## 2 Materials and Methods
### Element Geometric Transformation  
After the image segmentation, we obtained the geometric features of each element in the corresponding scene image. These geometric features can be used to demonstrate the structural relationship between various elements in RSSIs and to generate RSSIs.  
In order to generate a new RSSI that is distinct from the original RSSIs, it is necessary to transform the geometric features of the extracted elements for the purpose of reconstructing the entire RSSI. The transformation that we performed involves enlargement, reduction, translation, and symmetry (on both X- and Y-axes). To complete the mathematical description of feature transformation, we first abstracted the extracted feature as a polygon. We used a homogeneous coordinate matrix P to record the coordinates of each vertex counterclockwise from a vertex of the polygon. Assuming the coordinates of each vertex of a polygon to be \((x_{0},y_{0})\)\((x_{1},y_{1})\ldots(x_{n},y_{n})\), respectively, in the counterclockwise direction, the corresponding homogeneous coordinate matrix is given by  
\[\left(\begin{array}{cccc}x_{0}&y_{0}&1\\ x_{1}&y_{1}&1\\ \ldots&\ldots&\ldots\\ x_{n}&y_{n}&1\end{array}\right). \tag{1}\]  
Subsequently, we defined a \(3\times 3\) geometric transformation matrix \(T\), which is used for geometric transformation of polygons. The transformation matrices of the geometric transformation were implemented as follows.  
Enlargement and reduction were given by  
\[T=\left(\begin{array}{cccc}S_{x}&0&0\\ 0&S_{y}&0\\ 0&0&1\end{array}\right) \tag{2}\]  
where \(S_{x}\) and \(S_{y}\) represent the proportion of enlargement and reduction. When \(S_{x}\) and \(S_{y}\) are greater than 1, the operation is enlargement. When \(S_{x}\) and \(S_{y}\) are less than 1, the operation is reduction.  
The translation operation was performed as follows.  
\[T=\left(\begin{array}{cccc}1&0&0\\ 0&1&0\\ T_{x}&T_{y}&1\end{array}\right), \tag{3}\]  
where \(T_{x}\) and \(T_{y}\) represent the translation distance of features in \(x\) and \(y\) directions.  
Counterclockwise rotation was performed as follows.  
\[T=\left(\begin{array}{ccc}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{array}\right) \tag{4}\]  
where \(\theta\) is the counterclockwise rotation angle of the element.  
Clockwise rotation was performed as follows.  
\[T=\left(\begin{array}{ccc}\cos(-\theta)&\sin(-\theta)&0\\ -\sin(-\theta)&\cos(-\theta)&0\\ 0&0&1\end{array}\right), \tag{5}\]  
where \(\theta\) is the counterclockwise rotation angle of the element.  
The \(X\)-axis symmetry transformation is given by  
\[T=\left(\begin{array}{ccc}1&0&0\\ 0&-1&0\\ 0&0&1\end{array}\right), \tag{6}\]  
While \(Y\)-axis symmetry is given by  
\[T=\left(\begin{array}{ccc}-1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right), \tag{7}\]  
Assuming that the coordinates of each vertex of the transformed polygon are \((x_{0}^{\prime},{y_{0}}^{\prime})(x_{1}{}^{\prime},{y_{1}}^{\prime})\ldots \ldots\ldots(x_{n}{}^{\prime},{y_{n}}^{\prime})\), consequently, the homogeneous coordinate matrix P composed of them is given by  
\[\left(\begin{array}{ccc}{x_{0}}^{\prime}&{y_{n}}^{\prime}&1\\ {y_{0}}^{\prime}&{y_{1}}^{\prime}&1\\ \ldots&\ldots&\ldots\\ {x_{n}}^{\prime}&{y_{n}}^{\prime}&1\end{array}\right). \tag{8}\]  
After we perform these basic geometric transformations, we can compound these basic geometric transformations, and the corresponding geometric transformation formula is given by  
\[P^{\prime}=PT_{1}T_{2}\ldots \tag{9}\]  
Then, \(T_{i}\) is used to choose the corresponding transformation matrix according to the transformation needed. The significance of the formula is that the geometric transformation represented by \(T_{1}\) matrix is carried out first, and on this basis, the geometric transformation represented by \(T_{2}\) matrix is carried out, and so on.  
After the transformation, we could obtain the new structure of the reconstructed RSSI, which lays the foundation for the next section.