## 3 Method  
### Two-Dimensional Discrete Wavelet Transform  
Typically, for an image \(f(i,j)\), the definition of 2D Discrete Cosine Transform (DCT) can be expressed as:  
\[ \begin{split} f_{h,w}^{2d}&=\sum\limits_{i=0}^{H-1} \sum\limits_{j=0}^{W-1}x_{i,j}^{2d}\underbrace{\cos\left(\frac{\pi h}{H} \left(i+\frac{1}{2}\right)\right)\cos\left(\frac{\pi w}{W}\left(j+\frac{1}{2} \right)\right)}_{DCT\ weights}\\ s.t.h&\in\left\{0,1,\ldots,H-1\right\},w\in\left\{0, 1,\ldots,W-1\right\}\end{split}  \tag{1}\]  
where \(f_{h,w}^{2d}\) is the 2D Frequency domain, \(x_{i,j}^{2d}\mathbb{R}^{H\times W}\) is the input image, \(\left(\mathrm{H},\mathrm{W}\right)\) are the length and width of the image, respectively. FcaNet [18] demonstrates that global average pooling (GAP) is a special case of Discrete Cosine Transform (DCT). Assuming \(h=w=0\), Equation (1) is:\[ \begin{array}{ll}f_{0,0}^{2d}&=\sum\limits_{i=0}^{H-1}\sum\limits_{j=0}^{W-1}x_{ i,j}^{2d}\text{cos}\left(\frac{0}{H}\left(i+\frac{1}{2}\right)\right)\text{cos} \left(\frac{0}{W}\left(j+\frac{1}{2}\right)\right)\\ &=\sum\limits_{i=0}^{H-1}\sum\limits_{j=0}^{W-1}x_{i,j}^{2d}=GAP\Big{(}x^{2d} \Big{)}HW\end{array}  \tag{2}\]  
where \(f_{0,0}^{2d}\) is the lowest-frequency component in the 2D DCT, and it is proportional to global average pooling. In contrast to 2D DCT, 2D DWT possesses the capability of multi-resolution analysis, which involves decomposing the image into low-frequency and high-frequency components at various scales. This allows DWT to capture both the global and local features of the image simultaneously. Additionally, the DWT decomposition process divides the image into sub-bands with varying frequencies, allowing for image denoising by selectively retaining or discarding high-frequency components. For an image \(f(i,j)\), the 2-D wavelet decomposition into four sub-bands is represented as:  
\[ \begin{array}{ll}I_{\varphi}(i,j)\frac{H}{2}\times\frac{W}{2}=\frac{1}{ \sqrt{HW}}\sum\limits_{i=0}^{H-1}\sum\limits_{j=0}^{W-1}x_{i,j}^{2d}\text{ }_{H\times W}\cdot\varphi(i,j)\\ I_{\psi}^{k}(i,j)\frac{H}{2}\times\frac{W}{2}=\frac{1}{\sqrt{HW}}\sum\limits_{i= 0}^{H-1}\sum\limits_{j=0}^{H-1}x_{i,j}^{2d}\text{ }_{H\times W}\cdot\psi^{k}(i,j)\end{array}  \tag{3}\]  
where \(x_{i,j}^{2d}\varepsilon\mathbb{R}^{H\times W}\) is the input image, \((H,W)\) are the length and width of the image, respectively. \(I_{\varphi}\) represents A sub-band, which is the low-frequency component and has been proven to approximate global average pooling (GAP). \(k\in\{H,V,D\}\), where \(I_{\psi}^{H}\) is horizontal high-frequency components, \(I_{\psi}^{V}\) is vertical high-frequency component, \(I_{\psi}^{D}\) is diagonal high-frequency components. \(\varphi(i,j)\) and \(\psi^{k}(i,j)\) are the corresponding scaling functions of the sub-bands. The scale function \(\varphi(i,j)\) output corresponds to the low-frequency component of the original signal, and the wavelet function \(\psi^{k}(i,j)\) output corresponds to the high-frequency component of the original signal. The scale function of the orthogonal wavelet basis is separable, allowing specific one-dimensional filters to be applied separately to the rows and columns of the image to obtain the desired overall 2-D response. Therefore, the 2D DWT consists of the horizontal (\(i\)) and vertical (\(j\)) outputs of the 1D DWT, including a scale function \(\varphi(\cdot)\) and three wavelet functions \(\psi^{k}(\cdot)\). \[ \begin{array}{ll}\varphi(i,j)=\varphi(i)\varphi(j)\\ \psi^{H}(i,j)=\varphi(i)\psi(j)\\ \psi^{V}(i,j)=\psi(i)\varphi(j)\\ \psi^{D}(i,j)=\psi(i)\psi(j)\end{array}  \tag{4}\]  
where \((i,j)\) are the row and column of the 2D signal, respectively. \(\varphi(\cdot)\) and \(\psi(\cdot)\) is low-pass filter and high-pass filter, respectively. Figureinformation content and is mainly composed of noise signals. Reference [47] proposes that incorporating particular wavelet bands in complex scene recognition tasks can capture minor variations in the image, which is useful for detecting tiny objects.