## 2 Methods
### Focused Region Detection
\begin{aligned} \text{I}_{\text{src-down}}(\text{x},\text{y})&\text{k}=0\\ \text{DOWN}\left[\text{I}_{\text{gauss}}^{\text{k}-1}(\text{x}, \text{y})*\text{G}(\text{x},\text{y},\sigma)\right]&\text{k}>0 \end{aligned} ^{\prime},\right. \tag{4}\]  
Figure 2: The difference of Gaussians images derived from the adjacent frames with different focal regions of (**a**) lower-left part \(\left(\text{I}_{\text{dog1}}\right)\), (**b**) upper-right part \(\left(\text{I}_{\text{dog2}}\right)\). Next, construct the Laplacian pyramid. Up-sampling the k-th level image \(\rm I_{gauss}^{k}\) of the Gaussian pyramid, and make the difference between \(\rm I_{gauss}^{k-1}\) and the up-sampled image to obtain the \(\rm k-1\) layer Laplace image \(\rm I_{lp}^{k-1}\).This process can be expressed by Equation (5), \"UP\" represents up-sampling, which interpolates with zero padding. \[\rm I_{lp}^{k-1}(x,y)=\rm I_{gauss}^{k-1}(x,y)-UP\Big{[}\rm I_{gauss}^{k}(x,y )\Big{]} \tag{5}\]  
This step is applied to all layers of the Gaussian pyramid to obtain the \(\rm k-1\) levels of the Laplacian pyramid. The Laplacian pyramid contains most of the texture information at different scales and some edge information from the image. The top-level image of the traditional k-level Laplacian pyramid is the corresponding top-level image of the Gaussian pyramid. To further obtain more pronounced high-frequency details, the top-level image of the Gaussian pyramid is replaced with the difference of Gaussians image of the same size for reconstruction. The difference of Gaussians image \(\rm I_{dog}\) is down-sampled \(\rm k\) times to obtain \(\rm I_{dog}^{k}\), with the sample size matching that of the top-level image of the Gaussian pyramid. \(\rm I_{dog}^{k}\) is then used as the top-level image of the Laplacian pyramid for image reconstruction. Pyramid reconstruction can be expressed by Equations (6) and (7). \[\rm I_{dog}^{k}(x,y)= \begin{cases}\rm I_{dog}(x,y)&k=0\\ \rm DOWN}\Big{[}\rm I_{dog}^{k-1}(x,y)\Big{]}&k>0^{\prime}\end{cases}  \tag{6}\]  
\[\rm I_{lpr}^{k-n}(x,y)= \begin{cases}\rm UP\Big{[}\rm I_{dog}^{k}(x,y)\Big{]}+ \rm I_{lp}^{k-1}(x,y)&n=1\\ \rm UP\Big{[}\rm I_{lpr}^{k-n+1}(x,y)\Big{]}+\rm I_{lp}^{k-n}(x,y)&n>1^{\prime} \end{cases}  \tag{7}\]  
Starting from the top-level image \(\rm I_{dog}^{k}\), up-sampling is performed and added to the Laplacian pyramid image at the next lower level \(\rm I_{lp}^{k-1}\) to obtain the reconstructed top-level image \(\rm I_{lpr}^{k-1}\). This process is repeated by up-sampling \(\rm I_{lpr}^{k-1}\) layer by layer and adding the corresponding Laplacian difference images. This loop continues until the image reaches the lowest level, resulting in a high-frequency focused information image reconstructed using the difference of Gaussians image as the top-level image. In the image reconstruction process, the method based on Gaussian image difference enhances the high-frequency components, making texture and edge features more prominent and refined. However, the selection of pyramid levels requires a trade-off between computational efficiency and detail preservation: too many levels can disperse high-frequency information across multiple layers, not only increasing computation time but also reducing fusion quality, while too few levels may lose critical details, resulting in blurred fused images. Through multiple experimental verifications, setting \(\rm k=4\) for a four-level pyramid decomposition and reconstruction achieves the optimal balance for images with a resolution of 1920 \(\times\) 1080. For example, four-layer pyramids are constructed and restored for two different focused regions of the cell slices, resulting in the corresponding focus information maps \(\rm I_{lpr1}\) and \(\rm I_{lpr2}\), as shown in Figure 3. Compared to the original images, pixels with higher gray values in the focus information maps correspond to clearer pixels in the original image, while pixels with lower gray values correspond to more blurred pixels in the original image.