Residual Diffusion Bridge Model (RDBM)

• Residual Diffusion Bridge Model (RDBM) is a diffusion bridge that modulates stochastic noise based on the residual between clean and degraded images.
• It employs a generalized Ornstein–Uhlenbeck process with residual-dependent perturbation to focus restoration on highly degraded regions.
• RDBM unifies various bridge models under a common framework, improving image quality metrics like PSNR and SSIM in restoration tasks.

Searching arXiv for papers on Residual Diffusion Bridge Model and closely related diffusion bridge work. Residual Diffusion Bridge Model (RDBM) is a paired image-restoration diffusion bridge formulation that specializes generalized diffusion bridges by making the bridge perturbation amplitude residual-dependent rather than globally uniform. In the formulation introduced in “Residual Diffusion Bridge Model for Image Restoration” (Wang et al., 27 Oct 2025), the stochastic path connects a high-quality image $\mathbf{x}_0 \sim p_{HQ}(\mathbf{x})$ and its degraded counterpart $\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$, while the perturbation scale is set by the paired-image residual. The central claim is that conventional bridge models perturb all pixels globally, which can unnecessarily distort undegraded regions during restoration, whereas RDBM uses residual-modulated noise injection and removal so that strongly degraded regions receive larger stochastic treatment and intact regions are minimally disturbed (Wang et al., 27 Oct 2025). This places RDBM within the broader diffusion-bridge literature—especially DDBMs (Zhou et al., 2023), stochastic-interpolant-style bridge frameworks (Zhang et al., 2024), and SOC-based unifications (Zhu et al., 9 Feb 2025)—but with a specific emphasis on residual-adaptive restoration.

1. Generalized bridge formulation

RDBM is built from a generalized Ornstein–Uhlenbeck process with predefined amplitude $\boldsymbol{\pi}$: $$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$ Under the fixed ratio $\lambda = \sigma_t^2/(2\theta_t)$, the paper derives the corresponding generalized diffusion bridge through Doob’s $h$-transform as

$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$

where $\overline{\theta}_{s:t} = \int_s^t \theta_z dz$ (Wang et al., 27 Oct 2025). This yields a mean-arriving bridge whose terminal state is the degraded observation rather than a Gaussian prior.

The forward bridge has a closed-form solution,

$$\mathbf{x}_t = \boldsymbol{\mu} + (\mathbf{x}_0 - \boldsymbol{\mu})\frac{\sinh(\overline{\theta}_{t:T})}{\sinh(\overline{\theta}_{0:T})}+\int_0^t \sqrt{2\boldsymbol{\pi}^2\lambda\theta_s} \frac{\sinh(\overline{\theta}_{t:T})}{\sinh(\overline{\theta}_{s:T})} d\omega_s,$$

with mean and variance

$$E[\mathbf{x}_t] = \boldsymbol{\mu} + (\mathbf{x}_0 - \boldsymbol{\mu})\Theta_t, \qquad Var[\mathbf{x}_t] = \boldsymbol{\pi}^2\Sigma_t^2,$$

where

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$0

Accordingly,

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$1

and

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$2

are available analytically (Wang et al., 27 Oct 2025).

This generalized form places RDBM inside a wider bridge family. DDBMs also model endpoint-conditioned transport between arbitrary paired distributions, but do so by learning the score of a diffusion bridge derived from a base diffusion and paired endpoint samples $\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$3 (Zhou et al., 2023). More generally, the bridge-model design space can be parameterized through endpoint interpolation coefficients $\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$4, with Gaussian conditional kernels of the form

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$5

which suggests that RDBM is a particular analytically derived bridge inside a broader stochastic-interpolant framework (Zhang et al., 2024).

2. Residual modulation and adaptive restoration

The defining specialization of RDBM is

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$6

With this choice, the perturbation amplitude becomes residual-dependent (Wang et al., 27 Oct 2025). Pixels with small discrepancy between degraded and clean images receive little perturbation, while heavily degraded pixels receive larger perturbation. The method therefore performs adaptive noise injection and removal without requiring an explicit degradation mask.

The paper formalizes this through the pixelwise residual-to-noise ratio

$\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$7

If $\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$8, this depends on local residual magnitude and varies sharply across spatial locations. Under the RDBM choice $\boldsymbol{\mu} \sim p_{LQ}(\mathbf{x})$9, the pixelwise residual cancels and yields a spatially uniform time-dependent ratio,

$\boldsymbol{\pi}$0

with

$\boldsymbol{\pi}$1

and endpoint behavior $\boldsymbol{\pi}$2, $\boldsymbol{\pi}$3 (Wang et al., 27 Oct 2025). The paper interprets this as a smooth, monotonically decreasing residual-to-noise ratio. This suggests that the deterministic residual signal dominates early and stochasticity dominates later, while each pixel’s perturbation budget remains proportional to its degradation magnitude.

This residual modulation is the principal distinction between RDBM and earlier bridge models such as Brownian bridge, OU bridge, VE/VP bridge, and generic stochastic-interpolant bridge constructions, which the paper characterizes as using global perturbation scales (Wang et al., 27 Oct 2025). In the same spirit, the Brownian-bridge and DDBM families explicitly condition on endpoints but do not, in their default forms, scale perturbation pixelwise by paired-image residuals (Zhou et al., 2023). The design-space perspective of bridge models likewise separates path design from output parameterization, but does not by itself impose residual-adaptive perturbation (Zhang et al., 2024).

3. Reverse process and learning target

RDBM derives an analytical reverse update from the Gaussian forward marginals. The main deterministic reverse formula is

$\boldsymbol{\pi}$4

where the learned quantity is not plain $\boldsymbol{\pi}$5, but the residual-weighted noise $\boldsymbol{\pi}$6 (Wang et al., 27 Oct 2025).

The theoretical objective is a KL minimization,

$\boldsymbol{\pi}$7

which reduces to reverse-mean matching and then to residual-noise regression: $\boldsymbol{\pi}$8 Algorithm 1 in the paper uses an $\boldsymbol{\pi}$9 implementation loss,

$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$0

(Wang et al., 27 Oct 2025).

This target distinguishes RDBM from bridge formulations that predict a bridge score or a clean endpoint estimate. DDBMs learn the conditional bridge score

$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$1

through denoising bridge score matching,

$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$2

and often use an EDM-style predict-$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$3 parameterization rather than an explicit residual-weighted target (Zhou et al., 2023). In stochastic-interpolant bridge frameworks, denoiser, score, and normalized-noise parameterizations are interconvertible, but residual-weighted noise is again not the canonical output (Zhang et al., 2024).

A plausible implication is that RDBM’s choice of $$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$4 as the primary target aligns the learned variable more directly with the task-specific perturbation structure of restoration. That interpretation is explicit in the paper’s emphasis on adaptive treatment of degraded versus intact regions (Wang et al., 27 Oct 2025).

4. Unified analytical perspective and relation to other bridge families

One of RDBM’s stated contributions is to unify several bridge and transport families under the generalized bridge parameterization. With appropriate choices of $$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$5, $$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$6, and $$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$7, the paper identifies the following special cases (Wang et al., 27 Oct 2025):

Setting	Parameter choice
Flow Matching	$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$8
VE Bridge	$$d \mathbf{x}_t = \theta_t (\boldsymbol{\mu} - \mathbf{x}_t) dt + \boldsymbol{\pi} \sigma_t d \omega_t.$$9
VP Bridge	$\lambda = \sigma_t^2/(2\theta_t)$0
Brownian Bridge	$\lambda = \sigma_t^2/(2\theta_t)$1
OU Bridge	$\lambda = \sigma_t^2/(2\theta_t)$2
RDBM	$\lambda = \sigma_t^2/(2\theta_t)$3

This places RDBM in direct continuity with DDBMs, Brownian bridges, OU bridges, and flow-matching-style deterministic limits. DDBMs themselves already identify standard diffusion as a special case when the source endpoint is Gaussian noise, and show that OT-Flow-Matching or Rectified Flow arise in the noiseless VE-bridge limit (Zhou et al., 2023). The later comparative analysis of diffusion bridges and flow matching similarly recasts both under stochastic optimal control and argues that flow matching is the zero-drift special case of a diffusion-bridge dynamics (Zhu et al., 29 Sep 2025). In that sense, RDBM’s unification claim is consistent with a broader trend in the literature toward treating bridge models, stochastic interpolants, and flow-based transports as a continuous family.

Another relevant line of work is the SOC-based unification in UniDB, which interprets Doob-$\lambda = \sigma_t^2/(2\theta_t)$4-transform bridge models as the $\lambda = \sigma_t^2/(2\theta_t)$5 limit of a finite terminal-penalty control problem (Zhu et al., 9 Feb 2025). RDBM does not formulate a finite-$\lambda = \sigma_t^2/(2\theta_t)$6 objective, but its generalized mean-reverting bridge with residual-modulated diffusion is structurally compatible with that viewpoint. A plausible implication is that RDBM changes the local control geometry of restoration by scaling stochasticity with residual magnitude, whereas UniDB changes the global endpoint trade-off via terminal penalty (Zhu et al., 9 Feb 2025).

5. Inference, restoration procedure, and practical behavior

At inference, RDBM starts from the degraded image itself: $\lambda = \sigma_t^2/(2\theta_t)$7 It then runs a DDIM-style deterministic reverse sampler: $\lambda = \sigma_t^2/(2\theta_t)$8 where $\lambda = \sigma_t^2/(2\theta_t)$9 is predicted by the network conditioned on $h$0 (Wang et al., 27 Oct 2025). The paper reports that 10 timesteps is the best test-time NFE among those tested, with slightly worse performance beyond 10 steps, which it attributes to over-processing or drift under mixed degradations (Wang et al., 27 Oct 2025).

This source-anchored bridge sampling resembles other bridge formulations where inference begins from an observed endpoint rather than from isotropic Gaussian noise. DDBMs likewise condition on a given endpoint $h$1 and integrate a bridge reverse SDE or bridge probability flow ODE toward $h$2 (Zhou et al., 2023). More generally, bridge-model design work emphasizes that endpoint-conditioned sampling often requires carefully chosen stochasticity, since fixed-source conditions can otherwise lead to low diversity or over-deterministic reconstructions (Zhang et al., 2024).

RDBM’s stated practical intuition is narrower: because the network predicts residual-weighted noise, denoising activity concentrates where residuals are large, which reduces unnecessary reconstruction of already correct content (Wang et al., 27 Oct 2025). That distinguishes it from bridge samplers that remain endpoint-conditioned but spatially global in their perturbation pattern.

6. Empirical evaluation and significance

RDBM is evaluated on five restoration tasks: deraining, low-light enhancement, desnowing, dehazing, and deblurring, using datasets including Rain13K, DeRaindrop, LOL, VE-LOL-L, CSD, ITS_v2, D-HAZY, GoPro, and several real-world generalization sets (Wang et al., 27 Oct 2025). Metrics include PSNR, SSIM, NIQE, LPIPS, FID, and MetaIQA (Wang et al., 27 Oct 2025).

The paper reports that RDBM-L attains 31.04 dB PSNR and 0.917 SSIM on average, with about 1.55 dB average PSNR gain over prior universal restoration models (Wang et al., 27 Oct 2025). The most targeted ablation varies the modulation term $h$3:

$h$4 choice	Average PSNR / SSIM
$h$5	$h$6
$h$7	$h$8
$h$9	$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$0
$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$1	$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$2

These numbers are the main empirical support for the residual-modulated bridge claim (Wang et al., 27 Oct 2025). The paper also reports that the best schedule is cosine, the best stationary variance is $$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$3, and the best NFE is 10 (Wang et al., 27 Oct 2025).

The broader bridge literature offers useful context for interpreting these results. DDBMs already showed that endpoint-conditioned bridge transport can outperform standard conditional diffusion baselines on paired image translation tasks, especially in pixel space (Zhou et al., 2023). Bridge-model design studies further demonstrated that path design and sampler choice can strongly affect both sample quality and efficiency, even without changing the trained network (Zhang et al., 2024, Zheng et al., 2024). RDBM’s empirical contribution is therefore less the claim that bridges are useful per se, and more the claim that residual-adaptive perturbation is a particularly effective bridge specialization for restoration (Wang et al., 27 Oct 2025).

7. Conceptual boundaries and adjacent meanings of “residual bridge”

The term “Residual Diffusion Bridge Model” in (Wang et al., 27 Oct 2025) is distinct from older “residual-bridge” terminology in diffusion simulation. In “Residual-Bridge Constructs for Conditioned Diffusions” (Malory et al., 2016), a residual bridge is a proposal mechanism for conditioned SDE simulation formed by decomposing a target diffusion into an approximate diffusion plus a residual and then applying a modified diffusion bridge approximation to that residual. The same line of work compares residual proposals with guided proposals and characterizes them as auxiliary proposals rather than standalone generative models (Meulen et al., 2017). Those papers are directly relevant terminologically, but they address approximate conditioned-diffusion simulation in Bayesian inference rather than paired image restoration (Malory et al., 2016, Meulen et al., 2017).

A second ambiguity arises from “residual-based” bridge-like image models that are not formal diffusion bridges. For example, the residual-based efficient bidirectional diffusion model for dehazing defines dual residual-shifting Gaussian forward chains between haze-free and hazy images, but does not formulate a path-space bridge objective or Doob-$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$4-transform bridge (Liu et al., 15 Aug 2025). That work is bridge-like in mechanism but not identical in theory. RDBM (Wang et al., 27 Oct 2025), by contrast, explicitly derives generalized bridge SDEs and reverse processes.

Finally, recent bridge work has highlighted issues such as endpoint underfitting under DDPM-style bridge score matching and the benefits of bridge-specific target scaling (Gao et al., 27 May 2026). While that analysis is not part of RDBM, it suggests a broader principle: bridge models are sensitive to how their path variable and learning target are parameterized near endpoints. A plausible implication is that RDBM’s residual-weighted target may also be understood as an endpoint-aware parameterization choice, though the RDBM paper frames it primarily in terms of spatially adaptive restoration rather than target-noise alignment (Gao et al., 27 May 2026).

8. Assessment

RDBM is best understood as a residual-modulated member of the diffusion-bridge family. Its defining move is not merely to connect degraded and clean distributions, but to set the bridge stochasticity itself to the paired-image residual,

$$d\mathbf{x}_t = \theta_t\coth(\overline{\theta}_{t:T})(\boldsymbol{\mu}-\mathbf{x}_t)dt + \sqrt{2\boldsymbol{\pi}^2\lambda \theta_t} d\omega_t,$$5

thereby making restoration intensity spatially adaptive (Wang et al., 27 Oct 2025). In theoretical terms, it derives a closed-form generalized bridge, analytical forward marginals, and a deterministic reverse sampler. In algorithmic terms, it learns residual-weighted noise rather than an unconditional score or plain clean endpoint estimate. In empirical terms, it is positioned as a universal restoration model that preserves intact regions while concentrating reconstruction on degraded ones (Wang et al., 27 Oct 2025).

Within the larger arXiv literature, RDBM occupies a point where several strands intersect: DDBM-style endpoint-conditioned generation (Zhou et al., 2023), stochastic-interpolant bridge design (Zhang et al., 2024), SOC-based bridge unification (Zhu et al., 9 Feb 2025), and the older residual-bridge tradition in conditioned diffusion simulation (Malory et al., 2016). Its specific contribution is to reinterpret diffusion-bridge perturbation as a residual-dependent resource allocation mechanism for restoration.