## 3 The Establishment of Self-Architected Fuzzy Calibration Network
### Expansion of Data Sample  
The number of samples used to train the self-architected fuzzy network will directly influence the fineness of the network's description of the relationship between different variables, which in turn determines the precision of the calibration networks. Therefore,  
Figure 4: Measurement devices: (**a**) Strain sensor and temperature compensation device; (**b**) Demodulation instrument.  
expanding the training data sample capacity is of great significance for improving the precision of network calibration. According to the reconstruction theory discussed in Section 2, the relationship between strain measurement and reconstruction displacement is nonlinear. Therefore, this paper proposes to use the 3 rd B-spline function for data sample expansion.  
For a given \(n+1\) plane or spatial points \(P_{i}(i=0,1,\cdots,n)\), the definition for an \(n\)-th parameter segment on a B-spline curve is given by  
\[P_{k,n}(u)=\sum_{i=0}^{n}P_{i}G_{i,n}(u),\;ue[0,1] \tag{12}\]  
Here, \(P_{k,n}(u)\) is the \(k\)-th segment on the \(n\)-th degree B-spline curve, \(u\) is the data point parameter; \(G_{i,n}\) is the basis function of the \(n\)-th degree B-spline curve, and the polygon formed by its vertex \(P_{i}\) is called the B-spline curve feature polygon. According to the definition of the B-spline curve, the k segment of an \(n\)-th B-spline curve is only related to \(n+1\) control points signified by \(P_{i}\). The 3 rd B-spline curve is controlled by the 6 control points designated by \(P_{i}\) is composed of three B-spline curve segments, each is controlled by 4 points in \(P_{i}\), as shown in Figure 5.  
The B-spline curve is local. The change of one control point will only influence the adjacent curve segment and will not influence the trend of change on the entire curve. For \(n=3\), the 3 rd B-spline curve segment \(P_{i,3}(u)\), is given by  
\[P_{i,3}(u)=\sum_{i=0}^{3}P_{i}G_{i,3}(u),\;ue[0,1], \tag{13}\]  
where \(i=0,\;1,\;2,\;3\). The basis functions of the cubic B-spline curve are as follows:  
\[\left\{\begin{matrix}G_{0,3}(u)=\frac{1}{6}\left(-u^{3}+3u^{2}-3u+1\right)\\ G_{1,3}(u)=\frac{1}{6}\left(3u^{3}-6u^{2}+4\right)\\ G_{2,3}(u)=\frac{1}{6}\left(-u^{3}+3u^{2}+3u+1\right)^{,}\;u\epsilon[0,1]\\ G_{3,3}(u)=\frac{1}{6}u^{3}\end{matrix}\right. \tag{14}\]  
It can be written in matrix form as  
\[\left\{\begin{matrix}x_{u}=GX\\ y_{u}=GY\end{matrix}\right.\;ue[0,1] \tag{15}\]  
where \(G\) is the basis function matrix of the cubic B-spline, \(X\) and \(Y\) are the coordinate vectors of the control points in each section of the cubic B-spline curve, and \((x_{u},y_{u})\) are the coordinates of a point on the cubic B-spline curve. For continuous values of \(u\), a smooth B-spline curve can be determined. For \(n\) control vertices, a complete cubic B-spline curve can be obtained by only moving the control vertices \(n-3\) times. After successfully fitting the cubic B-spline curve, the data point parameter \(u\) is taken from 0 to 1 in a certain step,  
Figure 5: Schematic diagram of cubic B-spline curve.  
such that the corresponding number of points on the curve can be obtained, and the data can be expanded.