## 3 Method
### Frequency-Adaptive Fusion Module (FAF)
#### 3.1.1 Mathematical Model  
The design of the FAF module is based on a key insight: when the illuminance decreases, the variance of high-frequency noise increases exponentially, while the low-frequency signals shift nonlinearly due to Bayer demosaicing artifacts. Our method treats these intrinsically coupled phenomena as a unified problem. **Steerable Spectral Decomposition. ** Given multi-scale features \(F_{hr}\in\mathbb{R}^{C_{hr}\times H\times W}\) and \(F_{lr}\in\mathbb{R}^{C_{lr}\times\frac{H}{2}\times\frac{W}{2}}\), our adaptive decomposition learns the importance of frequency bands:  
\[ \begin{split}\mathcal{H}_{high}(F)&=F*\text{LoG}( \sigma_{l}(l))\\ \mathcal{H}_{low}(F)&=\text{GaussianBlur}(F,\text{ Adaptive}\Sigma(l))\end{split}  \tag{1}\]  
Figure 2: Architecture diagram of the FAF module. where LoG is the Laplacian-of-Gaussian operator, which is used to extract high-frequency components; GaussianBlur applies low-pass filtering; and Adaptive\(\Sigma(l)\) determines the size of the blur kernel based on the illumination level \(l\). The scale parameter \(\sigma_{t}(l)=\alpha_{t}\cdot e^{-\beta_{t}\cdot l}\) decays exponentially as the illumination decreases, modeling the physical behavior of photon-dependent noise. The parameters \(\alpha_{t}\) and \(\beta_{t}\) are calculated as follows:  
\[ \begin{split}\alpha_{t}&=\alpha_{base}+W_{\alpha} \cdot F_{\theta}(I)\\ \beta_{t}&=\beta_{base}+W_{\beta}\cdot F_{\theta}(I )\end{split}  \tag{2}\]  
where \(\alpha_{base}\) and \(\beta_{base}\) are initialized constants (set to 2.0 and 0.5, respectively, in our implementation), \(W_{\alpha}\) and \(W_{\beta}\) are learnable weight matrices, and \(F_{\theta}(I)\) is a global image descriptor extracted through a small convolutional network. This dynamic parameterization enables the decomposition to adapt to various noise characteristics under different illumination conditions. Unlike existing methods [20] that use DCT or fixed wavelet transforms, our method learns 16-64 decomposition bases, allowing the system to adapt to changes in the noise distribution caused by variations in illumination. **Scale-Frequency Attention. ** After the initial fusion \(F_{base}=\mathrm{Y}(\mathcal{H}_{high}(F_{hr})\oplus\mathrm{Y}(\mathcal{H}_{ low}(F_{lr})))\), where \(\mathrm{Y}\) represents a \(1\times 1\) convolution for dimension alignment, we refine the features as follows:  
\[A_{s}=\frac{\exp(W_{s}*F_{base}+b_{s})}{\sum_{s^{\prime}}\exp(W_{s^{\prime}}*F_ {base}+b_{s^{\prime}})} \tag{3}\]  
Each attention head \(A_{s}\) focuses on a frequency band \(s\in\{\mathrm{HF},\mathrm{MF},\mathrm{LF}\}\), achieving the following. **Contextual Suppression**: At 0.1 lux, the HF attention weight drops by 68% to suppress noise (Figure 3). **Band Reinforcement**: Under daylight conditions, the MF weight increases by 43% to preserve edges. Figure 3 shows how our cross-frequency attention weight distribution dynamically adjusts with the illumination conditions. The red markers highlight the optimal 3:7:0 ratio at the dominant feature layer (Level 4), which empirically suppresses high-frequency hallucinations by 37%  
Figure 3: Cross-frequency attention weight distribution across wavelet sub-bands. The \(\lambda_{1}\) (HF attention), \(\lambda_{2}\) (MF attention), and \(\lambda_{3}\) (LF attention) curves demonstrate our dynamic balancing mechanism derived from Theorem 1. Red markers highlight the optimal 3:7:0 ratio at the dominant feature layer (Level 4), empirically suppressing \(\dagger\)37% of the high-frequency hallucinations compared to the baseline. **Nonlinear Band Fusion. ** The final feature combines the spectral components through gradient-aware mixing:  
\[F_{fused}=\sum_{s}A_{s}\odot(\Phi_{s}(M_{s}(F_{base}))) \tag{4}\]  
where \(M_{s}\) is a frequency-specific dense block. \(\Phi_{s}\) employs switched batch normalization conditioned on the illumination level. \(\odot\) represents the Hadamard product.