored clean endpoint minimizes the objective after source-endpoint elimination, and its conditionally optimal source endpoint makes the pair the global minimizer of the joint PBD objective.The derivation first minimizes over the source endpoint, then over the reduced clean-side objective.Source: §D.7 · p. 26

• The nonnegative measurement and bridge-residual terms yield μ-strong convexity in (a, b), making Jt a coercive, strictly convex quadratic with at most one stationary point.Source: E Proof of Proposition 1: Unique… · p. 26
• For fixed b, Eq. (32) gives the unique conditional minimizer a⋆(b); minimizing the reduced objective gives b̄t, and Eq. (49) sets āt = a⋆(b̄t).Source: E Proof of Proposition 1: Unique… · p. 26
• Eqs. (7) and (8) jointly yield the unique global minimizer of the PBD objective, and every other endpoint pair incurs at least a quadratic objective increase in joint endpoint distance.Source: E Proof of Proposition 1: Unique… · pp. 26–27

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

• No contraction occurs when κ = 0 or t = 1, while at t = 0 both residuals are zero.At κ = 0 the source endpoint is unchanged; at t = 1 the bridge depends entirely on the clean endpoint, so source changes cannot reduce the residual.Source: F Proof of Proposition 2: Exact… · p. 28
• Source-side re-coupling exactly contracts the local endpoint-state consistency residual introduced by clean-side anchoring, without implying contraction of global reconstruction error.Contraction holds when κ > 0, t ∈ (0, 1), and the clean endpoint changes; the proof concerns the local bridge residual at the current sampling time.Source: F Proof of Proposition 2: Exact… · pp. 28–29

G.1 Dataset Descriptions

• AFHQ-Cat evaluations use the same 100 test images as related work, with images resized to 256 × 256; COCO restoration uses a fixed 100-image validation set, with training images resized to 128 × 128.Source: §G.1 · p. 29
• IXI-Brain uses T2-weighted MRI slices, PMUB uses T2-weighted prostate MRI slices, and X-Ray Hand uses 100 radiographs for testing.IXI-Brain reserves 100 slices as a candidate test set; PMUB has 7,219 training slices and 100 independent test slices from additional volumes; images are processed at 256 × 256.Source: §G.1 · pp. 29–30

G.2 Model Sources

• CelebA and AFHQ-Cat use PnP-Flow pretrained priors, while COCO and X-Ray Hand use Restora-Flow pretrained priors.CelebA and AFHQ-Cat use Gaussian sources, U-Net velocity networks, and Mini-Batch Optimal Transport Flow Matching; their training durations are 200 and 400 epochs, respectively.Source: §G.2 · p. 30
• IXI-Brain and PMUB use separately trained unconditional Flow Matching priors with U-Net velocity fields and linear interpolation between Gaussian source and clean endpoints.Both models train for 400 epochs at learning rate 1 × 10−4 and batch size 64, using 256 × 256 images normalized to [−1, 1].Source: §G.2 · p. 30

G.3 Restoration Task Settings

• Natural-image tests cover denoising, deblurring, super-resolution, random inpainting, and box inpainting, with dataset-specific scales, masks, and measurement-noise levels.Denoising uses σy = 0.2; random inpainting removes 70% of pixels. CelebA and COCO use 2× super-resolution and centered 40 × 40 masks, while AFHQ-Cat uses 4× and centered 80 × 80 masks. Measurement noise is σy = 0.05 for deblurring and σy = 0.01 for other inverse tasks.Source: §G.3 · pp. 30–31
• Medical-image tests include denoising, 2× super-resolution, random inpainting, and box inpainting, with task-specific noise levels and centered mask sizes.Denoising uses σy = 0.08, random inpainting removes 30% of pixels, and other tasks add Gaussian measurement noise with σy = 0.01. Box masks are 32 × 32 for IXI-Brain and X-Ray Hand and 60 × 60 for PMUB.Source: §G.3 · p. 31

G.4 Comparison Methods

• The comparison covers velocity-field correction, local-trajectory optimization, source-variable optimization, intermediate-state correction, and clean-endpoint updating under a shared evaluation protocol.Official implementations and recommended configurations are used; configurations for COCO, IXI-Brain, PMUB, and X-Ray Hand are selected by validation performance, and all methods use the same test images, degradations, metrics, and hardware.Source: Table 6 · p. 31, §G.4 · p. 31
• OT-ODE (Pokle et al., 2024) corrects velocity using measurement gradients; Flow-Priors (Zhang et al., 2024) alternates local trajectory optimization with constraints and prior; D-Flow (Ben-Hamu et al., 2024) optimizes the initial source.The methods use their official implementations or procedures and recommended configurations.Source: §G.4 · pp. 31–32
• PnP-Flow (Martin et al., 2025) alternates data-consistency updates and flow mappings; Restora-Flow (Hadzic et al., 2026) uses mask guidance and trajectory correction; Flower (Pourya et al., 2026) updates a flow-consistent clean endpoint using measurements.COCO and X-Ray Hand experiments use pretrained models provided by Restora-Flow.Source: §G.4 · p. 32

H.1 Effect of the Number of Sampling Steps

• At K = 100, PSNR peaks at 28.16 dB and SSIM at 0.820; at K = 200, LPIPS reaches its lowest value of 0.116.All three metrics improve consistently as sampling steps increase from K = 10 to K = 100, on AFHQ-Cat 4× super-resolution.Source: Table 7 · p. 32, §H.1 · p. 32

H.2 Sensitivity to the Bridge Re-Coupling Strength κ

• 34.51 dB PSNR, 0.962 SSIM, and 0.014 LPIPS are best at κ = 5, improving over κ = 0 by 0.79 dB, 0.012, and 0.007, respectively.The analysis fixes ρ = λ = 1 on CelebA 2× super-resolution; at κ = 8 and 10, PSNR falls to 33.94 dB and 33.30 dB, respectively.Source: Figure 7 · p. 33, §H.2 · p. 33

H.3 Trajectory-Level Analysis of PBD Components

• The analysis tracks measurement defect, clean- and source-side flow-prior deviations, and bridge residual across early, middle, and late sampling stages on four CelebA restoration tasks.Values are normalized by image dimensionality; comparisons include Clean-Side Only, Same-¯bt, Frozen Source, and ReBridge-Flow, with curves showing means and one standard deviation.Source: §H.3 · p. 33, Figure 8 · p. 34
• Across four tasks, ReBridge-Flow and Frozen Source show similar measurement and clean-side deviations, but synchronous source re-coupling lowers bridge residual most in Gaussian deblurring.ReBridge-Flow’s source-side deviation concentrates in the middle stage and approaches zero at both ends; clean-only correction or a fixed source leaves larger middle-stage mismatch.Source: §H.3 · p. 33, Figure 8 · p. 34
• ReBridge-Flow does not simply minimize measurement defect; its controlled source-side adjustment balances observation consistency, flow-prior preservation, and local bridge compatibility.Same-¯bt, Frozen Source applies the same clean-side anchoring while fixing the source endpoint, isolating the effect of source-side re-coupling.Source: §H.3 · pp. 33–34

H.4 Challenging Tasks

• All methods degrade from 4× to 8× as observed information falls; evaluation uses 10 fixed CelebA test images with measurement-space Gaussian noise σy = 0.01.Source: §H.4 · p. 34
• ReBridge-Flow achieves the best overall reconstruction quality at both 4× and 8×, with low GPU-memory use and faster inference than Restora-Flow and Flower.Source: §H.4 · p. 34, Table 8 · p. 35
• Qualitatively, ReBridge-Flow more consistently preserves identity, facial structure, and details such as hair and glasses than the compared methods.PnP-Flow over-smooths; Flower leaves some details blurred; OT-ODE and Restora-Flow are sharper but still deviate in contours and texture.Source: §H.4 · p. 34, Figure 9 · p. 35

I More Quantitative and Qualitative Results

• Figures 10–36 provide additional qualitative comparison results for ReBridge-Flow.Source: I More Quantitative and Qualitative… · p. 36
• Table 15 reports quantitative comparisons on IXI-Brain, PMUB, and X-Ray Hand across medical image restoration tasks, highlighting the best and suboptimal results.The reported metrics are PSNR, SSIM, and LPIPS.Source: I More Quantitative and Qualitative… · p. 36, Table 15 · p. 39