## 2 Methodology Setup
### Measurement Data Processing
If the interval between the two points is \(\Delta I\) after the base expansion, the interval length can be expressed as:  
\[\Delta l=\int_{L}F(r,y)ds, \tag{1}\]  
where \(L\) is the DOE's base curve, \(F(r,\,y)\), between points a and \(b\). In the data processing, the integral operation of the curve was approximated to the linear distance calculation of two points; then Equation (1) can be approximated as:  
\[\Delta l=\int_{L}F(r,y)ds\approx\sqrt{\left(y_{2}-y_{1}\right)^{2}+\left(r_{2} -r_{1}\right)^{2}}. \tag{2}\]  
Then, the coordinates of the two points on the unfolded surface could be expressed as \(A(r_{1},\,Y_{1})\) and \(B(r_{1}+\Delta l,\,Y_{2})\), and the diffraction structure on the base could be obtained by processing the model data points in turn, as shown in Figure 3. The unexpanded profile of the original DOE designed model is shown in Figure 3a, and Figure 3b shows the reference diffraction structure obtained by unfolded the base. There were similar geometric features in the model data and the measured data after the model base was unfolded, but they were located in two different Cartesian coordinate systems. For the purposes of evaluating the measured data, it was necessary to perform data registration to unify the two coordinate systems of two datasets. Before the data registration, the two datasets required a two-step solution. The first step was to remove the gross errors from the measured data using a median filter, because gross errors would seriously affect the accuracy of data registration. The size of the window was determined by the actual measured data and noise characteristics, because median filtering with large window sizes tends to eliminate a large amount of high frequency information in the measured data. Therefore, in order to ensure the integrity of the data information, and to filter out all the gross errors in the measured data, a small window size should be selected. Figure 4a is the raw measured data, and the data noise was reduced by median filtering, with a window size of 11, as shown in Figure 4b. Comparing these two datasets, it is clear that the median filter successfully removed the gross error data. The second step was moving the centroids of the measured data after filtering and the unfolded model data to the same position, providing initial position parameters using in data registration. Before the matching process, the centers of gravity of the two data sets were coincident as the preprocessing of the algorithm to improve the accuracy of the algorithm. Figure 3: Expanding the design model along the centerline of the base: (**a**) unexpanded surface and (**b**) unfolded surface. To realize the data registration, the Sparse iterative closest point (ICP) algorithm was used to match the design model data with the measured data. The basic idea was to find the nearest point in the geometric relationship in the two sets of data to form a matching point pair, and then to find the matrix parameters of rotation and translation of the rigid body transformation that make the geometric distance between all point pairs shortest, based on the least squares method. Then continuous iterative optimization until the distance correlation parameter between two pairs of datasets was less than a certain threshold. The Sparse ICP algorithm has been improved on the basis of traditional ICP. It can be described as follows:  
\[ \begin{array}{c}argmin_{Y}\sum\limits_{i=1}^{n}\left\|y_{i}-\mathbf{R}x_{i}-\mathbf{ T}\right\|_{2}^{p}+I_{Y}(y_{i})\\ argmin_{R,T}\sum\limits_{i=1}^{n}\left\|y_{i}-\mathbf{R}x_{i}-\mathbf{T}\right\|_{2}^ {p}+I_{SO(k)}(\mathbf{R})\end{array}  \tag{3}\]  
where \(\mathbf{R}\) is a rotation matrix, \(\mathbf{T}\) is a translation vector, \(x\) is the coordinates of point on the \(X\) surface, and \(y\) is the coordinates of point on the \(Y\) surface. R is restricted to the special orthogonal group \(SO(k)\) by the indicator function \(I_{A}\) (\(b\)) that evaluates to \(0\) if \(b\in A\) and to \(\infty\) otherwise.