Fourier Prior Guided Degradation Module (FGDM)

• The paper introduces FGDM, which leverages the structural guidance of Fourier phase to synthesize LR images with realistic degradation from HR inputs.
• It decomposes images into amplitude and phase, enhancing amplitude via AENet to mimic real-world degradation while preserving structure.
• Experiments demonstrate improved PSNR/SSIM metrics and show FGDM’s complementary role with RFDM in achieving state-of-the-art super-resolution results.

Searching arXiv for the cited paper to ground the article. arxiv_search(query="(Zhou et al., 10 Aug 2025)", max_results=5, sort_by="submittedDate") The Fourier Prior Guided Degradation Module (FGDM) is a component of the unsupervised real-world super-resolution method presented in "Unsupervised Real-World Super-Resolution via Rectified Flow Degradation Modelling" (Zhou et al., 10 Aug 2025). It is designed for the synthesis of realistic low-resolution (LR) images from unpaired LR and high-resolution (HR) data, with the specific aim of reducing the domain gap between synthetic degradations and real-world degradations. Within the overall framework, FGDM leverages structural information embedded in Fourier phase components to guide degradation modeling more precisely, and it operates together with the Rectified Flow Degradation Module (RFDM) to produce final synthetic LR images with real-world degradation characteristics (Zhou et al., 10 Aug 2025).

1. Problem setting and motivation

Real-world single-image super-resolution is described as difficult because the degradations that produce practical LR images are complex and unknown. Synthetic LR-HR pairs generated with standard degradations such as bicubic interpolation or Gaussian blur create a significant domain gap relative to real data, and existing methods struggle to generalize across that gap (Zhou et al., 10 Aug 2025).

FGDM is motivated by two observations. First, structural information in images is largely preserved in the Fourier phase, whereas degradation effects such as blur and loss of detail primarily affect the Fourier amplitude. Second, repeated up-down sampling operations on LR images progressively standardize their degradations, causing even differently degraded images to exhibit increasingly similar characteristics, although this process also risks information loss. In the formulation reported for FGDM, these observations motivate a module that uses phase information from real LR images as a prior for structural guidance while enhancing the amplitude component so that synthetic LR images retain realistic structure and reflect plausible real-world degradation (Zhou et al., 10 Aug 2025).

The immediate context is a comparison against prior strategies that either depend on simple repeated downsampling, as in UDDM, or rely on synthetic degradations that do not adequately match practical degradations. The description of FGDM suggests that its role is not merely to generate degraded inputs, but to constrain degradation synthesis with an explicit Fourier-domain prior so that the generated images remain structurally consistent with real LR observations.

2. Fourier prior and spectral decomposition

FGDM is built on a Fourier-domain decomposition of an image $I$:

$F(I) = A_I \exp(j\Phi_I),$

where $A_I$ is the amplitude spectrum and $\Phi_I$ is the phase spectrum (Zhou et al., 10 Aug 2025).

In the reported interpretation, $A_I$ is the real-valued spectrum that indicates the strength of frequency components, and $\Phi_I$ contains spatial structural information. The key prior used by FGDM is stated directly: structural content is encoded by the phase spectrum, while degradation effects primarily modify the amplitude spectrum (Zhou et al., 10 Aug 2025).

This prior defines the logic of the module. Rather than treating degradation synthesis as a purely pixel-domain transformation, FGDM separates structure and degradation into Fourier components and then recombines them asymmetrically: the phase from a real LR image supplies structural guidance, while the amplitude from a degradation-transformed intermediate image is enhanced to better match real LR degradation statistics. A plausible implication is that the method attempts to preserve real-world structural cues without requiring paired LR-HR supervision.

3. Workflow of FGDM

The FGDM pipeline begins with degradation-transformed LR (DT-LR) generation. Starting from an HR image, the method applies repeated up-down bilinear sampling, reported as $10\times$ in experiments, to generate an intermediate DT-LR image. This intermediate is intended to bridge the gap between HR images and unpaired real LR images. The description notes that this step reduces the influence of unknown degradations, but also risks information loss and over-smoothing (Zhou et al., 10 Aug 2025).

After DT-LR construction, Fourier transforms are applied to both the DT-LR image and a real LR image. For the DT-LR image $I_{\text{DT-LR}}$, the transform yields $A_{\text{DT-LR}}$ and $\Phi_{\text{DT-LR}}$; for the real LR image $F(I) = A_I \exp(j\Phi_I),$0, it yields $F(I) = A_I \exp(j\Phi_I),$1 and $F(I) = A_I \exp(j\Phi_I),$2 (Zhou et al., 10 Aug 2025).

The central operational choice is then defined as follows: the phase from the real LR image is used as a structural prior, and the amplitude from the DT-LR image is used as degradation content after enhancement. The enhancement is performed by an Amplitude Enhancement Network (AENet), which is described as using convolutions and Residual State Space Blocks (RSSB). Its purpose is to make the DT-LR amplitude better mimic the amplitude distribution of real LR images while preserving more structural and textural information than a plain DT-LR or a purely noisy image (Zhou et al., 10 Aug 2025).

The preliminary LR image produced by FGDM is reconstructed through inverse Fourier synthesis from the enhanced amplitude and the real LR phase. This decomposition-and-recomposition strategy is the defining mechanism of FGDM.

4. Mathematical formulation and learning objective

The module is specified with explicit Fourier-domain equations. For DT-LR and real LR images, the transforms are written as

$F(I) = A_I \exp(j\Phi_I),$3

and

$F(I) = A_I \exp(j\Phi_I),$4

The amplitude enhancement stage is defined by a learnable network:

$F(I) = A_I \exp(j\Phi_I),$5

Recomposition using the real LR phase is then expressed as

$F(I) = A_I \exp(j\Phi_I),$6

The training objective reported for AENet is an $F(I) = A_I \exp(j\Phi_I),$7 reconstruction loss between $F(I) = A_I \exp(j\Phi_I),$8 and the real LR images, with the stated purpose of ensuring that the synthesized image is structurally consistent and realistically degraded (Zhou et al., 10 Aug 2025).

These equations define FGDM as a guided spectral recomposition module rather than a generic image generator. This suggests a deliberate separation between the estimation of degradation-bearing spectral magnitude and the injection of structural priors through phase replacement.

5. Relation to RFDM and the full degradation pipeline

FGDM is not presented as an isolated module. In the overall method, LR images are processed by both FGDM and RFDM, producing final synthetic LR images with real-world degradation, and these synthetic LR images are paired with the given HR images to train off-the-shelf super-resolution networks (Zhou et al., 10 Aug 2025).

The division of labor between the two modules is explicitly described. FGDM addresses amplitude and structural mismatch by producing an initial pseudo-LR image that integrates realistic degradation through amplitude modeling with real-world structure through phase guidance. RFDM then addresses the residual distribution gap by using rectified flow, described as a continuous and invertible transformation, to map the distribution of pseudo-LR images from FGDM to the distribution of actual real-world LR images (Zhou et al., 10 Aug 2025).

The reported ablation results in Tab. 2 indicate that FGDM alone performs much better than RFDM alone, and that using both yields the best performance. Accordingly, FGDM should not be interpreted as a replacement for RFDM; the paper characterizes the two modules as complementary.

Component	Function	Effect
FGDM	Initial pseudo-LR synthesis	Provides structure-preserving, realistically degraded LR
RFDM	Distribution refinement via flow	Bridges residual gap to real LR distribution

A common misunderstanding would be to treat FGDM solely as a frequency-domain blur simulation stage. The reported design is narrower and more specific: it uses Fourier phase as a prior for structure and learned amplitude enhancement for degradation, after which RFDM further refines the resulting LR distribution (Zhou et al., 10 Aug 2025).

6. Experimental evidence and reported impact

The reported experiments associate FGDM with both improved LR realism and improved downstream SR performance. Figure 1 and the quantitative PSNR/SSIM comparisons in Tab. 1 are described as showing that LR images generated via FGDM and the full pipeline are much closer in appearance and statistics to real-world LR images than those generated by bicubic, Real-ESRGAN, UDDM, or Syn-Real. The description further states that FGDM preserves more structural details than aggressive downsampling or naive diffusion or adversarial methods (Zhou et al., 10 Aug 2025).

The most direct ablation on the Fourier prior is given in Tab. 4. Removing the Fourier phase prior degrades performance on RealSR from PSNR/SSIM $F(I) = A_I \exp(j\Phi_I),$9 with the Fourier prior to $A_I$0 without it. This result is presented as evidence for the necessity and effectiveness of Fourier phase guidance (Zhou et al., 10 Aug 2025).

For downstream SR, Tab. 1 reports that SwinIR, Real-ESRGAN, and StableSR trained on FGDM+RFDM-generated pairs achieve state-of-the-art PSNR, SSIM, LPIPS, and FID on RealSR and DRealSR datasets. Specific values quoted in the reported summary include SwinIR (Ours) with PSNR/SSIM on RealSR of $A_I$1, described as the best among all, and Real-ESRGAN (Ours) with FID on RealSR of $A_I$2, described as much lower than other methods. Visual results in Figs. 5 and 6 are described as showing fewer structural distortions, more realistic detail, and less artifacting than prior methods (Zhou et al., 10 Aug 2025).

These results place FGDM in a data-synthesis role: its contribution is evaluated not only by the realism of the intermediate LR images, but by the degree to which those images improve the training of existing super-resolution architectures.

7. Interpretation, scope, and practical significance

Theoretical and practical advantages are explicitly attributed to FGDM in the reported summary. On the theoretical side, it uses interpretable Fourier priors to disentangle structural information in phase from degradation information in amplitude, can leverage unpaired real-world LR and HR datasets in an unsupervised or weakly supervised setting, and enhances downstream data-driven degradation modeling while avoiding the pitfalls of diffusion noise or GAN instability (Zhou et al., 10 Aug 2025).

On the practical side, the method is described as generating synthetic pairs that generalize better to real test-time degradations, outperforming bicubic simulation, adversarial methods, and diffusion-only approaches in both distortion metrics such as PSNR and SSIM and perceptual metrics such as LPIPS and FID, and remaining agnostic to SR network architecture so that it can be plugged into existing pipelines (Zhou et al., 10 Aug 2025).

Within this framing, FGDM is best understood as a structurally guided degradation synthesis module. Its defining idea is not simply that Fourier analysis is useful for super-resolution, but that the phase of real LR images can be used as a prior to regularize degradation modeling in the absence of paired supervision. A plausible implication is that FGDM is most relevant in settings where synthetic degradation mismatch, rather than SR backbone capacity, is the dominant bottleneck.