Improving the accuracy of mirror measurements by removing noise and lens distortion

Telescope mirrors determine the imaging quality and observation ability of telescopes. Unfortunately, manufacturing highly accurate mirrors remains a bottleneck problem in space optics. One main factor is the lack of a technique for measuring the 3D shapes of mirrors accurately for inverse engineering. Researchers have studied and developed techniques for testing the quality of telescope mirrors and methods for measuring the 3D shapes of mirrors for centuries. Among these, interferometers have become popular in evaluating the surface errors of manufactured mirrors. However, interferometers are unable to measure some important mirror parameters directly and accurately, e.g. the paraxial radius, geometry dimension and eccentric errors, and these parameters are essential for mirror manufacturing. In this paper, we aim to remove the noise and lens distortion inherent in the system to improve the accuracy of a previously proposed one-shot projection mirror measurement method. To this end, we propose a ray modeling and a pattern modeling method. The experimental results show that the proposed ray modeling and pattern modeling method can improve the accuracy of the one-shot projection method significantly, making it feasible as a commercial device to measure the shapes of mirrors quantitatively and accurately.


Introduction
In the sixteenth century, the first astronomical telescope to use a refractive lens was invented by the great scientist Galileo. In 1789, Frederick William Herschel established the 1.22 m reflective telescope using specular mirrors. In 1948, the famous Hale telescope, which used a reflective mirror two hundred inches big, was established in San Diego. In 1976, a larger telescope 6 m in diameter and 25 m in length was made in Russia. Later on, more powerful telescopes with larger Telescope mirrors determine the imaging quality and observation ability of telescopes. Unfortunately, manufacturing highly accurate mirrors remains a bottleneck problem in space optics. One main factor is the lack of a technique for measuring the 3D shapes of mirrors accurately for inverse engineering. Researchers have studied and developed techniques for testing the quality of telescope mirrors and methods for measuring the 3D shapes of mirrors for centuries. Among these, interferometers have become popular in evaluating the surface errors of manufactured mirrors. However, interferometers are unable to measure some important mirror parameters directly and accurately, e.g. the paraxial radius, geometry dimension and eccentric errors, and these parameters are essential for mirror manufacturing. In this paper, we aim to remove the noise and lens distortion inherent in the system to improve the accuracy of a previously proposed one-shot projection mirror measurement method. To this end, we propose a ray modeling and a pattern modeling method. The experimental results show that the proposed ray modeling and pattern modeling method can improve the accuracy of the one-shot projection method significantly, making it feasible as a commercial device to measure the shapes of mirrors quantitatively and accurately.
Keywords: measurement, one-shot projection, telescope mirror, ray modeling, pattern modeling, interferometer (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. development of telescopes with high resolution and high imaging quality. In space optics, the accuracy of state of the art optics manufacturing technology has been unable to meet the technical requirements of the astronomical telescopes used in various NASA projects, e.g. the Astronomical Search for Origins (ASO), the Structure and Evolution of the Universe (SEU) and the Sun-Earth Connection (SEC). Research on new manufacturing techniques and technology in space optics is urgent and important for the development of the next generation of telescopes. One important factor that reduces the imaging accuracy of a telescope is mirror manufacturing error, which is defined as the difference between the practical parameters of the mirrors and their theoretical values. These parameters include the paraxial radius, geometry dimension, eccentric errors and surface errors. To improve manufacturing accuracy, techniques for measuring the 3D shapes of mirrors have become important, because the manufacturing errors of mirrors can be computed directly after their shapes are known. Although many non-contact measurement techniques [1][2][3][4][5][6][7][8][9][10] have been developed to measure mirror surfaces, none of them are capable of measuring the shapes of mirrors with adequate accuracy. Ever since the invention of the telescope, there has been a demand for accurate measurements of the 3D shapes of mirrors both qualitatively and quantitatively in order to directly calculate mirror manufacturing errors. Unfortunately, these kinds of techniques have been hindered by state of the art optical technology and machine vision technology.
In 1858, Jean Foucault invented a way of measuring telescope mirrors using a knife-edge test. Unfortunately, it could only measure spherical mirrors, and most telescope mirrors are aspherical instead of spherical. In addition, the Foucault method was only able to provide qualitative results instead of quantitative results. To yield quantitative results, it had to be combined with other methods, and great effort and considerable skills were needed to make accurate judgments. In 1922, Vasco Ronchi invented a different technique for measuring telescope mirrors, although it was unable to provide quantitative results either. Based on the previous methods, several new methods were proposed later, e.g. the star test, the Ross null test and the auto-collimation test. However, none of them were satisfactory. In the early 1970s, Karl Bath invented an interferometer to test telescope mirrors with quantitative results and it was recognized as the most informative method of the time. The Ceravolo interferometer is an alternative method which has similar performance. Unfortunately, both interferometers are best suited to testing spherical surfaces only. Later on, an interferometer was made by ZYGO, which used PV/RMS to evaluate the quality of the mirror. However, the complexity of the optics rendered PV and RMS incapable of describing the mirror quality adequately. Hence, the power spectral density (PSD), the SlopeRMS, the inverse Hartmann test and the structure function (SF) have been widely adopted in mirror quality evaluation [11][12][13][14][15][16][17]. In [15], the inverse Hartmann test was proposed for surface form measurement in spherical coordinates with increased dynamic range and resolution. However, its accuracy decreased when compared to that in rectangular coordinates. In [16], a tutorial about structural function analysis is presented and its advantages over Fourierbased methods were proved. In [18,19], researchers at the University of Arizona used a laser tracker to obtain the direct shape measurement for a GMT mirror and they achieved a measurement accuracy of 1/4 µm. In [20], ray tracing was used to measure the optical aberrations of aspherical lenses. All the above methods, except [20], were able to give quantitative results for the surface errors of mirrors, but only indirectly; the paraxial radius, geometry dimension and eccentric errors of the mirror were beyond their capabilities.
Although the authors in [20] claimed that their method could measure the profile of the surface, only a 1D profile was given in the experiments. In [21], a method was proposed to measure the 3D profiles of the mirrors with analytic solutions, with the potential to achieve zero error accuracy, provided that no noise or lens distortion existed. Unfortunately, the system noise and the radial lens distortion of the Pico laser projector that was used are severe. Consequently, the measurement accuracy of [21] is limited. To improve this, the Pico laser projector was replaced with an SNF laser, which is free of lens distortion and achieves better measurement accuracy [22]. However, the measurement accuracy is still not good enough, because the noise reduces it significantly. The SNF laser does not strictly obey the central projection from which some fundamental equations of the system are derived. Consequently, the reconstruction results of the system are highly distorted and it requires the least deformation principle [22] to re-calibrate the system, which is very time consuming.
In this paper, a ray and pattern modeling method is proposed to remove the noise and radial lens distortion as a whole by decreasing the degree of freedom of the multiple laser rays to one. The proposed method registers the captured pattern with the theoretical pattern that replaces the captured pattern during the reconstruction. Since the ray and pattern modeling method requires that projected rays obey the principle of central projection, it cannot be adopted by the system in [22], unless the SNF laser is used with a strictly central projection. Hence, we choose the Pico laser projection instead to generate the required laser rays. The mirror measurement system is designed according to the requirement of measuring the telescope mirrors. Due to the larger sizes of the mirrors, larger diffusive planes and a larger beam splitter are used with carefully selected distances. With the proposed ray and pattern modeling method incorporated into the designed system, this one-shot projection method was able to achieve 10 −13 mm measurement accuracy for telescope mirrors, which is super ior to most state of the art methods.
This paper is organized as follows. Section 2 describes the working principle of the mirror measurement system. In section 3, a noise analysis is provided and the fundamental pattern modeling method is proposed. The experimental results are given in section 4, and section 5 concludes the paper.

The system for the one-shot projection method
The designed system for the one-shot projection method is illustrated in figure 1. There are three planes p1, p2 and p3 with three cameras c1, c2 and c3 aimed at them, respectively. At the beginning of system calibration, the poses of the three cameras are estimated with a MATLAB calibration toolbox. The projection center of the projector is denoted as C and its symmetry point relative to the horizontal plane is C′, which denotes the projection center of the virtual camera. The horizontal plane p1 is the imaging plane of this virtual camera and it is defined as the reference plane, z = 0; its origin is at O. A set of laser rays is projected from point C and reflected by the mirror plane onto the diffusive plane. Based on the reflection law and central projection theorem, the equations of the reflected rays are the same as those projected directly onto the diffusive plane from point C′, which is the symmetric point of C relative to the mirror plane, as illustrated in figure 1. The central projection from C′ intercepts the mirror plane and the diffusive plane in the same way as light going through a pin-hole and an image is formed on the image plane. Hence, the mirror plane can be treated as the image plane of a virtual pin-hole camera. The interception points on the diffusive plane can be treated as objects whose image is formed on the mirror plane. With this virtual camera and another real camera forming an image of the set of interception points on the diffusive plane, the equation of the diffusive plane can be determined. This has been described in detail in [21,22]. With the equation of the diffusive plane, the homography between the camera and the diffusive plane and the camera coordinates of the interception points, the 3D world coordinate of the interception point can be computed. When the points on p3 are computed, they are mapped to p4. Then, two points intercepting one ray are obtained and the ray can be determined uniquely with analytical solutions. With the incident rays determined by camera c1, the interception points of the projected pattern on the mirror surface are computed as the intersections of the incident rays and their corresponding reflected rays. For system details, please refer to the references [21,22].

Analysis of the captured pattern
We choose the interception points on the reference plane z = 0 and show a zoomed-in view of them in figure 2. It can be seen that these five groups of points are not in straight lines as designed. In addition, the line appears to be curved if we connect the corresponding points on the same line. This is caused by noise and lens distortion. The noise in this imaging system is generated during image capturing. It is affected by different influential light sources as well as the automatic image processing, which has been affected by the unevenly distributed gray-scales of the laser points. The radial lens distortion is inherent in the projectors and cameras. Both the noise and radial lens distortion decrease the calibration accuracy and system measurement accuracy greatly; hence, they must be removed for better accuracy.

Ray modeling and pattern modeling
For the central projection, the angle between any two projected rays does not change and all the projected rays intersect  at the projection center. This is a fundamental property of the projector or camera, on which the proposed modeling method is based. The angle between any two rays can be computed by the following equation: x x y y z z x x y y z z x x y y z z x x y y z z ) and x y z , , j j j ( ) are the ith interception point and the jth interception point on the horizontal reference plane z = 0, as shown in figure 2.
In the ideally designed pattern, each bright point represents one projected ray. Along both row and column directions, the distance between two adjacent bright points is equal. The center of the designed pattern corresponds to the central projected ray of the projector. The proposed ray modeling method is based on the fundamental property of the projector, in which the angle between the central ray and any other ray is always fixed. Based on this property, we were hence able to determine the relative positions of the projected rays though their equations in the virtual coordinate system. The angle between the ith ray and the central ray can be computed by the following equation in the virtual coordinate system: where d i denotes the distance between the central point and the ith point, and D denotes the distance between the projection center of the projector and the interception plane that is vertical to the central ray. Both d i and D are unknowns to be solved.
From equation (2), it can be seen that the ratio of d i and D determines the angle; thus, the scale of them is not important. So, we assume the distance d i equals the pixel distance in the original pattern. Since the distance D is unknown, the following searching algorithm is proposed to find it: Step 1: Choose a set of points (45 points in this research) around the center of the projected rays. Compute the angles between the chosen rays and the central ray with equation (1).
Step 2: Choose the corresponding points from the pattern.
Compute the angles between the chosen points and the central ray based on equation (2) with an initial estimated value of D as 1000 in pixels.
Step 3: Compute the total difference of all the angles by the following equation: where i0 θ denotes the angle between the ith ray and the central ray.
Step 4: Find D that makes θ ∆ minimum.
D arg min .
After the distance D is found, i θ for each ray can be modeled by equation (2). They are denoted as modeled rays R i , 1, 2, .45 i m = … in this piece of research. To compare the modeled rays with the practically projected rays, we compute the angle between the central ray and any other ray by equations (1) and (2) respectively. The results are shown in figure 3. The computed angle of equation (1) is denoted as 'original' and the computed angle of equation (2) is denoted as 'modeled'. As can be seen, the angles obtained from these two different methods match well, but with obvious differences, caused by noise and lens distortion.
If the plane that intercepts the projected rays is known, the interception pattern can be determined. To find the interception plane, we propose the following pattern modeling method: Step 1: Use the plane ax by cz 1 + + = to intercept the modeled rays R i Step 2: For the captured pattern, compute the distances between the central point and the same set of points as those used in step 1 by the following equation: Step 3: Compute the total difference of all the distances by the following equation: Step 4: Find the optimal interception plane P a b c , , ( ) that makes d ∆ minimum: The intercepted points are computed in a virtual coordinate system instead of the world coordinate system. Registration is thus needed between the original points and the intercepted points to convert the coordinates correctly. We register the two sets of points based on the least square errors by finding the transformation matrix A that makes the sum of square errors, d r minimum: where ω is a constant and the transformation matrix A is defined as:  Figure 4 shows the modeled points after registration (in blue circles) versus the original points (in red dots). As can be seen, the registration method works very well. For the modeled points, each group is in straight lines, which indicates that the noise and radial lens distortion have been removed successfully. For a better view of the noise and radial lens distortion removal effect, the differences in the x and y coordinates for these 45 points before pattern modeling are plotted and shown in figures 5(a) and (b). The differences in the x and y coordinates for these 45 points after pattern modeling are plotted and shown in figures 6(a) and (b). It can be seen that the differences after pattern modeling vary regularly according to the designed pattern, and the noise (random variation) has been eliminated successfully. In addition, it can be seen that the differences in the adjacent points on the same lines constitute straight lines, which indicates that the lens distortion of the Pico laser projector and camera has been removed successfully.
After ray and pattern modeling, the captured patterns are replaced with the modeled patterns, and the system is calibrated in the same way as described in [21]. It is then possible to measure the mirror surface in real time using the calibrated system with a single projection. Figure 7 shows the system. Camera c1 is aimed at the horizontal screen, p1, which is placed on top of a metric lab jack whose height can be adjusted flexibly. The rays are produced by a Pico laser projector and they are reflected by the mirror surface onto the beam splitter that splits the rays into two. Half of the rays pass through and create an image on the diffusive plane, p2; half of the rays are reflected and create an image on the diffusive plane, p3. Two cameras, c2 and c3, synchronically record the images on the two diffusive planes, p2 and p3 at 60 frames s −1 . Figure 8(a) shows the designed pattern projected by the Pico laser projector onto a horizontal diffusive plane. The brightest point in the center denotes the central marker. A Dragonfly camera captures the projected pattern, as shown in figure 10(b). As can be seen, the line appears to be curved if we connect the bright points on the same line, which is caused by the lens distortion of both the Pico laser projector and the Dragonfly camera. These camera coordinates were transformed into world coordinates (as demonstrated in figure 2) and then modeled by the proposed method to remove both the noise and radial lens distortion. Figures 9(a) and (b) show the modeled coordinates (in red) versus the original coordinates (in blue). It can be seen that the modeled coordinates and the original coordinates match well. The aim of the pattern modeling method is to eliminate the random variations (noise) and radial lens distortion while keep the pattern ideal.

Experimental results
We use the flat mirror to evaluate the measurement accuracy of this one-shot projection method. We compute the root mean square errors between the reconstructed points and the modeled points by the following equation: where E x denotes the error in the x coordinate, E y denotes the error in the y coordinate and E z denotes the error in the z coordinate. (X Y Z , , We determine the homography between camera c1 and the horizontal reference plane z 0 = using the MATLAB calibration toolbox: where Z c is a scalar. f x and f y are the focal lengths in the x and y direction respectively. C C ,   parameters between the horizontal reference plane z 0 = and the image plane of camera c1. is determined by equations (3)-(15) as described in the above section, which could be summarized as: where M is the affine pattern modeling matrix.
Firstly, we compute the errors from the reconstruction of the flat mirror without pattern modeling. The measurement accuracy of the developed system is 0.2254 mm in the x coordinate, 0.1977 mm in the y coordinate and 0.0825 mm in the z coordinate. The reconstructed points versus the original points are shown in figure 10(a), where the blue circles denote the original points and the red crosses denote the reconstructed points.
Secondly, we compute the errors from the reconstruction of the flat mirror with pattern modeling of the world coordinates in p1, p2 and p3. The reconstruction measurement accuracy is 0.0332 mm in the x coordinate, 0.0278 mm in the y coordinate and 0.0113 mm in the z coordinate. The reconstructed points versus the original points are shown in figure 10(b), where the blue circles denote the original points and the red crosses denote the reconstructed points.
Thirdly, we reconstruct the convex mirror to show the strength of the proposed pattern modeling method visually.   Figure 11(a) shows one reconstructed convex mirror without pattern modeling; the noise ruined the reconstruction severely. Figure 11(b) shows the reconstructed convex mirror with three patterns modeled. As can be seen, there is a noise threshold that determines whether the proposed structured light method can work effectively. Only when all noise in the three patterns involved is eliminated can the proposed structured light method work well. From these experimental results, it was possible to verify both the strength of the proposed pattern modeling method and the one-shot projection method.
Fourthly, we evaluate the accuracy of the one-shot projection method in reconstructing a mirror with known form. Since the form of the mirror was known, it was possible to model the camera coordinates in c2 and c3 based on the known form and reduce the noise. We computed the errors from the reconstruction of the planar mirror with pattern modeling of the camera coordinates in c2 and c3 and the world coordinates in p1, p2 and p3. The measurement accuracy becomes 3.4681 × 10 −14 mm in the x coordinate, 6.6771 × 10 −14 mm in the y coordinate and 2.4653 × 10 −14 mm in the z coordinate respectively. At first glance, the measurement acc uracy seems extremely unlikely. A further thought reveals that when all the involved patterns are modeled, both the calibration stage and the reconstruction stage can be assumed to be free of noise and lens distortion. In addition, the reconstruction error is computed between the reconstructed points and the modeled points instead of the original points that could introduce random noise. The reconstruction error is close to zero, but not exactly equal to zero as in our simulation results with MATLAB [22], which is caused by the unideal hardware used here. Another adverse factor caused by the unideal hardware is shape distortion. Hence, the selected hardware should be as precise as possible to achieve the highest measurement accuracy. Since the telescope mirrors had known forms, e.g. spherical, planar and parabolic, we were always able to model the camera coordinates in c2 and c3 while measuring their shapes. Currently, the measurement acc uracy of the system for the planar telescope mirror is 10 −13 mm,  which indicates that the system is capable of measuring the flatness of the telescope mirror robustly. However, what we have achieved is only the first step in a long march. Our future work will include: (1) the adoption of better hardware to increase measurement accuracy further, and (2) evaluation of the accuracy of measuring other forms of telescope mirrors thoroughly.
To compare the measurement accuracy of this one-shot projection method with state of the art methods, we compared it with the state of the art literature for available quantitative results and show the comparisons in table 1. As can be seen, the measurement accuracy of the one-shot projection method is significantly better than the state of the art methods. Furthermore, some state of the art methods, e.g. [19], were unable to measure the 3D profiles of the mirror like this method could. Thus, the advantages of the one-shot projection method over other state of the art methods have been proved.

Conclusion
It is important and challenging to accurately measure the 3D shapes of mirrors for telescope manufacturing. This paper aims to improve a previously proposed one-shot projection method and allow it to be capable of accurately measuring the profiles of mirrors, which is essential for the direct measurement of manufacturing errors for the mirror. A ray and pattern modeling method was proposed to remove the noise and radial lens distortion by replacing the captured pattern with a theoretical pattern that was computed by registering the captured pattern with the designed pattern. The experimental results showed that the proposed ray and pattern modeling method were able to increase the measurement accuracy of the oneshot projection method from 0.1 mm to 10 −13 mm. Hence, the effectiveness of both the proposed modeling method and the one-shot projection method has been verified.