$γ$-Bridge: A Look-Parametric Diffusion Bridge

Technical Summary: $\gamma$-Bridge: A Look-Parametric Diffusion Bridge

Problem Statement
Multiplicative Gamma noise is a signal-dependent degradation inherent to coherent imaging systems, most notably in Synthetic Aperture Radar (SAR) despeckling. In SAR, an intensity image $Y$ is modeled as $Y = X \cdot N$, where $X$ is the underlying reflectivity and $N$ is unit-mean Gamma noise with variance $1/L$. The parameter $L$, known as the "equivalent number of looks," dictates the noise strength; $L=1$ represents single-look complex (SLC) data with low signal-to-noise ratio (SNR), while higher $L$ values correspond to averaged, cleaner observations.

Existing diffusion-based denoisers typically parameterize their forward processes using abstract signal-to-noise schedules rather than the physical look number $L$. Consequently, these models are tied to specific training noise levels and cannot independently control input ($L_{in}$) and output ($L_{out}$) noise levels at inference. Furthermore, real SAR data lacks paired clean ground truth, making supervised training on synthetic data prone to a domain gap when applied to heterogeneous real-world sensors. Current methods often require sensor-specific fine-tuning or rely on complex self-supervised assumptions that do not generalize well across different acquisition conditions.

Methodology
The authors propose $\gamma$-Bridge, a look-parametric diffusion bridge designed to restore Gamma-corrupted images by explicitly modeling the physical look number $L$ as the time axis of the diffusion process.

1. Look-Parametric Forward Bridge:
   Instead of an abstract noise schedule, the forward process is defined by a strictly monotone look schedule $L(t)$ connecting a clean limit ($L_{max}$) to the observed noise level ($L_{obs}$). The intermediate states follow exact Gamma marginals:
   $$x_t = \frac{x_0}{L(t)} G_t, \quad G_t \sim \text{Gamma}(L(t), 1)$$
   This ensures every intermediate state $x_t$ is a physically valid $L(t)$-look SAR image of the underlying reflectivity $x_0$.

2. Gamma–Lévy Decomposition and Reverse Posterior:
   The paper derives a closed-form reverse posterior $q(x_{t-1} | x_t, x_0)$ based on the additivity of independent Gamma variables (Lévy decomposition).

   • Stochastic Reverse: A reverse step involves adding an independent Gamma increment $Y_{t \to t-1} \sim \text{Gamma}(L(t-1)-L(t), 1)$ scaled by the clean estimate $\hat{x}_0$.
   • Deterministic Reverse: By taking the expectation of the stochastic step, a deterministic update rule is derived: $x_{t-1} = \alpha_t x_t + (1-\alpha_t)\hat{x}_0$, where $\alpha_t = L(t)/L(t-1)$. This form acts as an affine mixture between the current noisy state and the network's clean estimate.

3. Network Architecture and Training:

   • Log-Residual Parameterization: The network $G_\theta$ predicts a multiplicative correction in the log-domain: $\hat{x}_0 = x_t \cdot \exp(\Delta_\theta)$. This aligns with the analytical form of the Gamma noise and guarantees non-negativity of the output.
   • Training Objective: The model is trained on natural images corrupted with synthetic Gamma noise at $L_{obs}=1$. The loss function combines three terms:
     • Reconstruction Loss ($\mathcal{L}_{rec}$): $\ell_1$ loss between $\hat{x}_0$ and ground truth.
     • Radiometric Moment Matching ($\mathcal{L}_{ratio}$): Enforces that the residual ratio $x_{obs}/\hat{x}_0$ has the correct mean and variance for the target look number.
     • Two-Step Consistency Loss ($\mathcal{L}_{cons}$): A trajectory-level regularizer that ensures predictions at different steps of the reverse chain are consistent with each other, stabilizing multi-step inference in low-SNR regimes.

4. Inference Controls:
   The trained model supports two orthogonal controls at runtime:

   • Smart-Start (Input Control): Given an input with an estimated look number $\hat{L}_{in}$, the reverse chain is initialized at the corresponding bridge state $t^*$ where $L(t^*) \approx \hat{L}_{in}$, rather than always starting from $L_{obs}=1$.
   • Target-L Output Control: The reverse chain can be terminated at any step $t^*$ corresponding to a desired output look number $L_{out}$, allowing for flexible restoration quality.

Key Contributions

• Look-Parametric Diffusion Bridge: The first diffusion framework where the time axis is the physical look number $L$, providing exact Gamma marginals at every step and a direct physical interpretation of the diffusion process.
• Closed-Form Gamma–Lévy Posterior: Derivation of both stochastic and deterministic reverse transitions that preserve the Gamma structure without requiring clipping or heuristic constraints.
• Zero-Shot Transfer via Orthogonal Controls: A single model trained only on synthetic $L_{obs}=1$ data can perform zero-shot restoration across a full grid of $(L_{in}, L_{out})$ pairs. This is achieved by combining a homogeneous-patch look estimator with smart-start input control, eliminating the need for sensor-specific fine-tuning.
• Distributional Validation: Introduction of Beta-coupled ratio statistics to validate the fidelity of intermediate Gamma marginals during the reverse process.

Experimental Results

• Synthetic Benchmarks: On standard datasets (Set12, McMaster, Kodak24), $\gamma$-Bridge achieves leading or competitive PSNR and SSIM scores across various look numbers ($L_{obs} \in \{1, 2, 4, 8\}$). Notably, it demonstrates strong generalization to high-texture scenes (McMaster) where it outperforms sparse-prior methods.
• Real SAR Evaluation: The model was tested on data from six spaceborne and airborne sensors (Sentinel-1, TerraSAR-X, Gaofen-3, miniSAR, FARAD X/Ka-band) without any fine-tuning.
  • On spaceborne sensors, $\gamma$-Bridge achieved results comparable to or better than state-of-the-art methods (e.g., MONet, SAR2SAR) in terms of Equivalent Number of Looks (ENL) and edge preservation (EPI), while maintaining radiometric fidelity (Mean intensity).
  • On airborne sensors, performance was mid-pack, outperforming some baselines but trailing others on specific sensors, attributed to high-frequency structures and dark-tail intensities not fully captured by the synthetic training distribution.
• Multi-Step Inference: Experiments show that a 5-step deterministic reverse significantly improves performance over a 1-step baseline (+1.16 dB), though excessive steps (25) introduce slight over-shrinkage due to train-inference mismatch.
• Ablation Studies: Removing the closed-form posterior, observation conditioning, consistency loss, or the log-linear schedule resulted in significant performance degradation, confirming the necessity of each component for stable multi-step inference.

Significance and Claims
The paper claims that $\gamma$-Bridge establishes a new paradigm for Gamma image restoration by decoupling the restoration process from fixed noise schedules. Its primary significance lies in enabling zero-shot transfer across diverse SAR sensors and look numbers using a single model trained solely on synthetic data. By treating the look number as a controllable runtime variable, the method provides physically interpretable input and output controls absent in prior denoisers. The authors emphasize that while the method is not perfect for all high-resolution airborne scenarios (due to domain gaps), it successfully bridges the gap between controlled synthetic training and heterogeneous real-world deployment without requiring clean ground truth or sensor-specific adaptation. The work is presented as a step toward physically grounded, controllable diffusion models for non-Gaussian imaging problems.