# WIG-Net: Wavelet-Based Defocus Deblurring with IFA and GCN
## 3 Methods
### Wavelet Transform  
In two-dimensional Discrete Wavelet Transform (2D DWT), the input signal is divided into four filter bands: LL, LH, HL, and HH. Initially, a one-dimensional (1D) DWT is applied to each row of the image, resulting in a low-frequency component (L) and a high-frequency component (H) in the horizontal direction. Subsequently, another 1D DWT is performed on each column of the transformed data, generating four subbands:  
LL Subband: This subband contains low frequencies in both horizontal and vertical directions. LH Subband: This subband represents low frequency in the horizontal direction and high frequency in the vertical direction.  
HL Subband: This subband denotes high frequency in the horizontal direction and low frequency in the vertical direction.  
HH Subband: This subband represents high frequency in both horizontal and vertical directions.  
We employed 1D filters \(\phi\)(x) and \(\varphi\)(x) for filtering and horizontal downsampling of each column of the image. Subsequently, we employed two filters for filtering and vertical downsampling of each row. As a result, four sub-images, \(\mathrm{I_{LL}}\), \(\mathrm{I_{LH}}\), \(\mathrm{I_{HL}}\), and \(\mathrm{I_{HH}}\), could be computed. The 2D DWT can be represented as follows:  
\[\begin{split}\psi_{LL}(\mathrm{x},\mathrm{y})&=\phi (\mathrm{x})\phi(\mathrm{y})\\ \psi_{LH}(\mathrm{x},\mathrm{y})&=\phi(\mathrm{x}) \varphi(\mathrm{y})\\ \psi_{HL}(\mathrm{x},\mathrm{y})&=\varphi(\mathrm{x} )\phi(\mathrm{y})\\ \psi_{HH}(\mathrm{x},\mathrm{y})&=\varphi(\mathrm{x} )\varphi(\mathrm{y})\end{split} \tag{1}\]  
Conversely, during the reconstruction phase of 2D DWT, the inverse one-dimensional discrete wavelet transform was applied first to each column of the transformed result, followed by a similar operation on each row of the transformed data to obtain the reconstructed image. To summarize, the wavelet decomposition process of an image involves  
Figure 2: Schematic illustration of the proposed network architecture, which was built upon the U-Net framework, integrating wavelet transforms into the encoding stage and replacing conventional pooling layers. In the decoding phase, wavelet inverse transforms in lieu of upsampling layers were utilized. Furthermore, IFA modules and GCN layers between the encoding and decoding stages were introduced.  
separating the signal into low and high frequencies, and if necessary, the LL subband can be further decomposed until the desired level of detail is achieved.  
In this work, the pooling and upsampling layers in the U-net were substituted with discrete wavelet transform (DWT) and inverse discrete wavelet transform (IDWT), respectively. During the encoding phase, a series of convolutional operations and DWT were employed to extract informative features from the input image. Subsequent to DWT processing, the data size was reduced to one-quarter of its original size, whereas the number of channels was quadrupled (Figure 3). Unlike conventional pooling layers that may lead to the loss of some original information due to methods like merging or adjusting convolutional strides to reduce data size, our approach can mitigate the data loss caused by this step.