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子孔径拼接算法流程图如图1所示。
在对待测平面镜进行拼接检测前,首先需要根据待检测元件参数完成其待测子孔径规划。子孔径规划时需要考虑待测元件尺寸与检测子孔径尺寸,要求检测中各待测子孔径可以实现对检测平面镜的全口径覆盖,同时各相邻子孔径间存在一定比例的重叠区域,通常重叠区域比例不低于待测镜面全口径尺寸的25%[21]。
在完成对待测平面镜的子孔径规划后,即可对各规划子孔径进行干涉检测数据采集。假设在子孔径规划时,共规划检测子孔径数目为
$ N $ 。此时,以$ N $ 号子孔径所在检测坐标系为基准坐标系,则对于第$ i $ 个子孔径,其检测结果在该基准坐标系内可以表示为:$$ {{{\textit{z}}}}_i'({{x}},{{y}}) = {{\textit{z}}_i}({{x}},{{y}}) + \sum\limits_{k = 1}^L {{F_{ik}}{f_k}({{x}},{{y}})} $$ (1) 式中:
${z_i}({{x}},{{y}})$ 为第$ i $ 个子孔径对应的干涉检测结果;$ {F_{ik}} $ 为第$ i $ 个子孔径中各检测调整误差项所对应的调整系数;${f_k}({{x}},{{y}})$ 为子孔径检测调整误差项。对于光学平面镜拼接检测,共有三项调整项,分别如公式(2)所示:$$ \left\{ {\begin{array}{*{20}{c}} {{F_{i1}} = x} \\ {{F_{i2}} = y} \\ {{F_{i3}} = 1} \end{array}} \right. $$ (2) 在球面镜与非球面拼接时,可以直接对上述调整误差项进行对应扩展。利用全局优化最小二乘拟合方法,各子孔径相对于基准子孔径的调整系数可以通过公式(3)获得:
$$ \begin{split} \min =& \sum\limits_{i = 1...{{N}}} {\sum\limits_{j = 1...N}^{j \cap i} {{{(({{\textit{z}}_i}({{x}},{{y}}) + \sum\limits_{k = 1}^L {{F_{ik}}{f_k}({{x}},{{y}})} ) - ({{\textit{z}}_j}({{x}},{{y}}) +}}}}\\ &{{{{\sum\limits_{k = 1}^L {{F_{jk}}{f_k}({{x}},{{y}})} )^2}}} }\\[-15pt] \end{split} $$ (3) 其中公式(3)可以写为矩阵的形式:
$$ P = Q \cdot R $$ (4) 矩阵
$ P,Q,R $ 的行与列分别对应为$(({{N}} - 1) \cdot {{L}}) \times 1$ ,$(({{N}} - 1) \cdot {{L}}) \times (({{N}} - 1) \cdot {{L}})$ ,$(({{N}} - 1) \cdot {{L}}) \times 1$ ,则公式(4)中的矩阵$ P,Q,R $ 可以具体写为如下形式:$$ {P_{ij}}[{{k}}] = \sum\limits_{i \cap j} {{f_{ik}}({{x}},{{y}})({{{Z}}_i}({{x}},{{y}}) - {{{Z}}_j}({{x}},{{y}}))} $$ (5) $$ P\left\{ {i,1} \right\} = \sum\limits_{j = 1}^N {{p_{ij}}} $$ (6) $$ {{{Q}}_{ij}}({{m}},{{n}}) = \left\{ {\begin{array}{*{20}{l}} {\displaystyle\sum\limits_{i \cap j} {{f_{im}}({{x}},{{y}}){f_{jm}}({{x}},{{y}})\;\;\;\;\;\;i \ne j} } \\ { - \displaystyle\sum\limits_{k = 1,…,N}^{k \ne i} {\sum\limits_{i \cap k} {{f_{im}}{{({{x}},{{y}})}_{jm}}f({{x}},{{y}})\;\;\;\;\;\;i = j } } } \end{array}} \right. $$ (7) $$ Q\left\{ {i,j} \right\} = {Q_{ij}} $$ (8) $$ {R_i}[{{k}}] = {F_{ik}} $$ (9) $$ R\left\{ {i,1} \right\} = {R_i} $$ (10) 由公式(4)~(10)即可求解得出各子孔径相对基准子孔径坐标系的拼接调整项系数,利用上述调整系统,将各子孔径检测数据校正到全局基准坐标系中,即可获得待测平面镜的全口径拼接结果。在传统拼接算法中,对于各相邻子孔径间重叠区域的相位值,采用取各个重叠子孔径的平均值[21],即:
$$ w({x_j},{y_j}) = \frac{1}{m}\sum\limits_{i = 1}^m {{w_i}({x_j},{y_j})} $$ (11) 式中:点
$ ({x_j},{y_j}) $ 为m个子孔径重叠区域内的坐标点,在高精度镜面拼接检测时,这种取值方式通常会在全口径拼接结果中或残差结果中留下“拼痕”。为了在拼接结果或残差结果中消除拼痕,笔者提出了拼接因子用于重叠区域取值,对处于子孔径重叠区域的相位数据采取加权平滑算法。为了更简洁地描述上述方法,以两个子孔径为例解释算法,如图2所示。图2中,点
$ {O_1} $ 及$ {O_2} $ 分别为两个相邻检测子孔径的圆心,点$P^{'}$ 为子孔径重叠区域内任意一点。假设在干涉检测中,靠近圆心点处的检测数据可信度更高,则基于拉格朗日插值思想,重叠区域内任意一检测点$P^{'}$ ,其相位值可以写为公式(12)的形式:$$ {\textit{z}} = {{\textit{z}}_1} \cdot \frac{{\left\| {{X_{P^{'}}} - {X_{O2}}} \right\|}}{{\left\| {{X_{O1}} - {X_{O2}}} \right\|}} + {{\textit{z}}_2} \cdot \frac{{\left\| {{X_{P^{'}}} - {X_{O1}}} \right\|}}{{\left\| {{X_{O2}} - {X_{O1}}} \right\|}} $$ (12) 式中:
$ \left\| { \cdot \cdot \cdot } \right\| $ 为二阶范数;$ {{\textit{z}}_1} $ 与$ {{\textit{z}}_2} $ 为相邻子孔径1与子孔径2在点P'处的干涉检测结果;${X_{P^{'}}}$ ,$ {X_{O1}} $ 与$ {X_{O2}} $ 分别为点$P'$ ,$ {O_1} $ 与$ {O_2} $ 所对应的向量表示,即$$ \left\{ {\begin{array}{*{20}{c}} {{X_{P^{'}}} = ({x_{P^{'}}},{y_{P^{'}}})} \\ {{X_{O1}} = ({x_{O1}},{y_{O1}})} \\ {{X_{O2}} = ({x_{O2}},{y_{O2}})} \end{array}} \right. $$ (13) 其中,
$({x_{P^{'}}},{y_{P^{'}}})$ ,$ ({x_{O1}},{y_{O1}}) $ 与$ ({x_{O2}},{y_{O2}}) $ 分别为点$P'$ ,$ {O_1} $ 与$ {O_2} $ 在基准坐标系内的坐标表示。此时,考虑一般情况,假设点
$P'$ 为N个子孔径重叠区域内一点,则其相位值为:$$ {\textit{z}} = sum(w\_k \cdot {\textit{z}}\_k) $$ (14) 式中:
$ {\textit{z}}\_k $ 为点$P'$ 在第k个子孔径中对应的干涉检测相位值;$ w\_k $ 为相应的权重因子,$$ w\_k = \frac{{\left\| {{X_{P^{'}}} - {X_{O1}}} \right\|...\left\| {{X_{P^{'}}} - {X_{O(k - 1)}}} \right\|\left\| {{X_{P^{'}}} - {X_{O(k + 1)}}} \right\|\left\| {{X_{P^{'}}} - {X_{ON}}} \right\|}}{{\left\| {{X_{Ok}} - {X_{O1}}} \right\|...\left\| {{X_{Ok}} - {X_{O(k - 1)}}} \right\|\left\| {{X_{Ok}} - {X_{O(k + 1)}}} \right\|\left\| {{X_{Ok}} - {X_{ON}}} \right\|}} $$ (15) $$ {\textit{z}} = sum(w\_k \cdot {\textit{z}}\_k)/sum(w\_k) $$ (16)
Subaperture stitching testing to flat mirror based on weighting algorithm (Invited)
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摘要: 为了解决大口径平面反射镜高精度检测问题,建立了一种基于全局优化的子孔径拼接检测数学模型,同时提出了一种拼接因子用于重叠区域取值。基于上述方法,结合工程实例,对一口径为120 mm的平面反射镜完成拼接检测,检测中共规划了四个待测子孔径,为了对比文中所述算法与传统最小二乘拟合拼接算法的拼接性能,分别利用两种算法完成了待测平面镜的面形重构。实验结果表明,两种算法所得拼接结果光滑、连续、无“拼痕”,同时分别将两种算法所得拼接结果与全口径检测结果进行了对比分析,从传统拼接算法残差图中可以看到明显的“拼痕”,而加权拼接方法得到的拼接结果光滑、连续,同时其残差图的PV与RMS值分别为0.012λ与0.002λ,小于传统算法残差图的PV与RMS值,验证了算法的可靠性与精度。Abstract: To solve the problem of high-precision testing of large-diameter plane mirrors, a mathematical model of subaperture stitching testing based on global optimization was established, and a stitching factor was proposed for overlapping area values. Based on the above method, combined with engineering examples, the stitching testing of a plane mirror was completed with a diameter of 120 mm, and four subapertures to be tested were planned. In order to compare the stitching performance of the algorithm described in this paper with the traditional least-squared fitting stitching algorithm, two algorithms were used to complete the surface reconstruction of the plane mirror to be measured. The experimental results show that the stitching results obtained by the two algorithms are smooth, continuous, no "stitch marks". At the same time, the results of the two algorithms are also compared with the full-aperture testing results. In this paper, obvious "stitch marks" can be seen in the residual map of the traditional splicing algorithm, and the stitching results obtained by the algorithm method in this paper are smooth and continuous, while the PV and RMS values of the residual graph are 0.012λ and 0.002λ, respectively, which are less than the PV and RMS values of the traditional algorithm residuals chart, which verifies the reliability and accuracy of the algorithm.
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Key words:
- optical testing /
- interferometer /
- subaperture stitching /
- stitching factor
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