## 3 Method
### Spatial-Channel Coordinated Attention Mechanism (SCCA)
#### 3.2.1 Mathematical Model  
SCCA is based on a key insight: under conditions of insufficient illumination, the distribution of visual information of different frequency components is unbalanced. Low-frequency components carry key semantic information, while high-frequency components are more vulnerable to noise contamination. Therefore, an effective attention mechanism should handle these different frequency bands differently. **Entropy-Aware Channel Attention. ** Given the input features \(X\in\mathbb{R}^{C\times H\times W}\), we first calculate the frequency-separated moment matrices:  
\[M_{HF}=\mathbb{E}[X*K_{sobel}],\quad M_{LF}=\mathbb{E}[X*G_{\sigma}] \tag{6}\]  
where \(\mathbb{E}[\cdot]\) represents the expectation operator. \(K_{sobel}\) is the Sobel edge detection kernel, which is used to extract high-frequency components.\(G_{\sigma}\) is a Gaussian kernel with a standard deviation of \(\sigma\), which is used to capture low-frequency information. The moment matrices \(M_{HF}\) and \(M_{LF}\) represent the statistical characteristics of the feature maps in different frequency bands. The channel weights are predicted through dual-path mixing  
\[A_{c}=\Omega([MLP(M_{LF});DepthConv(M_{HF})]) \tag{7}\]  
Figure 4: Architecture diagram of the SCCA module. where \(\Omega\) is our innovative spectral gating unit:  
\[\Omega(x)=\frac{1}{1+e^{-(\alpha\cdot LPF(x)+\beta\cdot HPF(x))}}\]  
The parameters \(\alpha\) and \(\beta\) are dynamically predicted based on the input signal-to-noise ratio (SNR)  
\[\alpha =\sigma(w_{\alpha}\cdot SNR+b_{\alpha}) \tag{8}\] \[\beta =\tau\cdot\exp(-w_{\beta}\cdot SNR)+b_{\beta}\]  
where the SNR is estimated as follows:  
\[SNR=10\cdot\log_{10}\left(\frac{n_{signal}^{2}}{\sigma_{noise}^{2}}\right) \tag{9}\]  
\(\mu_{signal}\) is calculated as the average intensity of the input feature map, and \(\sigma_{noise}^{2}\) is estimated using the robust median absolute deviation (MAD) method in high-frequency components:  
\[\sigma_{noise}^{2}=\left(\frac{median(|X*K_{high}|)}{0.6745}\right)^{2} \tag{10}\]  
\(K_{high}\) is a high-pass filter kernel. The coefficients \(w_{\alpha}\), \(b_{\alpha}\), \(w_{\beta}\), \(b_{\beta}\), and \(\tau\) are learnable parameters that are optimized during training. This adaptive formula allows \(\alpha\) to control the contribution of high-frequency components (approaching 1 for a high SNR and 0 for a low SNR), while \(\beta\) modulates the low-frequency components using an exponential decay term that remains stable under severe noise conditions. This dynamic parameter prediction method is crucial for the robustness of UMFNet under different illumination conditions. **Decomposed Spatial Attention. ** We decompose the spatial attention into structural (LF) and textural (HF) components:  
\[ \begin{split} A_{s}^{LF}&=\sigma(W_{1}*DownSample(X ))\\ A_{s}^{HF}&=\tanh(W_{2}*(X\ominus MedianBlur(X)))\\ A_{s}&=LearnableSoftmax([A_{s}^{LF};A_{s}^{HF}]) \end{split}  \tag{11}\]  
where \(\sigma\) is the sigmoid activation function. \(\tanh\) is the hyperbolic tangent function. \(\ominus\) represents element-wise subtraction. \(LearnableSoftmax\) is a normalization function, which applies a learnable temperature parameter to the standard softmax. Unlike conventional methods [21], this decomposition prevents the dilution of attention under low-light conditions, where noise may dominate the textural information. **Theorem 2** (SCCA Optimality). : _SCCA minimizes the spectral confusion risk \(R=\mathbb{E}[||A_{ideal}-A_{pred}||^{2}]\) when_  
\[\frac{\partial L_{det}}{\partial A_{c}}=\lambda\frac{\partial L_{denoise}}{ \partial A_{s}} \tag{12}\]  
The proof utilizes nested cross-correlation analysis (Appendix A.2). **Phase-Aligned Fusion.