## 3 Methodology
### Multi-Feature Fusion Distance Measure Model of MFF-SLIC
#### 3.2.3 Distance Measure Model Construction of the MFF-SLIC  
Suppose the fusion eigenvector for pixels \(\bigl{(}I_{i},\,I_{j}\bigr{)}\) in image \(I\) with \(M\times N\) is denoted as \((L,a,b,x,y,\theta,\alpha,X,Y,Z)^{\mathsf{T}}\), then for digital images, the calculation formula for distance of color components \(\bigl{(}L_{i},a_{i},b_{i}\bigr{)}^{\mathsf{T}}\) and \(\bigl{(}L_{j},a_{j},b_{j}\bigr{)}^{\mathsf{T}}\) is represented as Equation (19).  
\[d_{Lab}=\sqrt{\bigl{(}L_{j}-L_{i}\bigr{)}^{2}+\bigl{(}a_{j}-a_{i}\bigr{)}^{2}+ \bigl{(}b_{j}-b_{i}\bigr{)}^{2}}, \tag{19}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n_{x}\) & \(n_{y}\) & \(n_{z}\) & **Dip (Direction)** & \(n_{x}\) & \(n_{y}\) & \(n_{z}\) & **Dip (Direction)** \\ \hline \(>\)0 & \(>\)0 & \(>\)0 & NE & \(<\)0 & \(>\)0 & \(>\)0 & NW \\ \(>\)0 & \(>\)0 & \(<\)0 & SW & \(<\)0 & \(>\)0 & \(<\)0 & SE \\ \(>\)0 & \(<\)0 & \(>\)0 & SE & \(<\)0 & \(<\)0 & \(>\)0 & SW \\ \(>\)0 & \(<\)0 & \(<\)0 & NW & \(<\)0 & \(<\)0 & \(<\)0 & NE \\ \hline \hline \end{tabular}
\end{table}
Table 1: The relation of the normal vector and the dip.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**The Normal Vector \(n=\bigl{(}n_{x}n_{y}n_{z}\bigr{)}\)**} & **Dip (Magnitude)** & **Dip (Direction)** & **Dip Angle** \\ \hline \(\bigl{(}n_{x}=0,n_{y}\
eq 0,n_{z}\
eq 0\bigr{)}\) & \(\begin{pmatrix}n_{y}\times n_{z}>0\\ (n_{y}\times n_{z}<0)\end{pmatrix}\) & – & N & \\ \(\bigl{(}n_{x}\times n_{z}>0\bigr{)}\) & – & S & & \\ \hline \(\bigl{(}n_{x}\
eq 0,n_{y}=0,n_{z}\
eq 0\bigr{)}\) & \(\begin{pmatrix}n_{x}\  
The calculation formula for distance of pixel planar position components is described as Equation (20).  
\[d_{xy}=\sqrt{\left(x_{j}-x_{i}\right)^{2}+\left(y_{j}-y_{i}\right)^{2}}, \tag{20}\]  
For the TLS point cloud, the local attitude (the dip angle \(\theta_{i}\) and \(\theta_{j}\), and the dip \(\alpha_{i}\) and \(\alpha_{j}\)) of each point could be calculated as Equation (21).  
\[\left\{\begin{array}{l}d_{\theta}=\left|\theta_{j}-\theta_{i}\right|\\ d_{\alpha}=\left|\alpha_{j}-\alpha_{i}\right|\end{array}\right., \tag{21}\]  
The distance for spatial location components of points in the TLS point cloud could be calculated as Equation (22).  
\[d_{XYZ}=\sqrt{\left(X_{j}-X_{i}\right)^{2}+\left(Y_{j}-Y_{i}\right)^{2}+\left( Z_{j}-Z_{i}\right)^{2}}, \tag{22}\]  
All above distances defined in various standards have different effects on the construction of multi-feature fusion distance measure model, which should be normalized according to some certain criterions. In this paper, the normalized distance measure model constructed by the eigenvector integrated with multiple features is described as Equation (23).  
\[D_{S}=d_{Lab}/m+d_{xy}/S+d_{\theta}/\Delta\theta+d_{\alpha}/\Delta\alpha+d_{ XYZ}/\Delta XYZ, \tag{23}\]  
where \(S\) means the grid interval of superpixels; \(m\) means the compactness control variable of superpixels; \(\Delta\theta\) and \(\Delta\alpha\) respectively means the maximum distance of local dip angle and dip.  
The TLS point cloud and digital images acquired independently are not always associated in a one-to-one relationship. The TLS point cloud is composed of a set of points distributing as a certain density, while the digital image consists of a series of pixels distributing densely and uniformly. For those pixels without corresponding points in the TLS point cloud, the spatial features could be obtained by use of the inverse distance weighting method, the expression of which is shown as Equation (24).  
\[\widetilde{Z}(x_{0},y_{0})=\sum_{i=1}^{n}\omega_{i}Z(x_{i},y_{i}),\text{ where }\omega_{i}=\frac{1}{d_{i}}/\left(\sum_{i=1}^{n}\frac{1}{d_{i}}\right), \tag{24}\]