Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models

Abstract: Diffusion bridge models have shown great promise in image restoration by explicitly connecting clean and degraded image distributions. However, they often rely on complex and high-cost trajectories, which limit both sampling efficiency and final restoration quality. To address this, we propose an Energy-oriented diffusion Bridge (E-Bridge) framework to approximate a set of low-cost manifold geodesic trajectories to boost the performance of the proposed method. We achieve this by designing a novel bridge process that evolves over a shorter time horizon and makes the reverse process start from an entropy-regularized point that mixes the degraded image and Gaussian noise, which theoretically reduces the required trajectory energy. To solve this process efficiently, we draw inspiration from consistency models to learn a single-step mapping function, optimized via a continuous-time consistency objective tailored for our trajectory, so as to analytically map any state on the trajectory to the target image. Notably, the trajectory length in our framework becomes a tunable task-adaptive knob, allowing the model to adaptively balance information preservation against generative power for tasks of varying degradation, such as denoising versus super-resolution. Extensive experiments demonstrate that our E-Bridge achieves state-of-the-art performance across various image restoration tasks while enabling high-quality recovery with a single or fewer sampling steps. Our project page is https://jinnh.github.io/E-Bridge/.

• The paper’s main contribution is E-Bridge, a framework that uses low-energy geodesic trajectories to achieve efficient and high-fidelity image restoration.
• It employs a closed-form, single-step solver driven by consistency objectives to reduce computational cost while maintaining state-of-the-art perceptual quality.
• Experimental results demonstrate improved metrics like LPIPS and FID, highlighting superior performance over traditional iterative diffusion and bridge models.

Energy-oriented Diffusion Bridge: A Manifold Geodesic Framework for Image Restoration

Introduction

The paper "Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models" (2604.10983) proposes a new framework, E-Bridge, for image restoration tasks leveraging foundational diffusion models. The authors systematically articulate that prevalent diffusion bridge models, while connecting degraded and clean data distributions, suffer from inefficient high-cost trajectories and redundant re-noising phases. These handicaps result in both sub-optimal sample quality and excessive computational overhead. Through a rigorously constructed energy-oriented approach, the paper establishes a novel bridge process traversing low-energy geodesic trajectories on the data manifold, paired with a single-step, consistency-driven solver. This enables task-adaptive, sample-efficient, and high-fidelity image restoration across a range of distortion types.

Background and Motivation

Diffusion models have become the preeminent class for generative modeling, particularly for image restoration tasks. Standard approaches—conditional diffusion and image-to-image bridge models—often utilize trajectories that do not minimize kinetic or control energy relative to the underlying data geometry, and most require slow iterative denoising starting from high-entropy (noise) states. Bridge models such as Brownian Bridge and Schrödinger Bridge variants provide more direct mappings but still enforce unnecessary re-noising and do not guarantee energetically minimal paths, with iterative solutions susceptible to computational bottlenecks and numerical instability.

Figure 1: Schematic depicting (a) standard diffusion (high-energy, lengthy path from noise), (b) conventional bridges (sub-optimal, include redundant re-noising), and (c) proposed E-Bridge (direct, low-energy geodesic using an entropy-regularized initialization).

Method

Geodesic Trajectory Formulation

E-Bridge explicitly formulates the restoration process as a transport problem across data manifolds—degraded to clean—using a stochastic process whose deterministic component (mean) traces a kinetic-energy-minimizing geodesic. Specifically, for a controllable horizon $T_0$, the expectation evolves linearly between clean and degraded marginals:

$$\mu(t) = \left[1 - \left(\frac{t}{T_0}\right)\right]\mathbf{X}_0 + \left(\frac{t}{T_0}\right)\mathbf{Y}$$

The process initialization is entropy-regularized—a convex mixture of the degraded image and Gaussian noise—by enforcing the time parameter $T_0$ as a tunable control of information/generation trade-off. This overcomes high-energy re-noising and enables precise adaptation to task severity.

Consistency-based Single-Step Solver

Departing from multi-step ODE/SDE-based sampling, E-Bridge derives a closed-form, single-step mapping designed via analytic inversion of the bridge process combined with a pretrained denoiser. The model is trained using a continuous-time consistency objective, enforcing that the mapping from any point on the geodesic to the clean endpoint is invariant, yielding stable, efficient, non-iterative restoration.

Task Adaptivity

The trajectory horizon $T_0$ is not statically defined but sampled from a continuous range during training. At inference, $T_0$ functions as a task-adaptive control knob—shorter horizons for tasks where the degraded input carries substantial structure (e.g., denoising), longer for severe or ill-posed degradations (e.g., super-resolution), thus modulating the information-entropy balance without explicit retraining or separate models.

Experimental Results

The authors evaluate the framework on super-resolution, denoising, raindrop removal, low-light enhancement, and demoiréing, using a large-scale pretrained backbone (Flux-dev). Performance is analyzed via PSNR, LPIPS, FID, NIQE, MUSIQ, and computational cost measured by number of function evaluations (NFE).

Figure 2: Visual comparison of E-Bridge with state-of-the-art models across multiple restoration tasks, demonstrating superior perceptual fidelity and realism.

Key findings include:

• E-Bridge achieves superior or competitive perceptual metrics (e.g., LPIPS, FID, NIQE, MUSIQ) with a drastic reduction in inference steps (often 1–10 NFE), markedly outperforming iterative solvers and bridge methods that require orders of magnitude more steps for similar quality.
• The control horizon $T_0$ empirically balances fidelity and generation, as larger values improve generative reconstructions for difficult tasks but can degrade fidelity where fine details are already present.
• The ablation studies verify that using the geodesic trajectory consistently surpasses both standard diffusion and conventional bridges in energetic cost and restoration quality, with a closed-form solver further accelerating inference without sacrificing perceptual outcomes.

  Figure 3: Additional qualitative results for image super-resolution on real-world naturally degraded data, demonstrating generalization and artifact-free reconstruction.

Implications and Future Directions

The work systematically demonstrates that manifold-geodesic-based restoration paths enable the efficient deployment of foundational denoising priors for a broad class of image restoration problems. The closed-form, consistency-based solver architecture dispenses with slow, numerically brittle iterative solutions, opening up practical deployment for real-time and resource-constrained scenarios—an area where classic diffusion models fall short.

A core theoretical implication is the explicit link established between entropy-regularized, adaptive initialization and data-consistent geodesic transport, pointing toward a more general theory of efficient generative restoration grounded in optimal transport and differential geometry. The utility of dynamic horizon control suggests a promising axis for spatial adaptivity (e.g., pixel-wise $T_0$ for spatially heterogeneous degradations) and model distillation for deployment on lightweight hardware.

Remaining limitations include the computational footprint of large generative backbones and the global nature of the $T_0$ parameter, especially in the presence of localized artifacts. Future work may target spatially adaptive controls and backbone compression techniques to further improve flexibility and scalability.

Conclusion

The E-Bridge framework advances the state of the art in image restoration by integrating geometrically optimal, energy-minimizing trajectories with a theoretically grounded, consistency-based single-step solver. This design unifies perceptual quality, efficiency, and adaptability, outperforming competing bridge- and flow-based models across numerous high-level tasks. The methodology and findings lay foundational groundwork for future research in energy-efficient, adaptive generative modeling for inverse problems.

Reference:

"Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models" (2604.10983)