Reversing Flow for Image Restoration

Conference: CVPR 2025
arXiv: 2506.16961
Code: None (Project Page available)
Area: Image Restoration / Generative Models
Keywords: Continuous Normalizing Flows, Image Restoration, Deterministic Degradation Path, Auxiliary Process, Entropy Conservation

TL;DR

ResFlow proposes modeling the image degradation process as a deterministic continuous normalizing flow (rather than a stochastic diffusion process). By resolving the irreversibility of degradation using auxiliary variables, it achieves reversible modeling. Implementing an entropy-preserving schedule, it completes high-quality image restoration in only 4 sampling steps, achieving SOTA on tasks such as desnowing, deraining, dehazing, denoising, and deblocking.

Background & Motivation

Background: Image restoration aims to recover high-quality (HQ) images from degraded low-quality (LQ) images. Score- and diffusion-based generative models are currently dominant, including DDRM, IR-SDE, I2SB, ResShift, RDDM, etc. They typically model degradation as a stochastic forward process and learn a reverse process to restore images.

Limitations of Prior Work: (1) Starting the reverse process from Gaussian noise is unnecessary and inefficient, as LQ images already contain abundant structural information; (2) Even if models like IR-SDE and I2SB incorporate LQ images into the forward process, they still treat degradation as a stochastic process of progressive diffusion, introducing extra complexity; (3) Stochasticity leads to low training and inference efficiency, typically requiring dozens to hundreds of sampling steps.

Key Challenge: Image degradation is fundamentally irreversible (information is erased), as manifested by declining mutual information—the data processing inequality (DPI) indicates that the mutual information between intermediate states and the HQ image monotonically decreases during degradation. However, a deterministic ODE describes a reversible process (preserving constant mutual information), creating a contradiction. Hence, one cannot directly model degradation using an ODE.

Goal: Model the degradation process as a deterministic path rather than a stochastic path to achieve fast and high-quality image restoration.

Key Insight: Analyze the irreversibility of degradation from an information-theoretic perspective, and introduce an auxiliary variable process \(\{\mathbf{y}_t\}\) to encode the information lost during degradation, making the augmented joint process \(\mathbf{z}_t = [\mathbf{x}_t; \mathbf{y}_t]\) a reversible ODE.

Core Idea: Compensate for the information loss caused by degradation using auxiliary variables—as \(\mathbf{x}_t\) approaches LQ and mutual information decreases, \(\mathbf{y}_t\) compensates for this loss to keep the total mutual information constant, thereby achieving reversible degradation modeling with deterministic flows.

Method

Overall Architecture

ResFlow defines an augmented state \(\mathbf{z}_t^T = [\mathbf{x}_t^T; \mathbf{y}_t^T]\), where \(\mathbf{x}_0 = \mathbf{x}_{HQ}\), \(\mathbf{x}_1 = \mathbf{x}_{LQ}\), \(\mathbf{y}_0 = 0\), and \(\mathbf{y}_1 \sim \mathcal{N}(0, I)\). A velocity field network \(\mathbf{v}_\theta(\mathbf{x}_t, \mathbf{y}_t, t)\) is trained via velocity matching. During inference, integration is performed from \(t=1\) (LQ + random \(\mathbf{y}_1\)) to \(t=0\), taking \(\hat{\mathbf{x}}_0\) as the restored result in only 4 steps.

Key Designs

1. Augmented Degradation Flow:

   • Function: Transforms the irreversible degradation process into a reversible ODE using auxiliary variables.
   • Mechanism: Observes that a deterministic ODE \(\partial \mathbf{z}_t / \partial t = \mathbf{v}(\mathbf{z}_t, t)\) preserves constant mutual information (Proposition 1), whereas the degradation process has decreasing mutual information. An auxiliary variable \(\mathbf{y}_t\) is introduced and coupled with the "uncertainty range"—as \(\mathbf{x}_t\) loses mutual information with HQ, \(\mathbf{y}_t\) carries the compensating information. \(\mathbf{y}_1\) is Gaussian noise (maximum entropy) and \(\mathbf{y}_0 = 0\) (no auxiliary information is needed when restoration is complete). During training, \(\mathbf{y}_t\) is independently coupled with \(\mathbf{x}_0\), but the trained velocity network produces a deterministic coupling during inference.
   • Design Motivation: Direct ODEs cannot model irreversible degradation; augmenting the state space allows the conservation of mutual information in ODEs to be correctly exploited.

2. Entropy-Preserving Schedule:

   • Function: Defines individual interpolation paths for the data and auxiliary components.
   • Mechanism: The data component follows a straight path \(\alpha_t^x = 1 - t\), \(\sigma_t^x = t\) (a geodesic in Euclidean space). The auxiliary component employs a non-linear schedule \(\sigma_t^y = \beta \cdot (1 - t + \beta)^{-1}\) (\(\beta = 10\)), keeping the total entropy constant throughout the process. This is based on the intuition that entropy should be conserved in a reversible process.
   • Design Motivation: Compared to parameterized estimation that directly predicts \(\mathbb{E}[\mathbf{z}_0 | \mathbf{z}_t]\), velocity matching avoids high discretization errors near \(t \to 0\).

3. Adaptive Loss Weighting:

   • Function: Balances training gradients across different timesteps.
   • Mechanism: As time approaches \(t=1\) (the LQ end), \(\mathbf{x}_t\) shares less mutual information with HQ, making velocity prediction harder. A loss weighting function \(\lambda(t) = (\cos(\frac{\pi}{2}(t-2)) + 1)^\gamma\) (\(\gamma = 1.75\)) increases the weight as \(t\) approaches 1, ensuring the model focuses more learning effort on difficult regions.
   • Design Motivation: Velocity estimation near the LQ end is more critical and difficult, requiring larger loss weights to compensate.

Loss & Training

Velocity matching loss: \(\min_\theta \mathbb{E}[\int_0^1 \lambda(t) \|\mathbf{v}_\theta(\mathbf{x}_t, \mathbf{y}_t, t) - \dot{\mathbf{z}}_t\|^2 dt]\). It matches target velocities directly without simulating the ODE, leading to efficient training. It uses the DDPM U-Net architecture, injecting timesteps via adaptive layer norm. It is trained on \(256 \times 256\) crops and tested at full resolution. It uses a uniform time schedule with 4-step sampling.

Key Experimental Results

Main Results

Synthetic Datasets:

Task	Metric	Prev. SOTA	Ours
Desnowing (Snow100K)	PSNR/SSIM/LPIPS	30.92/0.917/0.034	31.86/0.917/0.030
Deraining (Outdoor-Rain)	PSNR/SSIM	30.99/0.934	32.82/0.936
Dehazing (Dense-Haze)	PSNR/SSIM	17.07/0.63	17.12/0.59

Real-world Datasets:

Task	Metric	Prev. SOTA	Ours
Denoising (SIDD)	PSNR/SSIM	40.02/0.960	42.26/0.962
Dehazing (NH-HAZE)	PSNR	20.66	21.44
Deraining (LHP)	PSNR/SSIM	34.33/0.946	34.54/0.939

Ablation Study

Defocus Deblurring (DPDD Combined): Ours achieves the highest PSNR, outperforming Restormer (25.98) and FocalNet (26.18).

Key Findings

• Only 4 sampling steps are required to reach SOTA, which is more than an order of magnitude faster than diffusion methods (typically 50-100 steps).
• Deterministic paths are more suitable for image restoration than stochastic paths—since degradation is known, stochasticity is unnecessary.
• The entropy-preserving schedule performs better than the straight-line schedule.
• Auxiliary variables conceptually encode the information lost during degradation. Different sampling of \(\mathbf{y}_1\) during inference yields diverse restoration results, preserving the diversity of generative models.

Highlights & Insights

• The information-theoretic perspective is elegant—explaining why ODEs cannot directly model degradation and why auxiliary variables are needed through mutual information conservation/decay.
• The proof of Proposition 1 bridges ODEs and information theory, providing a solid theoretical foundation for ResFlow.
• Adapting the flow matching framework to image restoration is highly natural, completely avoiding the redundant stochasticity of diffusion models.
• The extreme efficiency of 4-step sampling is highly friendly to practical deployment.

Limitations & Future Work

• Random sampling of the auxiliary variable \(\mathbf{y}_1\) means restoration results can vary slightly each time, which may require extra handling for applications demanding deterministic outputs.
• The paper does not explore other low-level vision tasks such as super-resolution.
• For extreme degradation (where information is almost entirely lost), the compensation capability of the auxiliary variable might be insufficient.
• Integration with pre-trained diffusion models is unexplored—can this be adapted from a pre-trained Stable Diffusion model?

Related Work & Insights

• The success of Flow Matching / Continuous Normalizing Flow in generative modeling is transferred to image restoration.
• Compared to InDI (which incrementally estimates HQ), ResFlow's velocity matching avoids error sensitivity near the HQ target.
• The concept of using auxiliary variables to resolve irreversibility can be generalized to other ill-posed inverse problems.

Rating

• Novelty: 9/10 — The theory behind the information-theory-driven augmented flow design is beautiful and novel.
• Experimental Thoroughness: 8/10 — Covers 5 tasks across multiple datasets, though performance gains on some tasks are small.
• Writing Quality: 9/10 — Theoretical derivations are clear, and diagrams are intuitive.
• Value: 8/10 — Extreme efficiency of 4-step high-quality restoration is highly valuable for real-world applications.

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