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This paper presents a simple method for flat mirror tilt and piston measurement. The method only needs an LCD screen and a CCD camera. LCD screen displays structured light pattern, and camera takes the pattern’s virtual image (VI) reflected by the mirror. Pose relationship between camera coordinate system and VI coordinate system can be calculated by camera calibration. Through coordinate transition, the relative tilt and piston of the mirror can be obtained. This method is simple in both structure and principle, and experiments prove that it can measure the flat mirror tilt for about 1″ accuracy and piston for less than 1um accuracy. And under the guidance of our method, a mirror can be adjusted to a specified pose.
© 2014 Optical Society of America
1. Introduction
Higher resolution telescopes need larger aperture mirrors, which leads to more and more difficult manufacturing and measurement [1]. Some scientists introduced the method that stitch several secondary mirrors together to make a large primary mirror [2–4], this technology has reduced the difficulty of manufacturing large mirror optical systems, and simplified the supporting structures due to dramatically decreasing the weight of the primary mirror. However, the stitch technique requires that sub-mirrors must be adjusted to the same surface accurately, and maintained in a good state of stitching during using process. This raises new challenges to optical measurement and adjustment. Usually, mirror stitching process can be divided into rough stitching and precise stitching; different stages have different requirements on measurement accuracy.
Large flat mirrors are often used as reflecting correctors in telescope systems [5]. In the broader sense, they can be treated as free form surfaces, spherical or aspheric mirrors with special properties. Thus the study of flat mirror stitching is also a prelude for mirrors with other shapes. Flat mirror pose measurement can guide and help in mirror stitching, which includes pendulum pose (tilt/tip, here we use tilt for convenient) and displacement along the vertical axis (piston) measurement [6]. In the adjustment of optical or machine components, the angle measurement is very common and important nowadays [7]. Usually the mirror tilt angle is generally measured by laser interferometers [8] and photoelectric autocollimators [7, 9]. Both appliances perform high accuracy, but the setup of laser interferometer is difficult, massive and expensive, and requires harsh environment condition; photoelectric autocollimator is relatively easy to set up and convenient to read angles, but the measurement range is fairly small. Flat mirror piston can be measured by displacement interferometers [10], laser ranging systems [11], dial gauges [12], etc. In which the displacement interferometers perform the best accuracy (nanometer level) except for the massive setup and high price; laser ranging systems are easy to use, but the accuracy is in millimeter level, and the application is mainly focused on very long distance measurement; the accuracy of dial gauges is in micron level, while the stableness is weak, and there exists a certain mechanical hysteresis.
The paper presents a simple method for flat mirror tilt and piston measurement. The method only needs an LCD screen and a CCD camera. LCD screen displays crossed sine fringe pattern, and CCD camera takes the patterns’ VIs reflected by the mirror. The calibration characteristic points of the VIs are extracted by Fourier method, and then the spatial relationships between camera coordinate system and VI coordinate systems are determined by camera calibration. Through coordinate transition, and according to the reflection principle, relative pendulum poses and pistons of the mirror can be obtained. This method is simple in both structure and principle. The experiments show that it can measure pendulum pose and displacement of the flat mirror accurately; further, under the guidance of our method, a mirror can be adjusted to a specified pose.
2. Camera calibration
2.1 Camera model
Generally, nonlinear camera model considering aberration [13, 14] is used to describe the image formation, which is the center perspective projection linear camera model with lens distortion aberration. The camera image formation model is shown in Fig. 1. A spatial point p at location (X, Y, Z) in world coordinates and (u, v) in image coordinates is taken for example. For spatial points Z = 0, the 2D planar center perspective projection image relation is:
where is the non-zero scale factor; K means camera intrinsic parameters matrix; and represent for the normalized matrixes of u and v axis, respectively; and is the principal point coordinate. These are camera intrinsic parameters, which describing the internal geometric and optical characteristics of a camera. R and T are camera extrinsic parameters, representing the rotation matrix and translation vector between camera and world coordinate systems.Assume 3D calibration characteristic points are donated by , , , their corresponding image coordinates are donated by . Calculate R and T, for
where , is a orthogonal matrix; .2.2 Characteristic point extraction
The camera calibration accuracy heavily relies on the accuracy of calibration characteristic points. Comparing with commonly used calibration patterns, such as checkerboard pattern, Gaussian point grid pattern and so on, crossed sine fringe pattern based on Fourier fringe analysis performs best under noise and defocusing, and has the highest precision [15]. Thus we use crossed sine fringe pattern to extract characteristic points.
The intensity function of crossed sine fringe pattern (Fig. 2(a)) is:
generally, set a = 1/2, b1 = b2 = 1/4, φx0(x, y) and φy0(x, y) are the initial phases, respectively. px and py represent for the period of the stripe in two orthogonal directions.
Fig. 2 Characteristic point extraction. (a) Crossed sine fringe pattern. (b) Extracted characteristic points.
Characteristic points are locally brightest points, and the extraction process [16] is:
- (1) Through Fourier transform, frequency domain filtering and inverse Fourier transform, wrapped phase distributions in two orthogonal directions can be extracted.
- (2) By unwrapping the phases within a certain window size, we get the unwrapped phase distribution of u and v directions in the small region. Then 2D phases are fitted by second order polynomial fitting, and the locally brightest sub-pixel coordinates (Fig. 2(b)) can be obtained.
3. Mirror pose measuring principle
3.1 Tilt calculation
The tilt measurement principle is shown in Fig. 3. Intrinsic parameters of the camera are calibrated in advance. We capture reflected VIs when the mirror is at Pos. 1 and Pos. 2, and extract two sets of characteristic points, respectively. Then the rotation matrix R and translation vector T of the two VIs are calculated through extrinsic parameter calibration. Given an arbitrary pixel point q in the camera coordinate system, p1 and p2 are the corresponding points on VI 1 and VI 2, respectively. Then we have:
where subscripts are used to distinguish different VIs. The rotation matrix between two VIs is obtained from Eqs. (4):R12 can be defined by 3 Euler angles , , and , under the following decomposition [17]:
where rij is the entry from the ith row and the jth column of the matrix R12 and atan2(y, x) means the two-argument inverse tangent function. In our calculation model, and represent the VI tilt about x and y axis, respectively; is meaningless here, because the VI does not rotate about z axis. According to the principle of mirror reflection [18], the rotated angle of the mirror is a half of that of VIs, thus the synthesized mirror tilt angle can be obtained from:when and are small, Eq. (7) can be approximated to3.2 Piston calculation
If the mirror is displaced on z axis without any tilt, just as Fig. 4 shows. We can solve translation vector between two VIs from Eq. (4):
Provided that , are pistons of the coordinate system of the second VI in x, y, z directions, respectively. Under idealistic circumstance, tx and ty would be 0, because the VI does not move no matter how flat mirror moves in x and y directions. tz is used to calculate the piston Z of the flat mirror along z direction, according to the principle of reflection, Z = tz/2.
4. Experiment
4.1 Mirror tilt calibration by photoelectric autocollimator
Calibration setup of mirror tilt using a 2D photoelectric autocollimator is shown in Fig. 5(a). The 60 × 80mm test flat mirror is fixed on the circular table supported by 3 PZTs (Fig. 5(b)). Since the photoelectric autocollimator is worked horizontally, we used another flat mirror to deflect the light path. Driven by PZT, the test mirror tilts whilst the deflect mirror keeps fixedly. Relative angle change between 2 mirrors is the tilt angle of the tested mirror.
Fig. 5 Mirror tilt calibration setup. (a) Setup of tilt calibration. (b) Interior detail of PZT controlled 3 points support unit.
Figure 6 shows calibration results of photoelectric autocollimator, set the other two PZT elongations to a fixed value, elongate the third PZT substantially from 0um to 100um, and record each tilt angle corresponds to each elongation. Due to the deformation of machinery assembly, elongation value displayed on the control box is not exactly the real PZT elongation; here we take displayed elongation value as a reference scale for our experiment. Three times of calibration are carried out; Fig. 6(a) shows the relationship between elongations and angles calibrated by photoelectric autocollimator. Figure 6(b) shows the results of removing the mean value. We find the PV value of calibration error is under 4”, indicating that the result of photoelectric autocollimator calibration is reliable.
Fig. 6 Calibrated mirror tilt results. (a) Results of 3 calibrations. (b) Removing mean value from calibrated results.
4.2 Mirror tilt measuring experiment
Experiment setup for tilt measurement is described in Fig. 7, a SAMSUNG screen of 1280 × 1024 pixels displays crossed sine calibration pattern, and a Baumer camera of 1392 × 1040 pixels captures VIs of calibration pattern via the flat mirror. All our experiments are carried out under ordinary environment with light turned on.
Increasing the elongation of the calibrated PZT continuously, we get mirror tilt angles using our method. Figure 8(a) is the 5 repeated measurement results of the PZT, where the angle means the synthesized angle of x, y directions. Subtracting the mean value from measurement results, we get random errors, as shown in Fig. 8(b). RMS values of 5 random errors are 1.2230”, 0.9786”, 1.1221”, 1.3465”, 1.1370”, which proves the repeatability of our method is good.
Fig. 8 Tilt results of 5 measurements. (a) Results of 5 measurements. (b) Random error. (c) System error. (d) Error of 6th measurement.
Subtracting the mean calibration value appeared in Sec. 4.1, we find a relatively fixed distribution difference (Fig. 8(c)), which is probably caused by system assembly, inaccurate of camera intrinsic parameters, or characteristic point matrix chosen. All those factors are systematic, thus we set the mean value of the 5 distribution differences as system error.
Then the 6th measurement is carried out, remove the mean value of 5 measurements and system error above, the error of 6th measurement is obtained, see Fig. 8(d). The PV value is 7.0912”, and RMS is 1.2697”, which proves the accuracy of structured light reflection method can reach arc sec level for measuring flat mirror tilt, which is fairly enough for rough stitching measurement. In addition, the tilt measurement method can be used to measure angle tilts of mirrors or mechanisms, which is practical and usable at present.
According to the inherent property of camera calibration technique, as long as characteristic points been captured and extracted correctly, the spatial position of VI can be calculated out. For example, Fig. 9 shows two VI planes under camera centered view by camera calibration technique. The angle difference of those two VIs is 40.2527°, and the mirror tilt angle is 20.1263°; which is not the limit of our method by using camera calibration. Thus our tilt measurement range can be much larger than either laser interferometer ( ± 5°) [7] or optoelectronic autocollimator (about ± 450″).
4.3 Mirror piston measuring experiment
Experiment setup of piston measurement is shown in Fig. 10. The mirror piston is driven by an elevating stage, and displaced value is recorded using a digital dial gauge with the resolution of 1um, as a reference to evaluate the structured light projection method.
We make the mirror move up and the mirror tilts are kept unchanged as good as possible, Ten positions are chosen to take pictures of reflected crossed sine fringe pattern and the corresponding digital dial gauge readings are recorded. Piston results comparison between our method and dial gauge are shown in Table 1. It can be seen that, the two sets of data are generally fit; differences between two measurement applications are under 1um. Because the resolution and accuracy of the dial gauge is 1um, we can conclude that our method has at least the same accuracy as dial gauges. The rough stitching requirement on piston is about 2-3 wavelengths, which is 1-2um. Thus our method can fill the need of rough stitching.
Table 1. Piston measuring results
The crossed sine fringe has a good performance under camera defocusing [15]. According to our tests, the measurement range is about 30mm. Generally, the required piston range in the mirror stitching is less 5 mm, thus piston measurement range of our method should meet our need.
4.4 Mirror pose adjusting experiment
In mirror stitching measurement, an actual or virtual reference mirror is often needed to assist measurement and eliminate system error. The reference mirror is on the position of coordinate part of primary mirror. By adjusting a mirror to the position determined by reference mirror, stitching can be accomplished. Here we set a virtual reference mirror as a target position beforehand, and then drive the 3 PZTs and elevating stage to adjust the tilt and piston of the mirror. Using our method to measure the adjust tilt and piston value, we adjust the mirror as close as possible from the reference mirror. 3 times of adjustment are carried out; error between the target position and 3 adjusted positions is shown in Table 2. The best one is No. ② with piston error of 0.227um, and angle error of only 0.2353”. The result indicates that our method can be used to guide for adjusting a mirror to desired pose.
Table 2. Adjusting results
5. Conclusions
This paper presents a simple method for mirror tilt and piston measurement. The method only needs an LCD screen and a CCD camera. LCD screen displays structured light pattern, and CCD camera takes the patterns’ VIs reflected by the mirror. Based on well developed camera calibration technique, the pose relationship between the camera coordinate system and the VI coordinate systems can be obtained. Then the tilt angle and piston of the mirror can be calculated through coordinate transition and mirror reflection principle. This method is simple in both structure and principle, and has low requirement for the environment; experiments have shown its accuracy and feasibility, which meet the requirement of rough stitching adjustment. In addition, it has large measurement ranges in both tilt and piston measurements, which makes mirror adjustment more convenient. Because our method is easy, inexpensive and adaptive to the environment, we believe that it will be a prospective method in future projects of mirror stitching.
Acknowledgments
The authors acknowledge the support by the National Natural Science Foundation of China (61205007) and the Science and Technology Innovation Team of Sichuan Province (No. 2011JTD0001).
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