## 2 Materials and Methods
### Methods
#### 2.1.1 The Curvature-Based Method  
To measure how a surface bends in \(\mathbb{R}^{3}\) is to observe changes from each point on the surface. A grayscale image \(l(p\),\(q)\) has \(p\) rows and \(q\) columns, and each combination of rows and columns contains a grayscale value of 0-255. Then, the two-dimensional data \(\mathbb{R}^{2}\) is transformed into the three-dimensional data \(\mathbb{R}^{3}\). Row \(p\) and column \(q\) are transformed into the \(x\)-plane and \(y\)-plane, while the intensity value is transformed into the \(z\)-plane. Then, the curvature in this research is calculated from \(\mathbb{R}^{3}\) of \(l(p\),\(q)\) (Equation (1)), where the third dimension is not the height of the data but the intensity value of the images \(I\):  
\[I_{(p,q)}\to I_{(x,y,z)} \tag{1}\]  
\(I_{(x,y,z)}\) has a surface in space where each point on the surface is parameterized using two coordinates [41]:  
\[\mathbf{x}:\mathbb{R}^{2}\supseteq U\rightarrow\mathbb{R}^{3}:(u,v)\mapsto x(u,v) \tag{2}\]  
The first partial derivative with respect to \(\mathbf{u}\) and \(\mathbf{v}\) is  
\[\mathbf{x}_{1}=\left[\begin{matrix}\frac{\partial\mathbf{x}}{\partial u}&\frac{ \partial\mathbf{y}}{\partial u}&\frac{\partial\mathbf{z}}{\partial u}\end{matrix} \right];\,\mathbf{x}_{2}=\left[\begin{matrix}\frac{\partial\mathbf{x}}{\partial v}& \frac{\partial\mathbf{y}}{\partial v}&\frac{\partial\mathbf{z}}{\partial v}\end{matrix}\right] \tag{3}\]  
From the first partial derivative, we can calculate the first fundamental form (Equation (4)):  
\[I=\left[\begin{matrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{matrix}\right] \tag{4}\]  
Figure 1: The overall flowchart of the proposed idea. Connectors A, B, and C detail the curvature-based, scale-space, and discrete wavelet transform processes.  
where  
\[g_{ij}=x_{i}\cdot x_{j} \tag{5}\]  
The second partial derivative with respect to \(u\) and \(v\) is  
\[x_{11}=\frac{\partial x_{1}}{\partial u};\,x_{22}=\frac{\partial x_{2}}{\partial v };x_{12}=x_{21}=\frac{\partial x_{1}}{\partial v}=\frac{\partial x_{2}}{ \partial u} \tag{6}\]  
Similarly, from the second partial derivative, the second fundamental form (Equation (7)) can be calculated:  
\[\Pi=\begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix} \tag{7}\]  
where  
\[b_{ij}=x_{ij}\cdot n \tag{8}\]  
and  
\[n=\frac{x_{1}\times x_{2}}{\|x_{1}\times x_{2}\|} \tag{9}\]  
There are several essential curvatures for the surface in \(\mathbb{R}^{3}\), Gaussian curvature (\(K\)), mean curvature (\(H\)), and principal curvatures (\(X,N\)) (Equations (10)-(12)) [41]:  
\[K=\frac{b_{11}b_{22}-b_{12}^{2}}{s_{11}s_{22}-s_{12}^{2}} \tag{10}\]  
\[H=\frac{1}{2}\frac{g_{11}b_{22}-2g_{12}b_{12}+g_{22}b_{11}}{g_{11}g_{22}-s_{12 }^{2}} \tag{11}\]  
\[X,N=H\pm\sqrt{H^{2}-K} \tag{12}\]  
These concepts of curvatures help to distinguish different perspectives on a surface. While the mean curvature is an extrinsic measure, the Gaussian curvature is an intrinsic measure. The importance of these measures is that some surfaces can be extrinsically curved but intrinsically flat [42]. The principal curvatures are the shape operator's eigenvalues and the maxima and minima values of a normal curvature. The principal curvatures can be obtained from the Gaussian and mean curvatures (Equation (12)) [43]. The curvature-based method appears in Figure 1 as connector A. In this work, a curvature-based method offers characteristics due to its ability to measure the surface's curvature of the eyes and eyebrows.