I am following a papre suggesting a vertex descriptor based on normal and geometry among other features (2.1. Vertex distinctness).
I could not understand how do you extract a vertex's information such as a normal or shape from object that is similar to a simple 3-D point. I can see the definition of a vertex in polygon mesh includes color, texture and normal and I can definitely understand how I can generate a 3D point cloud with information regarding color and texture. I am still lost on the definition of a normal or geometry of a point.
Is there an intuitive description of these characteristics?
Given an unstructured (meshless) point cloud $P=\{p_i\}$, estimating the normals of $P$ usually requires an estimation of the underlying surface. That is, it is assumed that the $p_i$'s are samples from some underlying true manifold.
Here's one simple approach to estimating the normal $n_i$ at $p_i$. Fix $k\in\mathbb{N}$ and denote $N_k(p)$ as the $k$ nearest neighbours of $p$. Compute the centroid $$ q_i = \frac{1}{k}\sum_{p\,\in\, N_k(p_i)} p $$ and local covariance matrix $$ \Sigma(p_i) = \frac{1}{k-1}\sum_{p\,\in\, N_k(p_i)} (p-q_i)(p-q_i)^T $$ and then the eigenvectors $v_j(p_i)=\text{eigvec}_j(\Sigma(p_i))$. Then, in $\mathbb{R}^3$, $n_i = v_3(p_i)$. The two tangent vectors to the estimated surface are then $T_{i}^1=v_1(p_i)$ and $T_{i}^2=v_2(p_i)$.
Obviously, this approach has problems when two far-away parts of the surface (in terms of geodesic distance) are near each other in the ambient space, or when the sampling is too sparse or uneven.
See also here. For more sophisticated approaches, see, for example, Estimating Surface Normals in Noisy Point Cloud Data by Mitra and Nguyen.