## 2 The Four-Probe System  
The device of the new method, whose purpose is to eliminate the data processing error and retain a high lateral resolution, is shown in Figure 1. The measuring device is located between two measured profiles and moves along x-direction by a scanning stage. Four displacement sensors used as probes are distributed in pairs with different spacings on two sides of the measuring device and installed on the scanning stage. The probes here could be displacement sensors that utilize different principles, including contact type and non-contact type, to satisfy the service environment and measurement requirements. For convenience, the figures in this text are drawn with contact sensors. \(P_{3}\) is opposite to \(P_{1}\). The spacing between \(P_{1}\) and \(P_{2}\) is \(d_{1}\) and the spacing between \(P_{3}\) and \(P_{4}\) is \(d_{2}\). After stepping for \(s\), probes collect measurement data of two profiles. What must be noted is that \(d_{1}-d_{2}=s\), and \(d_{1}\) and \(d_{2}\) need to be integral multiples of the step distance \(s\). When probes collect \(N\) times of data during one scanning, the outputs of the probe \(P_{j}\) can be expressed as \(m_{j}(x_{i})\), where \(i=0,1,2,\cdots,N-1\), \(j=1,2,3,4\). Assume that the profile heights of two measured profiles are \(f(x)\) and \(g(x)\); when the y-direction straightness error, the error which affects the measurement result most, is defined as \(e_{y}(x_{i})\) and the yaw error of the device, the zero-adjustment error, and the sensor measurement error are ignored, the outputs of probes during the scanning can be written as:  
\[m_{1}(x_{i})=f(x_{i})-e_{y}(x_{i}), \tag{1}\]  
\[m_{2}(x_{i})=f(x_{i}+d_{1})-e_{y}(x_{i}), \tag{2}\]  
\[m_{3}(x_{i})=-[g(x_{i})-e_{y}(x_{i})], \tag{3}\]  
\[m_{4}(x_{i})=-[g(x_{i}+d_{2})-e_{y}(x_{i})]. \tag{4}\]  
The differential values without straightness error are defined as \(mf\) and \(mg\) and are calculated by the following equations:  
\[\begin{array}{l}mf(x_{i}+d_{2})=\frac{m_{2}(x_{i})+m_{4}(x_{i})-[m_{1}(x_{i} +d_{2})+m_{3}(x_{i}+d_{2})]}{s}\\ \hskip 14.226378 pt=\frac{1}{s}[f(x_{i}+d_{1})-f(x_{i}+d_{2})]\\ \hskip 14.226378 pt=f^{\prime}_{s}(x_{i}+d_{2})\end{array} \tag{5}\]  
\[\begin{array}{l}mg(x_{i}+d_{2})=\frac{m_{2}(x_{i})+m_{4}(x_{i})-[m_{1}(x_{i} +d_{1})+m_{3}(x_{i}+d_{1})]}{s}\\ \hskip 14.226378 pt=\frac{1}{s}[g(x_{i}+d_{1})-g(x_{i}+d_{2})]\\ \hskip 14.226378 pt=g^{\prime}_{s}(x_{i}+d_{2})\end{array} \tag{6}\]  
The reconstruction profiles are denoted \(Y_{f}(x_{i})\) and \(Y_{g}(x_{i})\), and using an integration process, the values of the reconstruction points are obtained by integrating the differential values in Equations (5) and (6):  
\[Y_{f}(x_{i})=mf(x_{i-1})\cdot s+Y_{f}(x_{i-1}), \tag{7}\]  
\[Y_{g}(x_{i})=mg(x_{i-1})\cdot s+Y_{g}(x_{i-1}). \tag{8}\]  
When reconstructing the measured profiles using Equations (5)-(8), it can be summarized that the data processing process of the new method is equivalent to an STP process in which the probe spacing and the step distance are both equal to \(s\). Therefore, the new method is free from the data processing error and guarantees a high density of sampling points, as \(s\) can be chosen independently without being limited by probe spacing. The ability of the STP method to eliminate the straightness error of the reference is retained. When the yaw error needs to be measured to reduce the reconstruction error, an additional autocollimator can be used with a target mirror mounted on the scanning stage. Therefore, ignoring the measurement errors of sensors, the new method can be used to realize exact reconstruction.