# Low-Resolution Target Detection with High-Frequency Information Preservation
## 3 Method
### Dilated Frequency-Aware Channel Correlation  
DFACC aims to preserve task-relevant, high-frequency information in features during the downsampling process. In the design of the DFACC component, we adopt the space-to-depth (STD) operation to obtain submaps from feature maps, \(x\in\mathbb{R}^{C\times H\times W}\), scaled by a scale factor of \(s\), as formulated in Equation (1):\[\left\{ \begin{array}{l}x_{(0,0)}=x[0:H:\mathrm{s},0:W:\mathrm{s},:],\ldots,\\ x_{(s-1,0)}=x[\mathrm{s}-1:H:\mathrm{s},0:W:\mathrm{s},:],\ldots,\\ x_{(0,1)}=x[0:H:\mathrm{s},1:W:\mathrm{s},:],\ldots,\\ x_{(s-1,1)}=x[\mathrm{s}-1:H:\mathrm{s},1:W:\mathrm{s},:]\\ \vdots\\ x_{(0,s-1)}=x[0:H:\mathrm{s},W-1:W:\mathrm{s},:],\ldots,\\ x_{(s-1,\,s-1)}=x[\mathrm{s}-1:H:\mathrm{s},W-1:W:\mathrm{s},:].\end{array} \right. \tag{1}\]  
In the following step, the STD operation will concatenate these submaps along the channel dimension to form feature maps with \(s^{2}C\) channels, as formulated in Equation (2):  
\[y=concat(x_{(0,0)},\ldots,x_{(s-1,0)},\ldots,x_{(0,s-1)},\ldots,x_{(s-1,s-1)}). \tag{2}\]  
To reduce the amount of redundant information in features and avoid unnecessary computational costs, feature fusion is necessary. However, conventional convolution-based feature fusion methods, such as \(1\times 1\) convolution, tend to prioritize the encoding of low-frequency information while weakening the encoding of high-frequency information. To prevent the loss of high-frequency detail caused by downsampling, it is necessary to develop a more effective approach to encode both low- and high-frequency information. Discrete cosine transform (DCT) [22], a widely used transformation technique, is faster and more computationally efficient than discrete Fourier transform (DFT) and allows for the rapid transformation of images into direct-current (DC) and alternating-current (AC) components in the frequency domain. It converts low-frequency information in feature maps into DC coefficients and captures high-frequency details as AC coefficients, making it easier to process information from different frequency domains. This aligns with our goal of effectively handling information from different frequency domains. Building upon this, we leverage it to guide the dimension reduction process in DFACC. Specifically, we begin by applying the DCT to an input feature, \(y_{(\cdot,\cdot,\varepsilon)}\in\mathbb{R}^{\frac{H}{2}\times\frac{W}{2}}\), to extract the corresponding frequency coefficients, \(e_{(\cdot,\cdot,\varepsilon)}\in\mathbb{R}^{\frac{H}{2}\times\frac{W}{2}}\), as formulated in Equation (3):  
\[e_{(h,\mu,\varepsilon)}=\sum_{p=0}^{H/2-1}\sum_{q=0}^{W/2-1}y_{p,q,\varepsilon }\underbrace{\left(\cos\left(\frac{\pi h}{H/2}(p+\frac{1}{2})\right)\cos\left( \frac{\pi w}{W/2}(q+\frac{1}{2})\right)\right)}_{\text{DCT weights}} \tag{3}\]  
where \(h\in\{0,1,\cdots,\frac{H}{2}-1\}\) and \(w\in\{0,1,\cdots,\frac{W}{2}-1\}\). Then, we feed the frequency coefficients into a linear transformation function, \(h(\cdot)\), to obtain multifrequency embeddings. The multifrequency embeddings are then passed through Global Average Pooling and Global Max Pooling separately and fed into a shared multi-layer perceptron (MLP), leading to multifrequency features with enhanced high-frequency representation. This whole processes is shown in Equation (4):  
\[g(\cdot)=\text{MLP}(GlobalMaxPool(\cdot),\theta)+\text{MLP}(GlobalAvgPool( \cdot),\theta) \tag{4}\]  
To reduce potential interference caused by channel fusion and preserve task-relevant high-frequency information in downsampling, we combine the two vectors obtained from the outputs of the MLP by adding them together and then apply the combined vector as high-frequency-aware channel weights to the downsampled features before the channel fusion process. This is formulated in Equation (5):  
\[\left\{ \begin{array}{l}y_{f}=h(\text{DCT}(y)),\\ y_{\text{DFACC}}=\text{ChannelFusion}(g(y_{f})*y),\end{array} \right.