Wavelet Fourier Diffuser (WFDiffuser)

• Wavelet Fourier Diffuser (WFDiffuser) is a hybrid diffusion framework that integrates wavelet decomposition for spatial localization with Fourier transforms for spectral structure analysis.
• It is applied in image synthesis, reinforcement learning, underwater image restoration, and low-light enhancement, showcasing versatile, domain-specific adaptations.
• The framework employs a multiresolution strategy, explicitly separating and modeling low- and high-frequency components to achieve enhanced performance in signal processing tasks.

Searching arXiv for the cited WFDiffuser variants and closely related papers. Wavelet Fourier Diffuser (WFDiffuser) is a designation used in several arXiv works for diffusion-based frameworks that combine wavelet decomposition with Fourier-domain processing in order to model low- and high-frequency structure explicitly. In the cited literature, the name appears in a hybrid spectral diffusion model for image synthesis, an offline reinforcement-learning framework for trajectory modeling, an underwater image restoration system under the variant name WF-Diff, and a low-light enhancement method abbreviated CFWD (Kiruluta et al., 4 Apr 2025, Luo et al., 4 Sep 2025, Zhao et al., 2023, Xue et al., 2024). Across these usages, the recurring pattern is a multiresolution representation in which wavelets provide spatial localization, Fourier transforms provide frequency-domain structure, and diffusion supplies the stochastic forward corruption and learned reverse denoising dynamics.

1. Terminological scope and research usage

The term does not denote a single fixed architecture in the current literature. Rather, the available arXiv usages describe distinct frameworks that share a wavelet-Fourier-diffusion design pattern. In image synthesis, "A Hybrid Wavelet-Fourier Method for Next-Generation Conditional Diffusion Models" introduces a framework named Wavelet-Fourier-Diffusion for high-quality, high-fidelity image generation with improved spatial localization (Kiruluta et al., 4 Apr 2025). In offline reinforcement learning, "Wavelet Fourier Diffuser: Frequency-Aware Diffusion Model for Reinforcement Learning" uses the same name for a trajectory model that decomposes sequences by Discrete Wavelet Transform and extracts frequency-domain features by Short-Time Fourier Transform (Luo et al., 4 Sep 2025). In underwater image restoration, "Wavelet-based Fourier Information Interaction with Frequency Diffusion Adjustment for Underwater Image Restoration" uses the variant WF-Diff for a two-stage restoration pipeline (Zhao et al., 2023). In low-light enhancement, "Low-light Image Enhancement via CLIP-Fourier Guided Wavelet Diffusion" presents CFWD, which combines wavelet diffusion, Fourier guidance, and CLIP-based supervision (Xue et al., 2024).

Paper	Setting	Core formulation
(Kiruluta et al., 4 Apr 2025)	Conditional and unconditional image generation	wavelet sub-band decomposition with partial Fourier steps
(Luo et al., 4 Sep 2025)	Offline reinforcement learning	DWT, STFT, and cross attention mechanisms
(Zhao et al., 2023)	Underwater image restoration	WFI2-net and FRDAM
(Xue et al., 2024)	Low-light image enhancement	CLIP-Fourier Guided Wavelet Diffusion

A common misconception is that WFDiffuser is merely a pixel-space diffusion model preceded by a frequency transform. The cited works instead place the decomposition and frequency interaction inside the modeling pipeline itself: as a hybrid spectral Markov chain in image generation, as cross-frequency conditioning for trajectory diffusion in offline RL, or as residual diffusion in wavelet space for restoration and enhancement.

2. Shared representational principle: wavelets for localization, Fourier analysis for spectral structure

The principal mathematical motif is a separation of signal content into low- and high-frequency components. In the image-generation formulation, an image $x_0 \in \mathbb{R}^{H \times W \times 3}$ is mapped by a multi-transform operator

$$W(x_0) = \bigl(x_0^{LF}, \{x_0^{HF,k}\}_{k=1\ldots K}\bigr),$$

where $x_0^{LF}$ is the low-frequency wavelet sub-band and $x_0^{HF,k}$ are the $K$ high-frequency sub-bands. The low-frequency band is further mapped to partial Fourier coefficients,

$X_0 = \mathcal{F}(x_0^{LF}),$

yielding the hybrid tuple $z_0 \equiv (X_0, \{x_0^{HF,k}\})$ (Kiruluta et al., 4 Apr 2025).

In the reinforcement-learning formulation, a fixed-length state sequence $\tau = (s_0,s_1,\ldots,s_{H-1})$ is decomposed by a 1-level Haar DWT into

$$(\tau_{low}, \tau_{high}) = \mathrm{DWT}(\tau), \qquad \tau = \mathrm{IDWT}(\tau_{low}, \tau_{high}),$$

with exact recovery because Haar is orthogonal (Luo et al., 4 Sep 2025). In the underwater restoration work, a 1-level 2D discrete Haar wavelet yields [ (I_{LL}, I_{