# Lenke Classification of Scoliosis Based on Segmentation Network and Adaptive Shape Descriptor
## 2 Methods
### Adaptive Shape Descriptor for Vertebrae
#### 2.4.3 Improving Shape Representation by Quantification of Tilt Angles  
The above two strategies represent the shape of scoliosis in terms of the overall contour and the local maximum curvature of the spine, respectively. In fact, the horizontal inclination of each spine block, that is, the tilt angle of the upper and lower boundaries, also plays an important role in the Lenke classification of scoliosis. An intuitive example can be found in Figure 9, in which the two spinal sequences have similar overall profiles, but their respective upper and lower boundaries have different tilt angles, which may result in them belonging to different Lenke classifications. In this section, we further improve the shape representation and matching based on the vertebra's horizontal tilt angle.  
Figure 8: Examples for key segments in the segmented spinal image.  
Specifically, suppose \(S{=}\left\{\mathbf{p}_{0},\mathbf{p}_{1},\ldots,\mathbf{p}_{m}\right\}\) denotes the left outline point set of the segmented spinal image \(X\), and a horizontal line \(\mathbf{l}_{i}\) is drawn through each \(\mathbf{p}_{i}\), then use \(\mathbf{l}_{i}\) as the basis and calculate the angle \(\theta_{i}\) of the upper or lower boundary corresponding to \(\mathbf{p}_{i}\) in the clockwise direction. After that, we define Formula (9) to calculate the horizontal tilt angle \(\alpha_{i}\) of each spinal block. It is worth noting that Formula (9) calculates the horizontal angle inclination direction of the spinal block, i.e., if \(\theta_{i}\leq\pi/2\), that means the angle \(\alpha_{i}\) is determined by \(\mathbf{l}_{i}\) intersecting the upper or lower boundaries in a clockwise direction, and in this case, \(\alpha_{i}\geq 0\); if \(\theta_{i}>\pi/2\); that means \(\alpha_{i}\) is determined by \(\mathbf{l}_{i}\) intersecting the upper or lower boundaries in a counterclockwise direction, with \(\alpha_{i}\) in this case being <0.  
\[\alpha_{i}=\left\{\begin{array}{l}\theta_{i},\text{ }if\text{ }\theta_{i}\leq\pi/2\\ -(\pi-\theta_{i}),\text{ }if\text{ }\theta_{i}>\pi/2\end{array},i=0,1, \ldots,m\right. \tag{9}\]  
Suppose \(X\) and \(Y\) are two segmented spinal images to be matched, \(S{=}\left\{\mathbf{p}_{0},\mathbf{p}_{1},\ldots,\mathbf{p}_{m}\right\}\) and \(S^{\prime}{=}\left\{\mathbf{p}^{\prime}{}_{0},\mathbf{p}^{\prime}{}_{1},\ldots,\mathbf{p} ^{\prime}{}_{m}\right\}\) are their outline point sets, respectively; \(C_{1}{=}\left\{x_{1},x_{2},\ldots,x_{m}\right\}\) and \(C_{2}{=}\left\{y_{1},y_{2},\ldots,y_{m}\right\}\) are their corresponding coded sequences, respectively. \(\alpha{=}\left\{\alpha_{0},\alpha_{1},\ldots,\alpha_{m}\right\}\) and \(\beta{=}\left\{\beta_{1},\beta_{2},\ldots,\beta_{m}\right\}\) are the horizontal tilt angles of \(X\) and \(Y\), respectively. Then, the dissimilarity between \(\alpha\) and \(\beta\) is defined as formula (10), and the dissimilarity between \(X\) and \(Y\) can be further improved as formula (11).  
\[DisTile\_angles(X,Y)=\sum_{i=0}^{m}\{|\alpha_{i}-\beta_{i}|\} \tag{10}\]  
\[DisSim_{3}(X,Y)=\sum_{i=1}^{m}\{w_{i}\cdot\min((x_{i}-y_{i}+n)\text{mod}n,(y_ {i}-x_{i}+n)\text{mod}n)\}+DisTile\_angles(X,Y) \tag{11}\]