Dual-approx Bridge: Deterministic Image Translation

• Dual-approx Bridge is a denoising Brownian bridge model that employs two neural approximators to achieve deterministic image-to-image translation.
• It uses one approximator to estimate the clean source and another to reconstruct reverse noise, balancing consistency with fine detail preservation.
• Empirical results on benchmarks like Cityscapes and DF2K show that with only 3 sampling steps, the method outperforms baselines in PSNR and SSIM.

Dual-approx Bridge is a denoising Brownian bridge model with dual approximators for deterministic image-to-image translation. It was introduced to address a specific tension in image-to-image tasks such as super-resolution: stochastic diffusion or bridge samplers often produce high-quality outputs but yield different results for the same input, whereas deterministic samplers such as PF-ODE-style approaches are consistent but often lose fine details and become blurry. The method therefore seeks to preserve both determinism and high fidelity by learning two neural approximators within a Brownian bridge formulation, producing outputs with negligible variance while remaining close to paired ground truth (Xiao et al., 29 Dec 2025).

1. Problem setting and conceptual motivation

The method is designed for deterministic image-to-image translation, where each input is expected to generate a consistent and predictable output that closely matches the ground truth. The paper frames this as a conflict between two established behaviors. On one side, SDE-based diffusion and bridge samplers inject randomness at every denoising step and therefore produce diverse but unstable outputs. On the other side, deterministic samplers eliminate randomness but often sacrifice texture and local detail, especially in bridge-based formulations (Xiao et al., 29 Dec 2025).

Dual-approx Bridge addresses this by retaining a Brownian bridge model while replacing the two quantities required in reverse-time sampling with learned approximations. This is the basis of the name: the method is “dual-approximator” because it uses one neural approximator for the forward diffusive process and one for the reverse process. The paper’s core claim is that this combination yields a deterministic bridge that remains faithful to ground truth without collapsing into the blur often associated with ODE-style deterministic sampling (Xiao et al., 29 Dec 2025).

A recurrent misconception is to view the model as merely a deterministic diffusion sampler. The paper argues against that interpretation. Its construction is explicitly tied to Brownian bridge dynamics, and the reverse trajectory is not obtained by simply removing noise from a standard diffusion process. Instead, the model learns a deterministic route through a bridge process by approximating both denoised content and bridge noise.

2. Brownian bridge formulation

Let $X_0$ denote the source image and $Y$ the target image. Dual-approx Bridge uses the Brownian bridge stochastic differential equation

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$

with the process pinned so that $X(1)=Y$ (Xiao et al., 29 Dec 2025).

The paper defines

$G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$

which gives the bridge marginal

$X_t \sim \mathcal{N}\big((1-t)X_0+tY,\; G(t)I\big),$

or equivalently

$X_t = (1-t)X_0+tY+\sqrt{G(t)}\,\epsilon,\qquad \epsilon\sim\mathcal{N}(0,I).$

For simplicity, the method sets $g(t)=1$, yielding $G(t)=t(1-t)$, and therefore

$X_t = (1-t)X_0+tY+\sqrt{t(1-t)}\,\epsilon.$

This produces a continuous stochastic path between source and target while keeping the bridge pinned at $Y$0 (Xiao et al., 29 Dec 2025).

The reverse-time SDE is written as

$Y$1

with score

$Y$2

After discretization, reverse sampling requires estimating two latent quantities at each step: the clean source state $Y$3 and the reverse Brownian increment. This observation is the formal reason the paper introduces two separate approximators rather than a single denoiser.

3. Dual approximators and training objectives

The first network is the forward approximator,

$Y$4

which learns to estimate the clean source component $Y$5 from the bridge state $Y$6. The second network is the reverse approximator,

$Y$7

which learns to estimate the standard Gaussian noise $Y$8 associated with the reverse-time Brownian increment (Xiao et al., 29 Dec 2025).

The paper states that a single approximator is not enough because the forward and reverse steps need different latent quantities from the same stochastic bridge dynamics. The forward model predicts denoised content, while the reverse model reconstructs the stochastic increment in a learned deterministic manner. This division is central rather than cosmetic: it is the main mechanism by which the method removes per-step randomness without discarding the bridge structure itself.

With $Y$9, the forward bridge sample is

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$0

The forward network minimizes

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$1

The paper notes that an optional conditional version $dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$2 can also be used (Xiao et al., 29 Dec 2025).

The reverse approximator is trained independently using

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$3

with the same bridge state

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$4

Both networks are trained on paired data. The paper also defines the bridge variance schedule for discrete implementation as

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$5

4. Deterministic bridge sampling

The paper discretizes the reverse SDE with $dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$6 as

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$7

Sampling then replaces $dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$8 by the forward approximation

$dX = -\frac{X-Y}{1-t}\,dt + g(t)\,dW, \qquad t\in(0,1),$9

and replaces the reverse noise increment with the output of $X(1)=Y$0 (Xiao et al., 29 Dec 2025).

Using the notation

$X(1)=Y$1

the paper derives the iterative deterministic update

$X(1)=Y$2

Operationally, sampling begins from $X(1)=Y$3. At each step, the model predicts $X(1)=Y$4, estimates noise using $X(1)=Y$5 except at the initial step, constructs bridge states, and updates $X(1)=Y$6. The final output is $X(1)=Y$7.

The only injected randomness appears at the very first step $X(1)=Y$8, where the coefficient on $X(1)=Y$9 is on the order of $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$0. The paper therefore characterizes the resulting outputs as near-deterministic with extremely low sampling variance. This is a crucial distinction from standard SDE samplers, which inject randomness at every denoising step.

5. Empirical behavior

The paper reports a characteristic tradeoff between realism-oriented and faithfulness-oriented metrics on Cityscapes. With 1000 steps, Dual-approx Bridge obtains FID 42.70, LPIPS 0.442, PSNR $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$1, and SSIM $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$2. With 3 steps, it obtains FID 56.07, LPIPS 0.523, PSNR $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$3, and SSIM $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$4. The paper’s interpretation is that more steps improve realism metrics such as FID and LPIPS, whereas fewer steps improve faithfulness metrics such as PSNR and SSIM. It therefore chooses 3 sampling steps as the default in later experiments because this setting gives strong robustness and efficiency with almost zero variation (Xiao et al., 29 Dec 2025).

Against Brownian bridge baselines on Cityscapes, the reported results are: SDE-based bridge with PSNR $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$5 and SSIM $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$6, PF-ODE-based bridge with PSNR $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$7 and SSIM $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$8, and Dual-approx Bridge with PSNR $G(t) = (1-t)^2\int_0^t \frac{g^2(s)}{(1-s)^2}\,ds,$9 and SSIM $X_t \sim \mathcal{N}\big((1-t)X_0+tY,\; G(t)I\big),$0. The qualitative summary given in the paper is correspondingly direct: SDE sampling is detailed but inconsistent, PF-ODE sampling is consistent but blurry, and Dual-approx Bridge is both detailed and consistent (Xiao et al., 29 Dec 2025).

For deterministic image-to-image translation on Cityscapes against deterministic baselines such as Pix2Pix, CycleGAN, and UNIT, the paper reports FID 48.70, PSNR 15.70, and SSIM 53.26\%. In super-resolution, trained on DF2K and tested on BSD100 and Urban100, the method reports LPIPS 0.1267 and PSNR 26.56 on BSD100, and LPIPS 0.1207 and PSNR 23.23 on Urban100. The paper emphasizes that these results are obtained with only 3 sampling steps, whereas many diffusion baselines use 1000 steps (Xiao et al., 29 Dec 2025).

6. Position within bridge-based generative modeling

Dual-approx Bridge belongs to a broader family of bridge-based image-generation methods, but its construction is materially different from other “dual” bridge formulations. In particular, DCDB proposes a Dynamic Conditional Dual Diffusion Bridge for ill-posed multi-task problems such as dehazing and visible-infrared fusion. DCDB uses two coupled bridges: an X-bridge for the main restoration or fusion task and a Y-bridge for dynamic condition generation, with both bridges sharing the same noise schedule (Huang et al., 3 Sep 2025).

This contrast clarifies the specific meaning of “dual” in Dual-approx Bridge. In DCDB, duality refers to two synchronized diffusion bridges. In Dual-approx Bridge, duality refers instead to two approximators operating within a single Brownian bridge: one for $X_t \sim \mathcal{N}\big((1-t)X_0+tY,\; G(t)I\big),$1 and one for reverse noise. The two models therefore solve related but distinct problems. DCDB diffuses the condition itself; Dual-approx Bridge learns a deterministic surrogate for the stochastic elements of a Brownian bridge (Huang et al., 3 Sep 2025).

A plausible implication is that Dual-approx Bridge should be understood less as a generic deterministic sampler and more as a bridge-specific approximation strategy. Its principal contribution is not merely to reduce randomness, but to do so while preserving the latent structure of Brownian bridge sampling. In the paper’s own framing, this is what allows it to approach the quality behavior of stochastic sampling while retaining the consistency required by deterministic image-to-image translation (Xiao et al., 29 Dec 2025).