## 2 Related Work  
### Basic Geometrical Features  
In this subsection, we discuss the two conventional geometrical feature-based relief extraction methods: height (depth)-based and curvature-based methods. The height-based method is one of the simplest and most prominent methods for relief extraction [8]. The height estimation of the new surface point is considered using a given path on the mesh surface with the surface normal vector. The height estimate \(\hat{h}(\mathbf{v}_{n})\) at \(\mathbf{v}_{n}\) on the surface along the path from \(\mathbf{v}_{0}\) to \(\mathbf{v}_{n}\) is given as  
\[\hat{h}(\mathbf{v}_{n})=h(\mathbf{v}_{0})+\sum_{i=1}^{n}((\mathbf{v}_{i}- \mathbf{v}_{i-1})\cdot\mathbf{n}_{B}(e_{i,i-1})), \tag{1}\]  
where \(\mathbf{v}_{0}\) is the reference point location on the surface mesh, and the vertices along the path are \(\{\mathbf{v}_{i}\}\), such that \(0\leq i\leq n\), and \(\mathbf{n}_{B}(e_{i,i-1})\) is the base normals associated with each corresponding edge \((e_{i,i-1})\) between two vertices. Assuming that the heights of the reliefs and background follow Gaussian distributions, respectively, the Gaussian mixture model is applied to determine the threshold that intersects the two height distributions. This implies that the accuracy of this technique relies on the selection of a good threshold value. Furthermore, depth cannot be an absolute measure for detecting reliefs because dents and cracks on the surface of the stele can also be deep, particularly in a severely weathered stele.  
For the curvature-based methods, the authors exploited the distribution of surface curvatures to extract reliefs from wall carving [7; 18]. Letting \(\kappa_{min}\) and \(\kappa_{max}\) (\(|\kappa_{min}|\leq|\kappa_{max}|\)) denote the minimal and maximal curvatures, respectively. In a relief region, it is expected that \(|\kappa_{min}|\approx 0\) and \(|\kappa_{max}|\geq 0\). The Frangi filter is modified and applied to measure the vesselness, Y, at each vertex within the surface mesh as [17]  
\[\mathrm{Y}(\mathbf{v}_{i})=\begin{cases}0,&\text{if }\kappa_{max}<0,\\ \exp\biggl{(}\frac{-A^{2}}{2\alpha^{2}}\biggr{)}\cdot\biggl{(}1-\exp\biggl{(} \frac{-B^{2}}{2\beta^{2}}\biggr{)}\biggr{)},&\text{otherwise},\end{cases} \tag{2}\]  
where \(A\) and \(B\) are \(\frac{\kappa_{min}}{\kappa_{max}}\) and \(\sqrt{\kappa_{min}^{2}+\kappa_{max}^{2}}\), respectively, and \(\alpha\) and \(\beta\) are parameters that are used to control the effect of the first and second term of the filter. From Equation (2), when \(\kappa_{max}\geq 0\), the first term identifies regions of interest relating to the local shape of the surface. This term becomes large when the local surface is curved like a valley. The second term is related to the structure of the local surface and increases as the valley becomes steep and deep.  
After the vesselness, Y, for all the vertices has been determined, vertices with non-zero values of Y are identified as reliefs and the zero values as background. However, the computation of the curvatures requires derivative operation, and thus is prone to be affected by noise from the scanning process and the roughness of the surfaces of the monuments. These noisy curvatures give inaccurate vesselness values because of the small marginal difference in the principal curvatures, which result in unsatisfactory reliefs.