FAST: Fourier Attention SNR-Prior Transformer

• The paper introduces FAST, which decomposes features into amplitude and phase spectra to fuse RGB and SNR information for enhanced underwater image restoration.
• It utilizes spectral interaction mechanisms to selectively amplify low-frequency informative content while suppressing high-frequency noise.
• Empirical results show significant gains in PSNR, SSIM, and improved texture and color restoration compared to methods using spatial SNR priors.

Searching arXiv for FAST and closely related Fourier-attention/SNR-prior transformer work. arXiv search: "Fourier Attention SNR-prior Transformer FAST SFormer" Fourier Attention SNR-prior Transformer (FAST) is a frequency-domain transformer module introduced in SFormer, a U-shaped dual-branch neural architecture for underwater image enhancement. FAST uses the SNR prior in the frequency domain, decomposing features into amplitude and phase spectra for better channel modulation, and combines spectral interactions with SNR cues to highlight key spectral components. Embedded at skip connections, it fuses an RGB feature stream with an SNR-guided branch, while the Frequency Adaptive Transformer (FAT) bottleneck merges low- and high-frequency branches using a gated attention mechanism (Tian et al., 26 Aug 2025).

1. Research context and motivation

Recent learning-based underwater image enhancement methods have advanced by incorporating physical priors into deep neural networks, particularly using the signal-to-noise ratio (SNR) prior to reduce wavelength-dependent attenuation. FAST is motivated by two stated limitations of spatial domain SNR priors: they cannot effectively separate cross-channel interference, and they provide limited help in amplifying informative structures while suppressing noise (Tian et al., 26 Aug 2025).

The move to the frequency domain is intended to address those limits by decomposing features into amplitude and phase spectra. In the formulation associated with FAST, this decomposition separates information-rich low-frequency and noise high-frequency components, enables targeted enhancement of amplitude and structural information, and allows more effective channel modulation and suppression of wavelength-dependent noise (Tian et al., 26 Aug 2025).

Within this framing, FAST is not presented as a generic Fourier block, but as a mechanism for coupling spectral representations with an explicit physical prior. The module therefore belongs to a line of work in which Fourier operations are used not merely for acceleration, but for task-specific inductive bias in restoration and enhancement.

2. Placement within the SFormer architecture

SFormer is described as a unified U-shaped architecture that integrates a conventional RGB stream with an SNR-guided branch. FAST modules are embedded at multiple skip connections between encoder and decoder, providing frequency-aware SNR-guided fusion at each semantic scale. At the network bottleneck, the Frequency Adaptive Transformer further aggregates global low- and high-frequency features (Tian et al., 26 Aug 2025).

The inputs to FAST at skip connection $i$ are the RGB feature map $x_i$ and the corresponding downsampled SNR prior map $s_i$. SNR maps $S$ are derived from the input and are designed to highlight regions with a low signal-to-noise ratio based on underwater physical properties. The single-channel SNR map is downsampled to match the spatial resolution of each encoder stage for compatibility with features (Tian et al., 26 Aug 2025).

This architectural placement makes FAST a cross-branch fusion operator rather than an isolated attention replacement. Its function is local to the skip pathway, but the intended effect is multi-scale: preserving spatial structures, color fidelity, and fine details while conditioning enhancement on SNR-informed degradation cues.

3. Spectral computation and attention mechanism

FAST begins with standard transformer-style projections. Each feature is projected into queries $Q$, keys $K$, and values $V$ via learned convolutions or linear layers. Both $Q$ and $K$ are then subjected to a Fast Fourier Transform, producing complex-valued spectra that are decomposed into amplitude and phase components (Tian et al., 26 Aug 2025).

For the RGB stream, associated with keys, the spectral decomposition is

$$A_x = \operatorname{Amp}(\mathcal{F}(K)), \qquad P_x = \operatorname{Pha}(\mathcal{F}(K)).$$

For the SNR stream, associated with queries, the decomposition is

$x_i$0

FAST then performs spectral interaction by element-wise multiplication of amplitude spectra and phase spectra:

$x_i$1

The recombined spectrum is mapped back to the spatial domain using inverse FFT:

$x_i$2

The resulting frequency-aware attention map modulates the value tensor through Layer Normalization and residual addition:

$x_i$3

followed by a lightweight MLP refinement:

$x_i$4

The residual connection is described as preserving low-frequency components, ensuring stability and facilitating gradient flow (Tian et al., 26 Aug 2025).

Operationally, the pipeline can be summarized as projection, FFT, amplitude/phase decomposition, spectral interaction, inverse FFT, value modulation, and residual-MLP refinement. In this design, the attention signal is synthesized from cross-branch spectral interaction rather than from a conventional spatial-domain $x_i$5 map.

4. Interpretation of amplitude, phase, and SNR guidance

FAST assigns distinct roles to amplitude and phase spectra. Amplitude is associated with structure and contrast, while phase is associated with edge and geometry. By interacting both amplitude and phase spectra, FAST is intended to ensure that structural geometry and global color/contrast are both optimally restored (Tian et al., 26 Aug 2025).

The claimed advantages over spatial SNR priors follow directly from this decomposition. Frequency decomposition, especially amplitude versus phase, enables separation of color or channel modulation and structure, which is difficult to achieve with spatial SNR priors. Since underwater noise mainly contaminates high-frequency bands such as fine textures and edges, the frequency domain naturally facilitates their suppression or enhancement. FAST attention can selectively preserve low-frequency salient structures and dampen high-frequency noise per the SNR prior (Tian et al., 26 Aug 2025).

This suggests that FAST should be understood less as a generic efficiency device than as a restoration-oriented prior injection mechanism. Its purpose is not only to model dependencies, but to modulate enhancement according to the interaction between meaningful image structures and noise characteristics encoded by the SNR stream.

5. Relation to other Fourier-based transformer designs

FAST should not be conflated with other Fourier-labeled transformer mechanisms, because the term “Fourier” spans several distinct design choices across sequence modeling, vision restoration, and theoretical attention acceleration.

Method	Core Fourier operation	Relation to FAST
Fourier Transformer (He et al., 2023)	DCT via FFT to progressively remove redundancies in hidden sequence	Compresses hidden sequences; self-attention blocks are unchanged
FFAT in ShadowRefiner (Dong et al., 2024)	Attention computed in the frequency domain for refinement	Frequency-domain attention for shadow removal refinement
FourierQK (Zeris, 8 Jul 2026)	Spectral preprocessing applied only to Q/K projections	Preserves full attention score structure without SNR prior

Fourier Transformer inserts parameter-free spectral filters between transformer layers, using the ready-made Fast Fourier Transform operator to perform Discrete Cosine Transformation and progressively remove redundancies in hidden sequence; its emphasis is long-range sequence efficiency and pretrained-model compatibility rather than SNR-guided restoration (He et al., 2023). ShadowRefiner’s Fast-Fourier Attention based Transformer computes attention in the frequency domain to refine shadow removal outputs and mitigate pixel misalignment, but it is presented for shadow detail refinement and color consistency rather than underwater enhancement with an explicit SNR-guided branch (Dong et al., 2024). FourierQK applies FFT-based spectral preprocessing only to learned query-key projections while preserving the full attention score structure, and attributes the benefit to global frequency-domain mixing rather than metric distortion (Zeris, 8 Jul 2026).

Other Fourier-based proposals differ even more substantially. FourierFormer replaces dot-product kernels with generalized Fourier integral kernels in a kernel-regression interpretation of attention (Nguyen et al., 2022). Fast RoPE Attention combines the polynomial method and the Fast Fourier Transform to approximate RoPE attention in almost linear time in the bounded entry regime (Alman et al., 17 May 2025). A separate direction, spectral dictionary learning, replaces self-attention with a learned global time-varying Fourier dictionary and per-token mixing coefficients (Kiruluta, 29 Apr 2025). Against this background, FAST occupies a specific niche: frequency-domain, SNR-guided feature fusion for underwater image enhancement.

6. Empirical performance and significance

SFormer was trained on 4,800 paired images from UIEB, EUVP, and LSUI, and is reported to surpass recent methods with a 3.1 dB gain in PSNR and 0.08 in SSIM, successfully restoring colors, textures, and contrast in underwater scenes (Tian et al., 26 Aug 2025). These results are reported for the full architecture, in which FAST and FAT are joint components.

The FAST-specific contribution is supported in the accompanying description by experiments and ablation studies stating that incorporating FAST leads to significant improvements in PSNR, SSIM, reduced LPIPS and $x_i$6. Visual results are described as demonstrating restoration of natural colors, sharpening of textures, and removal of noise and haze (Tian et al., 26 Aug 2025).

Within the SFormer design, FAST is therefore presented as the mechanism that enforces the SNR prior in the frequency domain at skip connections, while FAT handles bottleneck-level fusion of low- and high-frequency branches. A plausible implication is that the reported gains depend on this division of labor: FAST supplies local spectral guidance tied to degradation priors, and FAT supplies global contextual frequency fusion. In that sense, FAST represents a specialized synthesis of transformer attention, Fourier-domain processing, and physically motivated SNR guidance for underwater restoration.