# An Advanced Vehicle Body Part Inspection Scheme Based on Scattered Point Cloud Data
## 3 Materials and Methods
### Methods
#### 3.2.2 Denoising of PCD Based on a Hybrid Filtering Algorithm
\begin{array}{cc}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}& \sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})&\sum_{i=1}^{n}(x_{i}- \overline{x})(z_{i}-\overline{z})\\ \sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})&\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}&\sum_{i=1}^{n}(y_{i}-\overline{y})(z_{i}-\overline{z})\\ \sum_{i=1}^{n}(y_{i}-\overline{y})(z_{i}-\overline{z})&\sum_{i=1}^{n}(y_{i}- \overline{y})(z_{i}-\overline{z})&\sum_{i=1}^{n}(z_{i}-\overline{z})^{2}\\ \end{array} \right], \tag{12}\]

where \((a,b,c)\) is the normal vector of selected point. \[E(x)=\sum_{i=1}^{n}((x-\varepsilon_{i})^{T}n(x)^{2}G(x,\varepsilon_{i})) \tag{13}\]

\[n(x)=\frac{\sum_{i=1}^{n}n(\varepsilon_{i})G(x,\varepsilon_{i})}{\parallel \sum_{i=1}^{n}n(\varepsilon_{i})G(x,\varepsilon_{i})\parallel}, \tag{14}\]

where \(G\) is Gaussian function; \(x\) is projection point,; \(\varepsilon_{i}\) is the neighborhood of x. \[T=\left[ \begin{array}{cccc}1&0&0&d_{x}\\ 0&1&0&d_{y}\\ 0&0&1&d_{z}\\ 0&0&0&1\\ \end{array} \right] \tag{15}\]

\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ 1\\ \end{array} \right]=T\cdot\left[ \begin{array}{c}x\\ y\\ z\\ 1\\ \end{array} \right]. \tag{16}\]

When \(p\) is rotated in space, its motion is as follows:

\[R=\left[ \begin{array}{cccc}cos\beta\ cos\gamma+\sin\alpha\sin\beta\sin\gamma& -\cos\beta\sin\gamma+\sin\alpha\sin\beta\sin\gamma&sin\beta\cos\alpha&0\\ cos\alpha\sin\gamma&cos\alpha\cos\gamma&-\sin\alpha&0\\ -sin\beta\cos\gamma+\sin\alpha\sin\beta\sin\gamma&sin\beta\sin\gamma+\sin\alpha \cos\beta\cos\gamma&cos\alpha\cos\beta&0\\ 0&0&1\\ \end{array} \right] \tag{17}\]

\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ 1\\ \end{array} \right]=R\cdot\left[ \begin{array}{c}x\\ y\\ z\\ 1\\ \end{array} \right] \tag{18}\]

Therefore, when \(p\) moves arbitrarily in space, its motion process can be written as:

\[p^{\prime}=R\cdot p+T, \tag{19}\]

where \(T\) is the translation matrix in space; \(R\) is the rotation matrix; \(d_{x}\), \(d_{y}\) and \(d_{z}\) are the translation distances in the directions of \(X\), \(Y\) and \(Z\), respectively; and \(\alpha\), \(\beta\) and \(\gamma\) are the rotation angles on the \(X\), \(Y\) and \(Z\) axes, respectively. ## References

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