# Line Structure Extraction from LiDAR Point Cloud Based on the Persistence of Tensor Feature
## 3 Methodology
### The Refinement of Initial Tensor Using Different Dimensional Geometric Feature  
With tensor \(T\) computed through initial tensor voting process, the geometric feature in each dimension can be computed based on the tensor decomposition of \(T\). In \(N\)-dimensional space, \(T\) is presented by a \(N\times N\) dimensional matrix, and can be further decomposed by eigenvectors \(\Big{(}\overset{\rightarrow}{e}_{1},\overset{\rightarrow}{e}_{2},\cdots, \overset{\rightarrow}{e}_{N}\Big{)}\) and related eigenvalues \((\lambda_{1},\lambda_{2},\cdots,\lambda_{N})\), where \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\). Suppose the \(d\) dimensional geometric feature is going to be transmitted from point \(p(x_{1},x_{2},\ldots,x_{N})\) to \(q(x_{1},x_{2},\ldots,x_{N})\), the normal vector of normal subspace is denoted as \(v_{n}=\Big{(}\overset{\rightarrow}{e}_{1},\overset{\rightarrow}{e}_{2}, \cdots,\overset{\rightarrow}{e}_{d}\Big{)}\), the tangent direction vector is denoted as \(v_{t}=\Big{(}\overset{\rightarrow}{e}_{d+1},\overset{\rightarrow}{e}_{d+2},\cdots,\overset{\rightarrow}{e}_{N}\Big{)}\), and vector from \(p\) to \(q\) is denoetd as \(v=q-p\). Then, the normal vector \(v_{k}\), which is transmitted along a minimal circle path defined by points \(p\) and \(q\), meets the relations defined in Equation (4).  
\[v_{k}=v_{n}\cos(2\theta)-v_{t}\sin(2\theta) \tag{4}\]  
In Equation (4), angle \(\theta\) is computed based on cosine value and dot multiply operation using geometric relations of \(v_{t}\), and \(v_{k}\), i.e., \(\theta=a\cos(dot(norm(v_{k}),v_{t})\). Based on Equations (2)-(4), the \(d\) dimensional tensor feature \(T_{d}\) transmitted from \(p\) to \(q\) can be encoded as Equation (5).  
\[T_{d}=w(r,\theta)kron\Big{(}v_{k},v_{k}^{T}\Big{)} \tag{5}\]  
\(w(r,\theta)=e^{-(\frac{r}{p})^{2}}\big{(}\cos(\theta)\big{)}^{4}\) is the decay function with distance \(r\) and curvature information encoded as \(\theta\). It has been proven that the angle \(\theta\) will become 0 where dimension \(d\geq 2\), and there is no need to recompute \(\theta\) in each dimension, if \(v_{k}\) is projected to \(S_{d}\) (King [16]). Hence \(d\)-dimension tensor feature \(T_{d}\) can be represented as Equation (6).  
\[T_{d}=\left\{\begin{matrix}w(r,\theta)kron\big{(}v_{k},v_{k}^{T}\big{)}&d=1\\ w(r)\big{(}S_{d}-kron(v_{n},v_{n}^{T})\big{)}&d\geq 2\end{matrix}\right. \tag{6}\]  
Figure 2: The tensor voting process of the point with its neighborhood.  
On the other hand, the intensity \(s_{d}\) of \(d\) dimensional geometric feature can be represented using eigenvalues \((\lambda_{1},\lambda_{2},\cdots,\lambda_{N})\), as denoted in Equation (7).  
\[s_{d}=\left\{\begin{matrix}\lambda_{d}-\lambda_{d+1}&d<N\\ \lambda_{N}&d=N\end{matrix}\right. \tag{7}\]  
Based on Equations (6) and (7), tensor voted from \(p\) to \(q\) in each dimension can be computed by Equation (8).  
\[T^{p}=\sum\
olimits_{d=1}^{d\leq N}s_{d}T_{d} \tag{8}\]  
Finally, the tensor \(T^{q}\) of point \(q\) is voted by tensor \(T^{p}\) of each point \(p\) in p's neighborhood \(\Omega\), as denoted in Equation (9), and this process is called the tensor refinement. The reader can refer to [PERSON], et al. [6] for a detailed derivation of the tensor refinement process.  
\[T^{q}=\sum\
olimits_{p\in\Omega}T^{p} \tag{9}\]