## 2 Proposed Methodology
### Separable Attention Feature Fusion Module (SAT Fusion)  
To achieve efficient feature representation, we designed a feature fusion module based on an improved cross-attention mechanism, comprising two Separable Discrepancy Information Injection Modules (SDIIM) and four Separable Common Information Injection Modules (SCIIM), both utilizing separable cross-attention. Intra-modal features extracted by the gated linear unit were injected into the SDIIM for discrepancy extraction and into the SCIIM for common information fusion. The process is represented as follows:  
\[ \begin{array}{l}Z_{1}=SDIIM(Z_{s},Z_{p}),\\ Z_{2}=SCIIM(Z_{p},Z_{1}),\\ Z_{3}=SCIIM(Z_{s},Z_{2}),\\ Z_{4}=Z_{1}+Z_{3},\\ Z_{5}=SDIIM(Z_{4},Z_{m}),\\ Z_{6}=SCIIM(Z_{m},Z_{5}),\\ Z_{7}=SCIIM(Z_{4},Z_{6}),\\ F(s,p,m)=Z_{7}+Z_{4}\end{array}  \tag{6}\]  
where \(F(s,p,m)\) is the fusion feature containing the difference information and the common information of different spectral data obtained by the final fusion; \(Z_{s}\), \(Z_{p}\), and \(Z_{m}\) are the intra-modal features obtained from the spectral reflectance data, the spectral characteristic parameters, and the spectral similarity metric parameters extracted by the gated linear unit, respectively; \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) are the outputs of the first SDIIM, the first SCIIM, and the second SCIIM, respectively; \(Z_{5}\), \(Z_{6}\), and \(Z_{7}\) are the outputs of the second SDIIM, the third SCIIM, and the fourth SCIIM, respectively. SDIIM was used to extract the discrepancy information between different spectral data, the structure of which is shown in Figure 2.  
The spectral reflectance data and spectral characteristic parameters were first extracted for differential information. The traditional cross-attention was improved by adopting a separable cross-attention structure, which was assumed to be based on \(Z_{s}\) as \(K\), \(V\) and \(Z_{p}\) as \(Q\). The spectral features of different modalities were linearly transformed to generate query vectors (Query, \(Q\)), key vectors (Key, \(K\)), and value vectors (Value, \(V\)), respectively:  
\[ \begin{array}{l}Q=Linear_{Q}(LayerNorm(Z_{p})),\\ V=Linear_{V}(LayerNorm(Z_{s})),\\ K=Linear_{K}(LayerNorm(Z_{s}))\end{array}  \tag{7}\]  
After the linear transformation, \(Q\), \(K\), \(V\in R^{n\times d}\). The elements of the query vector \(Q\) were then normalized along the sequence dimensions by the Softmax function to obtain the context scores \(C_{s}\), where \(C_{s}\in R^{n\times 1}\). \[C_{s}=Soft\text{max}(Q) \tag{8}\]  
The context scores \(C_{s}\) were used to perform a broadcast element-by-element multiplication with elements in the key vector \(K\) and a weighted summation to compute the context vector \(C_{v}\). \(C_{v}\) was subjected to a broadcast element-by-element multiplication with the value vector \(V\) and activated by an activation function (Leaky ReLU) to obtain the public information \(SCM_{QV}\) between \(Q\) and \(V\). This process can be represented as follows:  
\[C_{v}=\sum_{i=1}^{n}C_{s_{i}}\odot K_{i} \tag{9}\]  
\[SCM_{QV}=Leaky\text{Rel}u(C_{v}\odot V) \tag{10}\]  
Then, the public information was removed to obtain information about the differences between \(Q\) and \(V\). This process can be represented as:  
\[SDIIM_{QV}=Linear(V-SCM_{QV}) \tag{11}\]  
To obtain complementary information from the spectral characteristic parameters, we injected the difference information into \(Q\), which can be expressed as:  
\[f_{add}=Q+SDIIM_{QV} \tag{12}\]  
\[f_{SDIIM_{QV}}=Leaky\text{Rel}u(LayerNorm(f_{add})+Linear(LayerNorm(f_{add }))) \tag{13}\]  
Figure 2: The architecture of the Separable Discrepancy Information Injection Module (SDIIM). where _LayerNorm_ (\(\cdot\)) denotes layer normalization and \(f_{SDIIM_{sp}}\) is the output of SDIIM.