## 2 Materials and Methods
### Tokenized KAN Block (Tok-KAN)  
The characteristics of landslide images are typically intricate and nonlinear, rendering traditional linear networks inadequate for capturing and processing their nonlinear features. This hinders the effective extraction of complex image details and patterns. The objective of this section is to introduce the Kolmogorov-Arnold network (KAN) into the U-Net++ framework, with the aim of enhancing the model's ability to learn and represent landslide image features. The multilayer perceptron (MLP), inherited from the KAN, is capable of modeling complex nonlinear relationships through a sequence of multi-level nonlinear transformations, thereby enabling it to deal with complex input data. Nevertheless, despite the MLP's exemplary performance in multilayer nonlinear transformations, the decision-making process is not readily comprehensible due to its dependence on a multitude of parameters when processing data, in addition to its status as a deep neural  
Figure 1: Architecture of the proposed LKN-UKAN network. The network consists of two main components: (1) the Tok-KAN (highlighted in grayish green), which enhances nonlinear feature modeling, and (2) the DLFFSKA module (highlighted in blue), which improves global perception through multi-scale feature fusion. network. Consequently, there are still some limitations in terms of parameter efficiency and interpretability. To overcome these shortcomings, the KAN is designed with a more efficient structure based on the Kolmogorov-Arnold representation theorem, which enables the network to improve its interpretability while maintaining its efficiency, thus capturing complex image features more effectively. This innovation opens up new possibilities for improving performance in landslide image segmentation tasks. Based on the efficiency and interpretability of the KAN, the U-KAN has been proposed as a novel network architecture, demonstrating remarkable performance in medical image processing tasks [45]. However, the relatively simple structure of the U-KAN is still insufficient to capture subtle differences in a landslide area. To compensate for this shortcoming, we introduce the Tok-KAN and incorporate it into the U-Net++ framework. The Tok-KAN further enhances the learning ability of landslide image features by effectively extending the KAN based on the nested skip connection and multi-scale feature fusion mechanism of U-Net++. This design enables the model to integrate information and capture details at a deeper level, thereby significantly improving the overall performance of the landslide image segmentation task. Figure 2 shows the overall structure of the Tok-KAN.  
Firstly, the tokenization process is a crucial step in the tokenized KAN (Tok-KAN) module [48]. By performing tokenization, the output features of the convolutional phrase \(\mathbf{X}_{L}\) are reshaped into a flattened sequence of 2D patches \(\left\{\mathbf{X}_{L}^{i}\in P^{2}\cdot C_{L}|i=1,2,\cdots N\right\}\), where each patch has a size of \(P\times P\) and \(N=(H_{L}\times W_{L})/P^{2}\) represents the additional feature augmentation. To achieve this transformation, we employ a trainable linear projection \(\mathbf{E}\in\mathbb{R}^{\left(P^{2}C_{L}\right)\times D}\) through a convolution layer with a kernel size of three, mapping the vectorized patches into a latent D-dimensional embedding space as described by Equation (1):  
\[\mathbf{Z}_{0}=\left[\mathbf{X}_{L}^{1}\mathbf{E};\mathbf{X}_{L}^{2}\mathbf{E };\mathbf{X}_{L}^{3}\mathbf{E};\cdots;\mathbf{X}_{L}^{N}\mathbf{E}\right] \tag{1}\]  
Secondly, the tokenized features are processed by the KAN layer, which further handles the tokenized input sequence. Similar to an MLP, a KAN with k layers can be described as a nested sequence of multiple KAN layers as shown in Equation (2).