## 2 Innovative Methodology
### Data Redundancy Identification  
In order to identify redundant data points in the compressed point cloud data flow from Step 2, it is first necessary to predict the current scan line in the unmeasured area. Herein, the prediction is realized by Hermite extrapolation [43], and a predicted curve is created. The data redundancy identification algorithm is detailed as follows:  
Figure 4 shows the schematic diagram of the data redundancy identification algorithm, in which line \(i\) is the current scanning line during the on-line measuring process, and \(P_{i,j}\) represents the \(j\)th point in scanning line \(i\). If \(j\geq 2\), a shape-preserving piecewise bicubic Hermite curve can be built to predict the shape and direction of the current scanning line; here, we name this the predicted curve, as shown in Figure 4. After that, suppose \(k\) is a positive integer and let \(1\leq k<j\) and the coordinates of point \(P_{i,k}\) be \((x_{k},y_{k},z_{k})\); then, a series of specific Hermite interpolation polynomials can be determined by  
\[\left\{ \begin{array}{l}H_{y}(x)=y_{k}\alpha_{k}(x)+y_{k+1}\alpha_{k+1}(x)+y^ {\prime}_{k}\beta_{k}(x)+y^{\prime}_{k+1}\beta_{k+1}(x)\\ H_{z}(x)=z_{k}\alpha_{k}(x)+z_{k+1}\alpha_{k+1}(x)+z^{\prime}_{k}\beta_{k}(x)+ z^{\prime}_{k+1}\beta_{k+1}(x)\end{array} \right., \tag{1}\]  
where  
\[\left\{ \begin{array}{l}\alpha_{k}(x)=\left(1+2\frac{x-x_{k}}{x_{k+1}-x_{k}} \right)\left(\frac{x-x_{k+1}}{x_{k}-x_{k+1}}\right)^{2}\\ \alpha_{k+1}(x)=\left(1+2\frac{x-x_{k+1}}{x_{k}-x_{k+1}}\right)\left(\frac{x-x _{k}}{x_{k+1}-x_{k}}\right)^{2}\\ \beta_{k}(x)=\left(x-x_{k}\right)\left(\frac{x-x_{k+1}}{x_{k}-x_{k+1}}\right) ^{2}\\ \beta_{k+1}(x)=\left(x-x_{k+1}\right)\left(\frac{x-x_{k}}{x_{k+1}-x_{k}}\right) ^{2}\end{array} \right., \tag{2}\]  
and the first derivatives of \(y^{\prime}_{k}\), \(y^{\prime}_{k+1}\), \(z^{\prime}_{k}\), \(z^{\prime}_{k+1}\) can be estimated by the following formulas. When \(1<k<j\):  
\[{y^{\prime}}_{k}={f_{y}}^{\prime}(x_{k})=\left\{ \begin{array}{l}0,if:\frac{y _{k+1}-y_{k}}{x_{k+1}-x_{k}},\frac{y_{k}-y_{k-1}}{x_{k}-x_{k-1}}<0\\ \frac{1}{2}\left(\frac{y_{k+1}-y_{k}}{x_{k+1}-x_{k}}+\frac{y_{k}-y_{k-1}}{x_{k }-x_{k-1}}\right),if:\frac{y_{k+1}-y_{k}}{x_{k+1}-x_{k}},\frac{y_{k}-y_{k-1}}{ x_{k}-x_{k-1}}\geq 0\end{array} \right., \tag{3}\]  
\[{z^{\prime}}_{k}={f_{z}}^{\prime}(x_{k})=\left\{ \begin{array}{l}0,if:\frac{z _{k+1}-z_{k}}{x_{k+1}-x_{k}},\frac{z_{k}-z_{k-1}}{x_{k}-x_{k-1}}<0\\ \frac{1}{2}\left(\frac{z_{k+1}-z_{k}}{x_{k+1}-x_{k}}+\frac{z_{k}-z_{k-1}}{x_{k }-x_{k-1}}\right),if:\frac{z_{k+1}-z_{k}}{x_{k+1}-x_{k}},\frac{z_{k}-z_{k-1}}{ x_{k}-x_{k-1}}\geq 0\end{array} \right.. \tag{4}\]  
When \(k=1\):  
\[{y^{\prime}}_{1}=\left\{ \begin{array}{l}0,if:d_{y},\frac{y_{2}-y_{1}}{x_{2} -x_{1}}<0\\ 3.\frac{y_{2}-y_{1}}{x_{2}-x_{1}}if:\left|{d_{y}}\right|>3\left|\frac{y_{2}-y_ {1}}{x_{2}-x_{1}}\right|\&\frac{(y_{2}-y_{1})(y_{3}-y_{2})}{(x_{2}-x_{1})(x_{3} -x_{2})}<0\end{array} \right., \tag{5}\]  
Figure 4: The schematic diagram of data redundancy identification. \[{z^{\prime}}_{1}=\left\{ \begin{array}{l}0,if:d_{z}\frac{z_{2}-z_{1}}{x_{2}-x_{1}}< 0\\ 3\frac{z_{2}-z_{1}}{x_{2}-x_{1}}if:|d_{z}|>3\left|\frac{z_{2}-z_{1}}{x_{2}-x_{1}} \right|\delta\frac{(z_{2}-z_{1})(z_{3}-z_{2})}{(x_{2}-x_{1})(x_{3}-x_{2})}<0 \end{array} \right.^{\prime} \tag{6}\]  
in which  
\[\left\{ \begin{array}{l}d_{y}=\frac{(x_{3}+x_{2}-2x_{1})(y_{2}-y_{1})}{(x_{2} -x_{1})(x_{3}-x_{1})}-\frac{(x_{2}-x_{1})(y_{3}-y_{2})}{(x_{3}-x_{2})(x_{3}-x_{ 1})}\\ d_{z}=\frac{(x_{3}+x_{2}-2x_{1})(z_{2}-z_{1})}{(x_{2}-x_{1})(x_{3}-x_{1})}- \frac{(x_{2}-x_{1})(z_{3}-z_{2})}{(x_{3}-x_{2})(x_{3}-x_{1})}\end{array} \right.. \tag{7}\]  
When \(k=j\):  
\[{y^{\prime}}_{j}=\left\{ \begin{array}{l}0,if:d_{y}\frac{y_{j}-y_{j-1}}{x_{j }-x_{j-1}}<0\\ 3\frac{y_{j}-y_{j-1}}{x_{j}-x_{j-1}}if:|e_{y}|>3\left|\frac{y_{j}-y_{j-1}}{x_{ j}-x_{j-1}}\right|\delta\frac{(y_{j}-y_{j-1})(y_{j-1}-y_{j-2})}{(x_{j}-x_{j-1})(x_{ j-1}-x_{j-2})}<0\end{array} \right.^{\prime} \tag{8}\]  
\[{z^{\prime}}_{j}=\left\{