Distortion analysis and focal length testing of off-axis optical system

离轴光学系统的畸变分析及焦距测量 Distortion analysis and focal length testing of off-axis optical system 1 corresp first-author E-mail： chengq@ciomp.ac.cn E-mail： chengq@ciomp.ac.cn 程 强（1986-），男，山西阳泉人，副研究员，硕士生导师，2008年于南开大学获得理学学位，2013年于中科院长春光机所获得工学博士学位，主要从事复杂光学曲面高精度检测技术，光学系统集成分析与像质提升技术等方面的研究。E-mail： chengq@ciomp.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228387&type= 22.01333237 32.00399780 程 强 （1986-），男，山西阳泉人，副研究员，硕士生导师，2008年于南开大学获得理学学位，2013年于中科院长春光机所获得工学博士学位，主要从事复杂光学曲面高精度检测技术，光学系统集成分析与像质提升技术等方面的研究。E-mail： chengq@ciomp.ac.cn chengq@ciomp.ac.cn 1 胡海翔（1990-），男，安徽淮南人，副研究员，硕士生导师，2012年于中国科学技术大学获得理学学位，2017年于中科院长春光机所获得工学博士学位，主要从事复杂光学曲面高精度制造技术等方面的研究。E-mail： hhx@ciomp.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228397&type= 22.01332855 32.00400162 胡海翔 （1990-），男，安徽淮南人，副研究员，硕士生导师，2012年于中国科学技术大学获得理学学位，2017年于中科院长春光机所获得工学博士学位，主要从事复杂光学曲面高精度制造技术等方面的研究。E-mail： hhx@ciomp.ac.cn hhx@ciomp.ac.cn 1 1 1 1 2 中国科学院 长春光学精密机械与物理研究所 中国科学院光学系统先进制造技术重点实验室，吉林 长春 130033 中国科学院 长春光学精密机械与物理研究所 中国科学院光学系统先进制造技术重点实验室，吉林 长春 130033 Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Key Laboratory of Optical System Advanced Manufacturing Technology， Changchun 130033， China Changchun Institute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Key Laboratory of Optical System Advanced Manufacturing Technology， Changchun 130033， China 中国科学院大学，北京 100049 中国科学院大学，北京 100049 University of Chinese Academy of Sciences， Beijing 100049， China University of Chinese Academy of Sciences， Beijing 100049， China 为了实现离轴光学系统的高精度畸变分析、测量及在轨图像的快速、高精度畸变校正，利用高阶畸变分析方法对离轴光学系统的相对畸变进行了拟合分析。首先，对离轴光学系统的理论畸变进行了分析，提出了等效焦距的概念，并利用高阶畸变分析方法对系统的相对畸变进行了拟合分析；接着，针对装调完成的某离轴光学系统完成了畸变测试及焦距测量；最后，利用实测结果完成了该离轴光学系统的物像对应关系的高精度解算。与ZEMAX光线追迹的结果相比，利用高阶畸变系数计算得到的等效焦距最大偏差为0.366 mm；对于某装调完成的离轴光学系统，实测等效同轴系统焦距的偏差仅为6.378 mm，相对偏差为0.073%。验证了该方法对于离轴光学系统畸变分析、测量及焦距测量具有正确性及精度保证，将为在轨图像的快速、高精度畸变校正提供可靠输入。 To realize the high-precision distortion analysis and measurement of an off-axis optical system and fast and high-precision distortion correction of on-orbit images， a high-order distortion analysis method is used to fit and analyze the relative distortion of the off-axis optical system. First， the theoretical distortion of the off-axis optical system is examined. Then， the concept of equivalent focal length is proposed， and the relative distortion of the system is fitted and analyzed using the aforementioned analysis method. The distortion testing and focal length measurement are then completed for the well-aligned off-axis optical system. Finally， the measured results are used to complete the object image correspondence of the off-axis optical system. The maximum deviation of the equivalent focal length calculated using the high-order distortion analysis method is 0.366 mm when compared with the results of the ZEMAX ray tracing. The measured focal length deviation of the equivalent coaxial system for an off-axis optical system is only 6.378 mm， and the relative deviation is 0.073%. The correctness and accuracy of this method for distortion analysis and measurement and focal length evaluation of the off-axis optical system are verified， thereby providing reliable input for the fast and high-precision distortion correction of on-orbit images. 离轴光学系统 畸变分析 焦距测量 等效焦距 高阶畸变系数 off-axis optical system distortion analysis focal length testing equivalent focal length high order distortion coefficient 1 引 言 1 随着空间光学遥感技术的快速发展，为满足空间对地观测高分辨、大视场的成像需求，离轴光学系统已逐渐成为空间光学遥感器的重要发展趋势。离轴光学系统具有更多的设计自由度，且没有中心遮拦、调制传递函数高等诸多优点，可满足空间光学遥感器对地观测的高分辨率、大幅宽需求 ［ 1 1 - 4 4 1-4 ］ 。 在离轴光学系统的畸变分析及测量方面，北京空间机电研究所针对超宽视场的离轴光学系统开展了全视场畸变的一致性校正技术研究，相关研究成果成功应用于两台超宽视场离轴系统的装调校正，取得了良好的效果 ［ 5 5 ］ ；中科院长春光机所针对离轴三反光学系统，开展了镜头像面畸变标定及相机畸变测量等方面的研究及工程实践，并应用于某测绘产品镜头的高精度标定 ［ 6 6 - 9 9 6-9 ］ ，但其鲁棒性不够理想，针对工程实际中的随机误差解算精度和稳定性不足。此外，浙江大学 ［ 10 10 ］ 、吉林大学 ［ 11 11 ］ 、中科院上海技术物理研究所 ［ 12 12 ］ 也对光学系统的畸变分析及测量技术开展了深入研究。 本文介绍的某离轴光学系统，采用时间延迟积分电荷耦合器件（Time Delay Integration Charge Coupled Device， TDICCD）推扫成像，光学系统视场角为0.37°×3.5°，相机最大相对畸变设计值为1.413%。光学系统的畸变将直接影响成像的几何位置精度，对系统的成像定位造成影响。 本文对离轴光学系统的理论畸变进行了分析，提出了等效焦距概念，对离轴光学系统的相对畸变系数进行了分离，并针对装调完成的某离轴光学系统，利用多维约束非线性优化方法完成了畸变分析和焦距测量，对相机实际焦平面上特征点的物像对应关系进行解算，提高了系统畸变及焦距的解算精度和稳定性，为相机在轨图像的快速、高精度畸变校正提供准确输入。 2 反射光学系统的理论畸变 1 畸变广泛存在于各类光学系统中。光学系统产生畸变的原因是在一对物、像共轭平面上，垂直放大率随视场角的变化而变化，不再保持常量，导致像相对于物失去了相似性。因此，畸变虽然不影响成像的清晰度，却直接影响成像的几何位置精度。 随着光学系统视场角的变大，系统的畸变也随之变大。对于空间光学遥感器而言，其口径和视场一般都比较大，且成像精度要求较高，因此必须采取措施来减小甚至消除畸变带来的影响。 传统的同轴反射光学系统如Ritchey Chretien（R-C）系统，其所有光学镜面都关于系统光轴回转对称，因此对于视场角相同的物点，不论方向如何，其垂轴放大率都是一样的。这种情况下，畸变只与像高有关。像高相同的轴外物点，其径向畸变量是相同的。在像面上，畸变图像关于光学中心呈中心对称。此时随初级畸变系数的不同，畸变分别呈现出枕形畸变和桶形畸变。 而对于离轴反射光学系统，与传统的R-C系统相比，主、三镜离轴放置，不关于光轴对称。因此光学系统的垂轴放大率不但随视场角大小而改变，而且随轴外点的离轴角而变化。对轴外物点成像时，如果它们的主光线方向不同，即使其视场角一样，其垂轴放大率也可能不一样。因此在像面上，畸变图像不再关于光学中心对称，而与离轴光学系统的视场角和离轴角有关，需要针对离轴光学系统的物方二维角度开展系统畸变的仿真分析。 2.1　离轴光学系统的理论畸变 2 对离轴光学系统理论畸变的表达式进行推导。定义 <math id="M1"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228545&type= 9.05933285 3.97933316 为物方坐标， <math id="M2"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228547&type= 10.92199993 3.55599999 为像方坐标。已知系统畸变的表达式为： <math display="block" id="M3"><mi>H</mi><mo>=</mo><mi>h</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228408&type= 29.46400070 3.97933316 ， 其中： <math id="M4"><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228411&type= 3.30200005 2.79399991 为像高， <math id="M5"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228415&type= 1.94733346 2.79399991 为物高， <math id="M6"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228593&type= 2.28600001 3.64066648 为三阶畸变系数， <math id="M7"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228595&type= 1.94733346 3.64066648 为五阶畸变系数。 将系统畸变的表达式转化为直角坐标的形式，即： <math display="block" id="M8"><mi>H</mi><mo>=</mo><mroot><mrow><msup><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo>,</mo><mi>h</mi><mo>=</mo><mroot><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228426&type= 47.75199890 5.24933338 . 则有： <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><mi>X</mi><mo>=</mo><mi>H</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>h</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi><mo>=</mo></mtd></mtr><mtr><mtd><mtext>    </mtext><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mi>x</mi></mtd></mtr><mtr><mtd><mi>Y</mi><mo>=</mo><mi>H</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>h</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi><mo>=</mo></mtd></mtr><mtr><mtd><mtext>   </mtext><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mi>y</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228430&type= 59.85933304 23.36800003 ， 将（3）中两式展开，可得： <math display="block" id="M10"><mtable columnalign="left"><mtr><mtd><mi>X</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>x</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>γ</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>γ</mi><mi>x</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr><mtr><mtd><mi>Y</mi><mo>=</mo><mi>y</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>y</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mi>y</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>γ</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>γ</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228435&type= 69.34200287 9.48266602 . 对 公式（4） 公式（4） 求一阶偏导数： <math display="block" id="M11"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>X</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">5</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>X</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">2</mn><mi>β</mi><mi>x</mi><mi>y</mi><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>y</mi><mo>+</mo><mi>x</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>Y</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">2</mn><mi>β</mi><mi>x</mi><mi>y</mi><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>y</mi><mo>+</mo><mi>x</mi><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>Y</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">5</mn><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228439&type= 68.49533844 38.94666672 . 选择离轴光学系统的物方轴外某视场内的某点 <math id="M12"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228443&type= 11.51466751 4.91066647 ，在其邻域内进行一阶展开，可得： <math display="block" id="M13"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr><mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>X</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>∂</mo><mi>X</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>Y</mi></mrow><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>∂</mo><mi>Y</mi></mrow><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></msub><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>y</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mi>B</mi></mtd></mtr><mtr><mtd><mi>C</mi></mtd><mtd><mi>D</mi></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>y</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228447&type= 141.22399902 22.35199928 ， 其中： <math id="M14"><mi>A</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228450&type= 69.68067169 5.16466665 ， <math id="M15"><mi>B</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228454&type= 48.93733215 5.16466665 ， <math id="M16"><mi>C</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228457&type= 48.93733215 5.16466665 ， <math id="M17"><mi>D</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228461&type= 70.35800171 5.16466665 。 式（6） 式（6） 中，理想像点的位置为 <math id="M18"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228466&type= 13.12333298 4.48733330 。考虑非线性关系，在 X 和 Y 方向的放大率不同，并且有数值大小相等的交叉项存在。其中，主要部分为矩阵中的左上和右下元素，主要影响光学系统在 x 和 y 方向的放大率，次要部分是左下和右上部分，主要影响系统的偏流角度。公式可进一步表达为： <math display="block" id="M19"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr><mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>x</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>y</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>y</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>y</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228470&type= 60.45199966 9.39799976 ， 其中： <math display="block" id="M20"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd></mtr><mtr><mtd><msub><mrow><mi>l</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd></mtr><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>θ</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">6</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228474&type= 128.27000427 40.21666718 . 2.2　离轴光学系统的等效焦距 2 在上述推导的基础上，定义等效焦距为物方 <math id="M21"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228479&type= 11.51466751 4.91066647 邻域所对应的像面上的大小，则两个方向的等效焦距分别为： <math display="block" id="M22"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>f</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>=</mo><mi>f</mi><mo>·</mo><mroot><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd></mtr><mtr><mtd><msub><mrow><mi>f</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>=</mo><mi>f</mi><mo>·</mo><mroot><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">5</mn><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi>β</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228482&type= 130.30200195 15.49400043 ， 其中， <math id="M23"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228486&type= 2.45533323 3.64066648 为等效同轴系统焦距。 由 公式（7） 公式（7） 和（9）可知，只要解算得到离轴光学系统的高阶畸变系数 <math id="M24"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228593&type= 2.28600001 3.64066648 和 <math id="M25"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228595&type= 1.94733346 3.64066648 ，即可完成离轴光学系统的畸变分析及焦距测量。 2.3　离轴光学系统的相对畸变分离 2 结合ZEMAX软件中的光线追迹数据，利用多项式拟合的方法，解算得到离轴光学系统的理论高阶畸变系数。 选取某特殊视场（例如：视场角0°，离轴角4.15°），对离轴和视场方向的角度在全视场范围内进行归一化，进一步结合ZEMAX软件中的像高数据，利用Matlab中的polyfit函数对归一化后的角度量矩阵和实际像高矩阵进行五阶多项式拟合，将系统的相对畸变1.413%进行分离，最终得到光学系统的三阶畸变系数 <math id="M26"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228593&type= 2.28600001 3.64066648 和五阶畸变系数 <math id="M27"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228595&type= 1.94733346 3.64066648 ： <math display="block" id="M28"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.013</mn><mtext> </mtext><mn mathvariant="normal">26</mn></mtd></mtr><mtr><mtd><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0.000</mn><mtext> </mtext><mn mathvariant="normal">86</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228503&type= 22.69066620 9.90600014 . 为了验证上述高阶畸变系数拟合的正确性和精度，将 <math id="M29"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228505&type= 2.28600001 3.64066648 和 <math id="M30"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228507&type= 1.94733346 3.64066648 代入等效焦距 公式（9） 公式（9） ，在典型的离轴角和视场角的工况下，与ZEMAX光线追迹的结果进行对比，二者的结果对比如 表1 表1 所示。 离轴角 Y /（°） 视场角 X/ （°） 方向 五阶畸变系数/mm ZEMAX光线追迹/mm 绝对偏差/mm 2 3.96 2 0 X 8 836.990 8 836.983 0.007 Y 9 019.132 9 019.185 -0.053 2 3.96 2 0.5 X 8 841.366 8 841.340 0.026 Y 9 020.857 9 020.887 -0.03 2 3.96 2 1 X 8 854.559 8 854.453 0.106 Y 9 026.046 9 026.009 0.037 2 3.96 2 1.5 X 8 876.756 8 876.514 0.242 Y 9 034.740 9 034.596 0.144 2 3.96 2 1.75 X 8 891.323 8 891.447 -0.124 Y 9 040.422 9 040.688 -0.266 2 4.33 2 0 X 8 855.021 8 855.017 0.004 Y 9 076.584 9 076.771 -0.187 2 4.33 2 0.5 X 8 859.478 8 859.787 -0.309 Y 9 078.388 9 078.554 -0.166 2 4.33 2 1 X 8 872.910 8 872.790 0.12 Y 9 083.816 9 083.920 -0.104 2 4.33 2 1.5 X 8 895.508 8 895.235 0.273 Y 9 092.909 9 092.917 -0.008 2 4.33 2 1.75 X 8 910.336 8 909.970 0.366 Y 9 098.851 9 098.803 0.048 表1 表1 中可以看出，当采用五阶畸变进行拟合时，最大偏差为+0.366 mm，相对偏差为+0.004 1%，拟合精度基本满足工程实际需求，接下来将利用上述系统畸变的仿真分析方法对该离轴光学系统的镜头畸变及焦距进行测量。 3 离轴系统的畸变及焦距测量方法 1 离轴光学系统的畸变测量原理如 图1 图1 所示。将精细刻画和标定过的网格板精确地放置在离轴光学系统的焦平面上，并且使其刻面的中心与光轴重合。转动精密转台使其上的网格板各刻线到测位，在物方用精密测角仪在转台上绕着被测离轴光学系统的入瞳旋转观测，测量网格板上不同刻线的实际像对应的物方二维视场角。 具体步骤如下： （1）经纬仪1对准离轴光学系统的基准棱镜，经纬仪2放在离轴光学系统的入光口处，利用两个经纬仪互瞄，不断调整经纬仪2的角度，使得经纬仪2瞄准离轴光学系统的中心视场，即 <math id="M31"><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228520&type= 9.90600014 3.97933316 ， <math id="M32"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228522&type= 3.13266683 3.72533321 为系统的离轴角； （2）将精细刻画和标定过的网格板大致放在系统的焦平面上，不断调整网格板的姿态，使得离轴光学系统入光口处的经纬仪2能够精确地瞄准网格板的中心叉丝，且经纬仪2水平、竖直扫描时能够与网格板上的特征线完全重合，此时经纬仪2已精确地放置在离轴系统的焦平面上； （3）将经纬仪2瞄准网格板上的中心叉丝并清零二维角度，利用六维调整机构对离轴光学系统的姿态进行高精度调整，使得经纬仪分别瞄准网格板上的各刻线，以系统的中心视场 <math id="M33"><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228524&type= 9.90600014 3.97933316 为基准，在系统工作视场范围内，记录网格板上 <math id="M34"><mi>M</mi><mo>×</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228569&type= 11.59933376 2.87866688 个不同位置时经纬仪的二维角度 <math id="M35"><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mo>⋯</mo><mo>,</mo><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228528&type= 70.61199951 4.82600021 。 已知离轴反射系统的离轴角为 <math id="M36"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228530&type= 3.13266683 3.72533321 ，进一步假设工作视场为 <math id="M37"><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><mi>α</mi><mo>×</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228532&type= 17.44133186 3.64066648 ，即离轴角方向的工作视场为（ <math id="M38"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228534&type= 12.36133385 3.72533321 ）~（ <math id="M39"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228536&type= 12.36133385 3.80999994 ），视场角方向的工作视场为 <math id="M40"><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228539&type= 7.87400007 3.64066648 ~ <math id="M41"><mi mathvariant="normal">Δ</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228541&type= 4.48733330 3.64066648 ， 则该离轴系统工作视场内的最大空间立体角为 <math id="M42"><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228543&type= 33.10466385 5.24933338 ，不论针对物方角度还是像方角度，都应采用工作视场内的最大空间立体角进行归一化，即在公式中， <math id="M43"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228545&type= 9.05933285 3.97933316 为归一化后的物方角度， <math id="M44"><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228547&type= 10.92199993 3.55599999 为归一化后的像方角度。物方角度归一化的计算过程为： <math display="block" id="M45"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228549&type= 52.23933029 23.96066666 ， 其中：物方真实角度为 <math id="M46"><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228551&type= 10.66800022 4.82600021 ，则归一化后的物方角度为 <math id="M47"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228553&type= 9.05933285 4.82600021 。 同理，像方角度归一化的计算过程见公式，像方真实角度为 <math id="M48"><mfenced separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228555&type= 12.27666664 3.97933316 ，则归一化后的物方角度为 <math id="M49"><mfenced separators="|"><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228557&type= 10.92199993 3.97933316 。 <math display="block" id="M50"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>X</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mi>Y</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228559&type= 52.83200073 23.96066666 . 将公式进一步变形为： <math display="block" id="M51"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">{</mo><mi>X</mi><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">[</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">{</mo><mi>Y</mi><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">[</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228561&type= 66.63265991 13.12333298 . 假设等效同轴系统焦距为 <math id="M52"><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228563&type= 3.04800010 3.89466691 ，则可求得所对应的理论像高为： <math display="block" id="M53"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mn mathvariant="normal">1,1</mn></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mn mathvariant="normal">1,1</mn></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>N</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>N</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>=</mo><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>·</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228565&type= 31.83466530 26.67000008 . 已知网格板上的各刻线对应的实际二维像高值为 <math id="M54"><msub><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mn mathvariant="normal">1,1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mn mathvariant="normal">1,1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>N</mi></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>N</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228567&type= 39.11600113 4.14866638 ，综合考虑离轴光学系统 <math id="M55"><mi>M</mi><mo>×</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228569&type= 11.59933376 2.87866688 组理论像高和实际像高，定义评价函数为公式，当评价函数取得最小值时，即可解算得到等效同轴系统焦距为 <math id="M56"><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228571&type= 3.04800010 3.89466691 、三阶畸变系数 <math id="M57"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228593&type= 2.28600001 3.64066648 和五阶畸变系数 <math id="M58"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228595&type= 1.94733346 3.64066648 ： <math display="block" id="M59"><mi>M</mi><mi>F</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>M</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow/></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow/></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228577&type= 65.87065887 8.97466660 . 4 实验结果与分析 1 针对评价函数 <math id="M60"><mi>M</mi><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228579&type= 6.09600019 2.87866688 中，采用多维约束非线性优化方法，可以解算得到等效同轴系统焦距为 <math id="M61"><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228581&type= 3.04800010 3.89466691 ，三阶畸变系数 <math id="M62"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228583&type= 2.28600001 3.64066648 和五阶畸变系数 <math id="M63"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228585&type= 1.94733346 3.64066648 ，物方二维角度和网格板上的像高数据见 表2 表2 ，解算结果见 式（16） 式（16） ，与理论等效同轴系统焦距8 750 mm相比，实测结果的偏差仅为6.378 mm，相对偏差为0.073%。 2 序号 2 物方角度/（°） 2 实际像高数据/mm X Y X Y 1 4.237 -1.532 8 656.761 -237.195 2 3.983 -1.194 7 615.960 -184.485 3 4.240 -0.852 6 656.760 -131.775 4 3.985 -0.512 5 615.961 -79.065 5 4.241 -0.170 6 656.759 -26.355 6 3.984 0.170 8 615.962 26.355 7 4.239 0.511 7 656.761 79.065 8 3.985 0.853 8 615.961 131.775 9 4.242 1.193 0 656.762 184.485 10 3.982 1.535 2 615.960 237.195 <math display="block" id="M64"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">8</mn><mtext> </mtext><mn mathvariant="normal">756.378</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></mtd></mtr><mtr><mtd><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.013</mn><mtext> </mtext><mn mathvariant="normal">258</mn></mtd></mtr><mtr><mtd><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0.000</mn><mtext> </mtext><mn mathvariant="normal">863</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228589&type= 32.17333603 15.32466698 . 在上述研究成果的基础上，针对在轨图像畸变校正的应用需求，还开展了探测器上特征点对应地面实际位置的应用研究。如 表3 表3 所示，针对每片探测器上的3个特征点，基于上述实测的等效同轴系统焦距 <math id="M65"><msup><mrow><mi>f</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228591&type= 3.04800010 3.89466691 、三阶畸变系数 <math id="M66"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228593&type= 2.28600001 3.64066648 和五阶畸变系数 <math id="M67"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37228595&type= 1.94733346 3.64066648 ，由像方角度可以计算得到物方角度，最终计算得到对应的地面实际位置，探测器上3个特征像元对应的地面位置如 图2 图2 所示（第1片探测器的第1个特征点标为：点1-1，其余表示相同）。将该方法推广至离轴光学系统所有的探测器，即可得到离轴系统整个探测器对应的真实地物坐标，为相机在轨图像的畸变校正提供重要依据，可实现相机在轨图像准实时畸变校正。 探测器序号 特征点1 特征点2 特征点3 1 （-210.315，667.8） （-237.195，667.8） （-264.075，667.8） 2 （-157.605，627） （-184.485，627） （-211.365，627） 3 （-104.895，667.8） （-131.775，667.8） （-158.665，667.8） 4 （-52.185，627） （-79.065，627） （-105.945，627） 5 （0.525，667.8） （-26.355，667.8） （-53.235，667.8） 6 （53.235，627） （26.355，627） （-0.525，627） 7 （105.945，667.8） （79.065，667.8） （52.185，667.8） 8 （158.655，627） （131.775，627） （104.895，627） 9 （211.365，667.8） （184.485，667.8） （157.605，667.8） 10 （264.075，627） （237.195，627） （210.315，627） mm 5 结 论 1 本文对离轴光学系统的理论畸变进行了分析，利用高阶畸变系数对系统的相对畸变进行了分离，并针对某装调完成的离轴光学系统，完成了畸变分析测试和焦距测量，与ZEMAX光线追迹的结果相比，等效焦距最大偏差为0.366 m，精度满足工程实际需求；针对某装调完成的离轴光学系统完成了畸变分析测试和焦距测量，实测等效同轴系统的焦距偏差仅为6.378 mm，进一步针对相机焦平面上的特征点完成了物像对应关系，为相机在轨图像的快速、高精度畸变校正提供了准确的输入。 本文初步实现了离轴光学系统的畸变分析、测试及焦距测量，下一步将综合考虑网格板的刻画精度及经纬仪的测角精度，进一步改进优化算法，提高其解算的鲁棒性，以期在未来几十米甚至一百米量级的长焦距离轴光学系统中得到更为广阔的应用前景。 参考文献 罗敬 ， 王金鑫 ， 鞠国浩 ， 等 . 主动随机送风条件下大口径离轴三反光学系统的计算机辅助装调 ［J］. 光学 精密工程 ， 2022 ， 30 （ 7 ）： 802 - 812 . doi: 10.37188/OPE.20223007.0802 http://dx.doi.org/10.37188/OPE.20223007.0802 主动随机送风条件下大口径离轴三反光学系统的计算机辅助装调 LUO J ， WANG J X ， JU G H ， et al . Computer-aided alignment for large-aperture off-axis three mirrors optical system with artificially random air flow ［J］. Opt. Precision Eng. ， 2022 ， 30 （ 7 ）： 802 - 812 . （in Chinese） . doi: 10.37188/OPE.20223007.0802 http://dx.doi.org/10.37188/OPE.20223007.0802 Computer-aided alignment for large-aperture off-axis three mirrors optical system with artificially random air flow 王国良 ， 刘金国 ， 龙科慧 ， 等 . 离轴三反航天测绘相机像移对成像质量的影响 ［J］. 光学 精密工程 ， 2014 ， 22 （ 3 ）： 806 - 813 . doi: 10.3788/ope.20142203.0806 http://dx.doi.org/10.3788/ope.20142203.0806 离轴三反航天测绘相机像移对成像质量的影响 WANG G L ， LIU J G ， LONG K H ， et al . Influence of image motion on image quality of off-axis TMA aerospace mapping camera ［J］. Opt. Precision Eng. ， 2014 ， 22 （ 3 ）： 806 - 813 . （in Chinese） . doi: 10.3788/ope.20142203.0806 http://dx.doi.org/10.3788/ope.20142203.0806 Influence of image motion on image quality of off-axis TMA aerospace mapping camera 郭疆 ， 孙继明 ， 邵明东 ， 等 . 离轴三反航天测绘相机焦距的计算 ［J］. 光学 精密工程 ， 2012 ， 20 （ 8 ）： 1754 - 1758 . doi: 10.3788/ope.20122008.1754 http://dx.doi.org/10.3788/ope.20122008.1754 离轴三反航天测绘相机焦距的计算 GUO J ， SUN J M ， SHAO M D ， et al . Calculation of focal length for off-axis TMA aerospace mapping camera ［J］. Opt. Precision Eng. ， 2012 ， 20 （ 8 ）： 1754 - 1758 . （in Chinese） . doi: 10.3788/ope.20122008.1754 http://dx.doi.org/10.3788/ope.20122008.1754 Calculation of focal length for off-axis TMA aerospace mapping camera 王孝坤 . 一种高分辨力空间相机的光学设计 ［J］. 光学学报 ， 2015 ， 35 （ 1 ）： 329 - 337 . doi: 10.3788/aos201535.0122001 http://dx.doi.org/10.3788/aos201535.0122001 一种高分辨力空间相机的光学设计 WANG X K . Optical design of a high resolution space camera ［J］. 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Changchun ： Jilin University ， 2019 . Research on Precision Measurement Technology of Optical System Distortion 陈杨 ， 王跃明 . 大像场离轴三反望远镜畸变特性分析与设计 ［J］. 光学学报 ， 2013 ， 33 （ 2 ）： 193 - 197 . doi: 10.3788/aos201333.0222003 http://dx.doi.org/10.3788/aos201333.0222003 大像场离轴三反望远镜畸变特性分析与设计 CHEN Y ， WANG Y M . Design and distortion characteristics analysis of the large-image-field off-axis three-mirror telescope ［J］. Acta Optica Sinica ， 2013 ， 33 （ 2 ）： 193 - 197 . （in Chinese） . doi: 10.3788/aos201333.0222003 http://dx.doi.org/10.3788/aos201333.0222003 Design and distortion characteristics analysis of the large-image-field off-axis three-mirror telescope 10.37188/OPE.20223022.2839 O439 A 1004-924X（2022）22-2839-08 程强,胡海翔,李龙响等.离轴光学系统的畸变分析及焦距测量[J].光学精密工程,2022,30(22):2839-2846. 程强,胡海翔,李龙响等.离轴光学系统的畸变分析及焦距测量[J].光学精密工程,2022,30(22):2839-2846. CHENG Qiang,HU Haixiang,LI Longxiang,et al.Distortion analysis and focal length testing of off-axis optical system[J].Optics and Precision Engineering,2022,30(22):2839-2846. CHENG Qiang,HU Haixiang,LI Longxiang,et al.Distortion analysis and focal length testing of off-axis optical system[J].Optics and Precision Engineering,2022,30(22):2839-2846. 中国科学院青年创新促进会资助项目（No.2022213）；国家自然科学基金资助项目（No.12003034，No.12003035，No.11903035，No.12103054，No.61805243，No.6207031149） 2022-04-25 2022-06-14 2022-11-25 2022-11-24 光学精密工程 22 30