## Abstract

A stereovision-assisted sub-aperture stitching (SAS) approach is proposed to solve the positioning problem of tested optics. The principle of stitching is analyzed and the overlapping calculation problem is simplified. Then, a binocular stereovision system is integrated into a test configuration. By measuring the coordinates of marks attached to the fixture in different positions, the stereovision system obtains the position and pose data of the optics to provide the initial values for optimization algorithm. The key benefit of stereovision is that the SAS test is easily performed without a precision positioning system. A simulation and practical examples are given to demonstrate the performance of this method. The stitching result shows good agreement with the full-aperture result.

© 2010 OSA

## 1. Introduction

Sub-aperture stitching is an effective way to extend the lateral and vertical dynamic range of a conventional interferometer. Large optics, particularly convex optics, can be especially challenging to test using conventional methods. Although sub-aperture stitching can be challenging, it is an attractive alternative to expensive large-aperture interferometers [1,2]. In this method, a precise platform with six degrees of freedom is needed to implement the complex adjustment of the tested optics. Generally, there are two types of adjusting mechanisms: automated and semi-automated. In the first, a fully integrated and automated adjustor is used. For example, QED Technologies Company has developed a series of interferometer workstations that can perform high-accuracy automated sub-aperture stitching [3–5]. These workstations are based on a six-axis precision stage system built by Schneider Optic-machines [5]. The stage system can automatically bring each sub-aperture into null adjustment and give the accurate position and pose data of the optics [5]. Despite its convenience, the high production cost of such automated system prohibits its wider application. In semi-automated mechanisms, several motion mechanisms, including electrical and manual components, are combined to adjust the optics [6–10]. However, this type of system is complex, and its center height is usually much higher than that of the interferometer, making the system very sensitive to environmental vibration. Furthermore, due to existing manual operations, it is impossible to obtain precise position and pose parameters of the tested optics. Although the positioning error can be partly compensated by the optimization algorithm [11], the compensation ability of the algorithm is limited. In fact, the more accurate the initial values of these parameters, the better the result of the optimization algorithm.

Many of the stitching methods mentioned rely on relatively accurate sub-aperture positioning. We propose using a simpler, less accurate motion system for positioning the sub-apertures. The accuracy of the stitched result, however, suffers when the sub-aperture position is not known well. Therefore we include a stereovision system to determine the approximate position of the sub-apertures prior to performing the stitching operation. The minor errors in the position data will be overcome by the optimization process of the stitching algorithm. This method combines the use of a manual adjustor and of a stereovision positioning system. It does not require significant changes to the interferometric test system, thus ensuring the stability of the system. It is simple and conveniently integrated with different interferometers, enabling the SAS test to be easily performed by using this method. This paper is organized as follows. In Section 2, the theory and the algorithm description of our stereovision-assisted stitching are given. In Section 3, the impact of different levels of positioning error on stitched results is shown by simulation. In Section 4, we demonstrate the performance of our stereovision system and report on the 15-subaperture testing of a convex sphere. Finally, in Section 5, our conclusions are presented.

## 2. Theory

#### 2.1 Principles of the SAS test

Taking a convex surface as an example, a typical SAS is shown in Fig. 1(a)
. The relationship between the *i*-th sub-aperture coordinate (${O}_{i}$) and the global coordinate (${O}_{g}$) is shown in Fig. 1(b). ${O}_{i}$ and ${O}_{g}$are constructed at the centers of the *i*-th and the central sub-apertures, respectively. The transformation matrix ${g}_{i}$ between ${O}_{i}$ and ${O}_{g}$is described in Eq. (1):

According to the test geometry of the Fizeau interferometer [11], the phase data $(u,v,\phi )$ of the sub-aperture and the object coordinates $(x,y,z)$ are related by Eq. (2):

*l*denotes the scale of lateral coordinates. Then, $(u,v,\phi )$ is transformed to a global point $(X,Y,Z)$ using Eq. (1) and Eq. (2). Here, ${r}_{ts}$is known and

*l*is calibrated. ${r}_{bs}$and the six parameters of ${g}_{i}$ are obtained by the stereovision system.

#### 2.2. Overlapping calculation and parameter optimization

The six parameters of the transformation matrix obtained by the stereovision system are usually further optimized by minimizing the mismatch in the overlapping area. The overlapping correspondence in the overlapping area needs to be found in advance to calculate the mismatch. Now that all phase data are transformed to their corresponding global 3D coordinates, the overlapping correspondence problem can be simplified as follows.

First, a uniform grid is defined on the X-Y plane. Then, all of the points in the *i*-th and *j*-th sub-apertures are projected onto the X-Y plane. For the sake of conciseness, only the grid points falling in the overlapping area are shown as dot points in Fig. 2(a)
. Consider the *k*-th grid point ${p}_{k}$ in the overlapping area, as shown in Fig. 2(b). Its coordinate in the Z direction can be individually interpolated from the data of the *i*-th and *j*-th sub-apertures. Usually, the coordinates so interpolated are different; the difference can be described by Eq. (3).

*σ*is the root mean square (RMS) of $erro{r}_{i,j}$ and ${\sigma}_{0}$ is the RMS value of the difference between the two sub-apertures and the nominal surface in Z direction, which can be calculated by Eq. (5).

#### 2.3 Binocular stereovision system

To set up the binocular stereovision system, two CCD cameras and some marked points are used. The setup is illustrated diagrammatically in Fig. 3(a) . The principles of calibration and measurement have been provided in our previous work [12]. It is worth mentioning that the coordinate system (${O}_{c}$) of the stereovision system does not coincide with that of the stitching system (${O}_{g}$). ${O}_{c}$ is usually established on the left camera, whereas ${O}_{g}$ is attached to the center of the central sub-aperture. The transformation matrix between ${O}_{c}$ and ${O}_{g}$ is obtained as described in the following paragraph.

As shown in Fig. 3(b), the coordinates of the marked points on the workstation are known. Because the fixture is self-centered, the center of the marked points coincides with that of the optics. The center height of the optics is also known. Thus, the coordinates of each marked point in ${O}_{g}$ can be computed. Once the 3D coordinates of each marked point in the central sub-aperture test are obtained by the stereovision system, the transformation matrix between ${O}_{c}$ and ${O}_{g}$ can be calculated. During the SAS test, the 3D coordinates of the marked points are recorded while the sub-apertures' phase maps are being acquired individually. Finally, the transformation matrix between each sub-aperture and the central sub-aperture is calculated using the quaternion method.

## 3. Simulation

The purpose of the simulation was to demonstrate the advantages of the stereovision. In particular, we simulated a stitching optimization with two different levels of sub-aperture positioning error. The first assumed sub-apertures with random positioning error of position and angle of 5 mm and 5 degrees, respectively. This is representative of the errors obtained in a relatively imprecise manual positioning system. The second assumed errors of only 0.1 mm and 0.1 degrees, representative of the expected accuracy of the stereovision system. Figure 4 shows the results of the stitching optimization under these two different sub-aperture positioning error conditions.

In the simulation, the tested optics is a concave paraboloid surface with an aperture of 250 mm and a twice-focal length of 1600 mm. As shown in Fig. 4(a), a total of 19 sub-apertures are acquired by a 4” interferometer with a transmission sphere radius of 1500 mm. Sub-aperture data and position parameters with two different levels of errors are input to the stitching algorithm and further optimized by minimizing the mismatch in the overlapping area. After the stitching is finished, the differences between the stitching results and the nominal surface in the Z direction are output to evaluate the quality of the stitching results. These differences can be thought of as the residual errors following the optimization process. Figure 4(b) displays the error map of stitching without stereovision. This map is not circular, indicating that the stitching result is invalid. In contrast, when the stereovision measurement system is adopted, the residual error of the stitching is greatly reduced, as shown in Fig. 4(c). In the latter case, the residual error is much smaller than the manufacturing error of a general transmission lens, indicating that a stitching result with satisfactory accuracy can be obtained under this condition.

## 4. Experiments

Two experiments were carried out to test the performance of our stereovision-assisted stitching process. In the first, the accuracy of the stereovision system was tested to ensure that the expected sub-aperture positioning knowledge would be achieved. In the second, a test spherical surface was measured using both sub-aperture stitching and a standard full-aperture test for comparison. The experimental setup for the SAS test is shown in Fig. 5(a) . It consists of a Fizeau interferometer (ZYGO GPI XP 4”), two CCD cameras and a 5-axis adjustor with a one-dimensional rotary mechanism to clamp and rotate the optics.

#### 4.1 Stereovision system performance

The accuracy of the stereovision system was evaluated using an electronic control platform (shown in Fig. 5(b)) with translational accuracy of 5 um and rotational accuracy of 0.001 degree. A model with marked points arranged similarly with the actual was fixed on the platform and moved in the X direction using a step length of 5 mm while 30 pairs of images were captured. The 3D coordinates of all marked points in different positions were computed by stereovision. Then the movement distances were obtained; after subtraction of translational values of the platform, this process resulted in measurement errors for every position, as shown in Fig. 6(a) . Next, the model was rotated around the Z axis at 10-degree intervals, and 36 pairs of images were captured. In a manner similar to that used with the distance measurement, the error of each angle was calculated. The results are shown in Fig. 6(b).

#### 4.2 Application to the SAS test of a sphere

In this experiment, the aperture and curvature radius of the convex sphere are 100 mm and 425 mm, respectively. The sphere was measured by the 4” Zygo interferometer with an f/10.7 transmission sphere whose nominal accuracy is 0.05 wave (PV). As shown in Fig. 7(a) , fifteen sub-apertures were obtained individually. The sub-apertures were first stitched together using each sub-aperture's approximate position and pose parameters. The stitching result is shown in Fig. 7(b). The map shows that some parameters, such as the lateral offset and the rotation angles, were not compensated well. After the parameters were solved by the stereovision, the sub-apertures data were stitched again, as shown in Fig. 7(c).

To verify the correctness of the stereovision-assisted stitching results, a cross-test was carried out. The sphere was tested by the Zygo interferometer with a 6” transmission sphere whose nominal accuracy is 0.025 wave (PV). Its full-aperture map is shown in Fig. 8(a) . Note that Fig. 7(c), which shows the stitching result, is slightly different from Fig. 8(a). The averaging of large amounts of sub-aperture data makes the stitching result smoother. For further comparison, the difference between the results obtained using the two methods is shown in Fig. 8(b) which has the same Z scale with Fig. 8(a). It can be seen that the residual error and the manufacturing error of the transmission sphere in the SAS test (PV is 0.049 wave and RMS is 0.007 wave) are at the same level. Therefore, the residual error is acceptable and the stitching result is valid.

## 5. Conclusion

The stereovision system is introduced to measure the position and pose data of the optics in the SAS test. The experimental results show that the practical positioning accuracy of this system is better than 0.08 mm in translation and 0.04 degree in rotation. The simulation shows that this system provides effective initial values for stitching algorithm, and a reconstructed full-aperture data map with satisfactory accuracy is acquired by combining this method with an optimization algorithm. An SAS test on a sphere verified that the optimal stitching result is in good agreement with the full-aperture result.

## Acknowledgments

This work was supported by the National High Technology Research & Development Program of China (2008AA04Z121), the National Natural Science Foundation of China (50975228), and the Creativity Supporting Plan (CXY1007).

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