Fig. 1. Calculation of position errors of marking points, where the triangles are is the positions of marking points, (x, y) is actual coordinates of marking points, and (x', y') is distorted coordinates

Fig. 2. Flow chart of reverse fitting distortion method and traditional method

Fig. 3. Flow chart for optimal calculation of marking points

Fig. 4. Optimal marking point positions when Zernike polynomials with different terms are used to correct circular optical element, in which the figure in the top of the graph is the conditional number. (a) Use the first three terms; (b) use the first four terms; (c) use the first six terms; (d) use the first eight terms; (e) use the first ten terms; (f) use the first eleven terms

Fig. 5. Optimal marking point positions when Zernike polynomials with different terms are used to correct rectangular optical element, in which the figure in the top of the graph is the conditional number. (a) Use the first three terms; (b) use the first four terms; (c) use the first six terms; (d) use the first eight terms; (e) use the first ten terms; (f) use the first eleven terms

Fig. 6. Simulation results of surface shape and distortion. (a) Randomly generated circular surface; (b) simulation of distorted irregular shape

Fig. 7. Fitted error matrices in x and y directions. (a) Error of ideal shape in x direction; (b) error of distorted shape in x direction; (c) error of ideal shape in y direction; (d) error of distorted shape in y direction

Fig. 8. Comparison of correction accuracy of different methods. (a) Original surface shape; (b) the result of subtracting original surface shape from the result of reverse fitting and optimal marking point correction; (c) the result of subtracting original surface shape from the result of reverse fitting and ordinary marking point correction; (d) distorted surface shape; (e) the result of subtracting original surface shape from the result of traditional fitting and optimal marking point correction; (f) the result of subtracting original surface shape from the result of traditional fitting and ordinary marking point correction

Fig. 9. Correction grid comparison of four methods, in which blue lines are original grids, and black lines are the grid after distortion correction. (a) Original grid; (b) correction result of reverse fitting and optimal marking points; (c) correction result of reverse fitting and ordinary marking points; (d) distorted grid; (e) correction result of traditional fitting and optimal marking points; (f) correction result of traditional fitting and ordinary marking points

Fig. 10. Setup of off-axis mirror testing

Fig. 11. Distortion correction results of off-axis mirror, in which black dots are ideal marking point positions, and red dots are actual centroid positions of marking points. (a) Before correction; (b) after correction

Fig. 12. Results before and after processing and triangles in the figures indicate the positions of the used marking points. (a) Before processing; (b) after processing