## 2 Materials and Methods
### Methods
#### 2.3.2 SegNet Neural Network  
SegNet is a deep convolutional neural network architecture widely used in remote sensing image analysis. Its structure consists of an encoder and a decoder; the encoder progressively extracts image features through convolution and pooling layers, while the decoder up-samples by using pooling indices, effectively restoring spatial resolution [69]. Compared to traditional deconvolution methods, SegNet's upsampling process is more lightweight and precise, enabling it to maintain computational efficiency while accurately reconstructing image details. This makes it particularly suitable for segmenting urban parcels in large-scale remote sensing images. In identifying urban spatial boundaries, SegNet is trained on annotated data, using a cross-entropy loss function to optimize the model, allowing it to distinguish urban areas from non-urban regions. After predicting new remote sensing images with the trained model, threshold processing is applied to generate binary images, followed by denoising and smoothing of the results. This allows for quantitative analysis of urban spatial range, supporting studies on urban expansion and spatial layout [70]. Overall, SegNet, as a deep convolutional neural network model, demonstrates significant advantages in processing large-scale remote sensing imagery. Compared to traditional methods, SegNet preserves pooling indices, enabling efficient feature extraction while accurately restoring the spatial resolution of images. Given the multi-centered and complex characteristics of the Pearl River Delta urban agglomeration, SegNet efficiently identifies core cities and peripheral functional zones, providing stronger support for the dynamic identification of spatial ranges [71]. The formula for SegNet is as follows.  
Encoder Part:  
\[Z^{(l)}=f(W^{(l)}*X+b^{(l)}) \tag{2}\]  
where \(Z^{(l)}\) is the feature map of the \(l\)-th layer, \(W^{(l)}\) is the convolution kernel of the \(l\)-th layer, \(*\) is the convolution operation, \(b^{(l)}\) is the bias term of the \(l\)-th layer, and \(f\) is the nonlinear activation function.  
\[P^{(l)}=MaxPool(Z^{(l)}) \tag{3}\]  
Here, \(P^{(l)}\) is the pooled feature map.  
Decoder Part:  
\[Z^{(l)}=Unpool(P^{(l)},pooling\ indices) \tag{4}\]  
where \(Z^{(l)}\) is the up-sampling result at the \(l\)-th layer. The up-sampled feature map is smoothed and refined through convolution:  
\[X^{(l)}=W^{(l)}*Z^{(l)}+b^{(l)} \tag{5}\]  
Final Output:  
\[\hat{g}_{i}=Softmax(X^{(l)}) \tag{6}\]  
Here, \(\hat{g}_{i}\) is the predicted class for pixel \(i\), and \(l\) is the final layer.  
In urban spatial range identification tasks, SegNet uses high-resolution remote sensing images as input data, covering large-scale urban areas with each pixel labeled (e.g., buildings, roads, green spaces). These data are fed into the SegNet model, where the encoder extracts deep-level features through convolution and pooling operations, simultaneously reducing spatial resolution. The decoder then uses saved pooling indices for up-sampling to restore the image resolution. After training, the model can generate pixel-level segmentation maps from remote sensing images, precisely extracting areas of buildings, streets, and other urban infrastructure. This enables analysis of urban spatial layout and expansion characteristics.