## Abstract

Detection accuracy is an important performance indicator of ground-based telescopes and is affected mainly by pointing error, geometric distortion of the optical system, and parameter errors caused by machining error and installation error. To improve detection accuracy, a modified algorithm based on a simulated annealing algorithm is proposed in this paper; this algorithm is able to correct pointing, derive a geometric distortion solution, and re-estimate some parameters of telescopes simultaneously. The efficiency of the proposed method is verified by using the observation data of the telescope, whose aperture is 600 mm under two distortion models (the physical model and polynomial fitting model). The results show that the method presented in this paper can effectively solve the problem of nonconvergence of the distortion solution with a pointing error. The final angle error under the polynomial fitting model is 1.07″, and the pixel error is 0.06 pixels; the errors under the physical model are 1.08″ and 0.07 pixels. The correction effect under the two distortion models is basically the same, but the averaged operation speed based on the physical model is 19.45% faster.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The detection accuracy of ground-based telescopes is an important performance indicator. With the progress of manufacturing techniques, although telescopes have better performance, the errors resulting from mechanical structures, optical systems, and control systems still seriously restrict the performance of telescopes. Therefore, pointing, geometric distortion, and certain parameters of telescopes must be solved to improve the performance of telescopes.

The essential task of distortion estimation is to find the functional relationship between measured star positions and those in the distortion-free reference frame. Therefore, the traditional correction algorithm includes mainly three parts: first, using the positions of stars measured with space-based telescopes (such as the Hubble Space Telescope) as distortion-free positions; second, determining the measured positions of stars; and third, using a distortion model and an estimation algorithm to solve for distortion coefficient.

In 2003, Anderson and King used the data of ω Centauri to acquire an improved distortion solution for the Hubble Space Telescope’s WFPC2 [1]. In 2006, Anderson *et al.* used the data of NGC 6121 (M4) and NGC 6397 to characterize the wide field imager (WFI) geometric distortion [2]. In 2010, Yelda *et al.* used on-sky data of M92 to solve the geometric distortion and achieved a smaller residual distortion in their solutions [3].

Currently, many software packages can easily measure the star position [4–6]. In 2000, Anderson and King proposed the concept of effective point-spread function (ePSF) to simplify the calculation process of PSF and improved the calculation accuracy of star positions; ePSF can measure the position of a single reasonably bright star in an image with a precision of 0.02 pixels, and the accuracy can be further improved by multiple observations [7]. In 2011, Zhang proposed three star point centroid algorithms based on gray value and compared those algorithms through simulation. Their results showed that the centroid method with a threshold is superior to the other algorithms in anti-noise performance and accuracy, and the accuracy can be up to 0.04–0.05 pixels [8].

In 2003, Anderson and King found an improved distortion solution for the WFPC2 of the Hubble Space Telescope, where they used two third-order polynomials to solve the distortion in the $x$ axis and $y$ axis separately [1]. In 2010, similar work was done on the Large Binocular Camera (LBC), at the Large Binocular Telescope (LBT) [9]. In 2010, Xu and Chu used the third-order Zernike polynomial to fit the distortion coefficient [10]. It should be noted that the coefficients of the Zernike polynomial and third-order polynomial can be uniquely determined from each other, which are actually the same kind of fitting model.

For telescopes with good performance, the pointing error and other factors are often not taken into account when calculating the geometric distortion of optical systems. These algorithms have been proved to work well in certain cases. However, in our cases, the telescope has an un-ignored pointing error and optical system parameter errors. To ensure detection accuracy, some improvements have been made in our work: (1) we use a heuristic searching algorithm to correct the pointing error and geometric distortion. (2) During the manufacturing and installation process, some parameters of the telescope change slightly; therefore, the re-estimation of these parameters is needed. (3) Considering that we are eager to correct the direction of the telescope in any pointing direction, the star catalogue is used to replace the positions of stars measured with space-based telescopes. (4) We use the distortion physical model to reduce the number of distortion coefficients to be fitted (18 distortion coefficients are reduced to seven distortion coefficients) and improve the efficiency of the algorithm. With the physical model, the constraint relationship between the $x$-axis and $y$-axis distortion of the image is preserved, which helps subsequent research. The results of our experiment show that the angle error is 170.26″ and the pixel error is 11.73 pixels before correction. Through the correction of the algorithm in this paper, the errors converge to 1.08″ and 0.07 pixels. Meanwhile, the running speed of the algorithm in the physical model is 19.45% faster than that in the polynomial fitting model.

The sections of this paper are arranged as follows. Section 2 discusses the method of star position calculation. Section 3 describes two distortion models, which are compared in this paper. Section 4 introduces the simulated annealing algorithm. Section 5 analyzes the result of our experiment. Section 6 concludes this paper with a summary and discussion of future directions.

## 2. POSITION CALCULATION

#### A. Star’s Distortion-Adjusted Position

Being influenced by factors such as the diffraction limitation, the image of the star on the CCD is usually a spot. In this paper, the centroid method with threshold is used to calculate the centroid [8], and the calculated centroid is utilized as the distortion-adjusted position of the star. The star’s distortion-adjusted positions can be calculated by the following formula:

#### B. Star’s Distortion-Free Position

The star’s right ascension and declination at any time can be accurately calculated. However, for the sake of comparison, it is necessary to convert the star’s position from celestial coordinates to pixel coordinates. The whole process consists mainly of three steps: (1) using the star catalogue to calculate the star’s current right ascension and declination; (2) using the geodetic coordinates of the telescope to calculate azimuth angle and elevation angle (AE); and (3) converting the star’s position from angular coordinates to pixel coordinates, and finally the distortion-free star position is obtained.

Calculating the distortion-free positions is a complicated process. In order to obtain a high-precision position, it is necessary to consider the influence of various factors, e.g., earth rotation parameters (EOPs), atmospheric refraction, light deflection, aberration, etc. It is not easy work but fortunately benefits from the International Astronomical Union’s Standards of Fundamental Astronomy (SOFA). Our work can be significantly reduced because SOFA can solve the AE by calling several functions [11]. Therefore, we need only to convert the AE coordinates to the pixel coordinates.

Figure 1 shows the projection relationship of the miss distance in the image plane of the image sensor, where ${P}_{1}$ represents the position of the star on the celestial sphere, ${O}_{1}$ is the optical center, and $P$ and $O$ mean the images formed by ${P}_{1}$ and ${O}_{1}$ on the image sensor, respectively.

Formula 3 can be easily deduced from Fig. 1 [12]:

Some telescopes have an image rotation mechanism to improve image quality; the coordinates after rotation can be calculated by formula 4:

Transformation from image coordinates to pixel coordinates is shown in formula 5:

where $X$ and $Y$ are the pixel coordinates of the star $P$; ${X}_{0}$ and ${Y}_{0}$ are the pixel coordinates of point $O$; and ${p}_{x}$ and ${p}_{y}$ are the pixel sizes of the image sensor in the $x$ and $y$ directions, respectively.It can be seen from formula 3 to formula 5 that although the pointing and some parameters of the telescope are not directly related to optical distortion, the pointing error and parameter errors directly affect the calculation of distortion-free star positions. This is why we need to correct pointing and to re-estimate corresponding parameters of our telescope.

## 3. DISTORTION MODEL

#### A. Physical Model

The distortion of an optical system consists of the following three main types: radial distortion, tangential distortion, and thin prism distortion [13].

### 1. Radial Distortion

The vertical magnification of the optical system at different fields of view (FOVs) is different, which results in radial distortion and is the main source of distortion. The expression of radial distortion is shown in formula 6:

### 2. Tangential Distortion

The coaxial error of the optical components in the optical system leads to tangential distortion, and the expression of this type of distortion is shown in formula 7:

### 3. Thin Prism Distortion

The thin prism distortion is caused by the tilt of the optical component and the CCD, which is equivalent to inserting a thin prism into a nontilted optical component. The thin prism distortion can be expressed as

Then, the final expression of the physical distortion model is

#### B. Polynomial Fitting Model

The polynomial fitting model has a simple form; however, there are lots of distortion coefficients that need to be solved. The distortion expression under the third-order polynomial is shown in formula 10:

Performance of the polynomial fitting model under different orders was tested by Cox *et al.* [14]. Their results showed that the second- or third-order model is better than other orders. In addition, a large number of studies have been carried out under the third-order model. To facilitate comparison, the third-order model is also used in this paper to compare with the physical model. Obviously, there are only seven distortion coefficients in the physical model, which is far less than 18 in the third-order polynomial fitting model.

## 4. SIMULATED ANNEALING ALGORITHM

The simulated annealing algorithm is a heuristic search algorithm that can be used to find the optimal estimate of multiple parameters (strictly, a suboptimal estimation that is very close to the optimal one). The flow chart of the algorithm is shown in Fig. 2.

Here, the index function $J(\omega )$ is used to calculate the position error between the distortion position and the distortion-free position of the star under current estimated parameters. Its expression is shown in formula 11; $t$ represents the “temperature” of the simulated annealing algorithm; $\mathrm{Drop}(t)$ represents the falling function of “temperature”; the Metropolis Criterion can offer the algorithm a certain probability to accept the inferior solution, thus ensuring that the algorithm jumps out of the local minimum:

## 5. RESULTS

The flow chart of the whole algorithm is shown in Fig. 3. The star identification module is used to establish the corresponding relationship of the star between the image and catalogue. Then, the distortion-free positions are calculated by the method in Section 2. Meanwhile, the positions extracted from the star map are used as distortion-adjusted positions. These two positions are used as input of the simulated annealing algorithm, which can solve the optimal estimation of the pointing, distortion, and some parameters of telescopes, and finally, the correction result is obtained.

The diameter of our telescope is 600 mm with a focal distance of 830 mm. In addition, it employs a single 16-bit ($2040\times 2040$) chip, with a reference pixel-scale of 5.94″/pix (0.024 mm/pix), providing a total FOV of $\sim 3.37\xb0\times 3.37\xb0$.

We list the initial input as well as the parameters to be estimated in Table 1.

We use the simulated annealing algorithm to accomplish the optimal estimation of the above parameters. To ensure that the algorithm finds a set of solutions close enough to the optimal one, the solution is tested in an extra iteration. When the reduction of the index function of the adjacent iterative process satisfies the termination condition, the search for the optimal solution is terminated.

We filter the star image to better separate stars from background; meanwhile, the CCD noise can be effectively suppressed by filtering. The original image and filtered image are shown in Fig. 4.

We choose 649 stars as the calibration source to correct the errors in our telescope. The position of these stars are shown in Fig. 5. It can be seen that our calibration source effectively avoids the area where the CCD noise is larger and its distribution in the image is relatively uniform.

The error of star position under initial conditions is shown in Fig. 6. In Fig. 6, it is obvious that the error gradually increases from the upper-left corner to the lower-right corner of the image. Nevertheless, it can be concluded from the distortion model that the geometric distortion of the optical system is approximately symmetrical with respect to the optical center. However, the distribution of the error shown in Fig. 6 is due mainly to the existence of the pointing error, which is equivalent to introducing a systematic error in the calculation of the position of the star. To further explain the importance of pointing correction, we solve the distortion coefficients without correcting the pointing error at first.

As seen in Fig. 7, the simulated annealing algorithm can search the optimal solution rapidly and basically converge after only one iteration. In the second iteration, the RMSE of AE and XY did not decrease significantly. However, the RMSE of the AE is still nearly 89.23″, while RMSE of XY is 6.30 pixels. Therefore, for our telescope with a large pointing error, it is impossible to achieve a satisfied distortion solution if we do not correct the pointing simultaneously. Under such a condition, the result is far beyond acceptable limits.

To solve this problem, we use the simulated annealing algorithm to estimate parameters in Table 1. We compare the performance of the two distortion models under the same input and configuration. The angular error and pixel error in each iteration are shown in Fig. 8.

We can draw some conclusions from Fig. 8: (1) the algorithm can effectively search for the optimal solution for both models; (2) the algorithm converges after three iterations; (3) the operation speed of the algorithm under the physical model is obviously faster than that of the polynomial fitting model. For contrast, the final estimation results of the parameters under the two models are shown in Table 2.

According to the modified parameters of the two distortion models, we replot the distribution of the errors after correction at the corresponding positions.

Comparing Fig. 6 with Fig. 9, we can find that our algorithm efficiently determines an available optical distortion solution under the condition of a large pointing error. Even in the edge part of the image where the large distortion existed, a good correction result is also attainable. Meanwhile, Figs. 9(a) and 9(b) indicate that there is no obvious difference in the correction results between the two distortion models, but the algorithm under the physical model has an obvious advantage in running time while preserving the relationship of the distortion in the $x$ and $y$ directions.

## 6. CONCLUSION AND DISCUSSION

Distortion estimation is of great significance to improve the performance of telescopes [1,2,9,15]. However, when the pointing and some optical system parameters contain large errors, directly solving the distortion coefficient cannot achieve a good correction effect. To solve this problem, we present an algorithm that can simultaneously correct the telescope pointing, solve for the distortion coefficient, and re-estimate the parameters of the optical system. The proposed method has three main advantages. First, it can effectively solve the problem of nonconvergence of distortion correction under a large pointing error. Second, we use the physical model instead of the traditional polynomial fitting model to estimate the geometric distortion. Since the physical model has only seven parameters to be estimated, it has a faster running speed. The results show that the physical model used in this paper is 19.45% faster than the polynomial fitting model. Last but not least, the distortion coefficients of the $x$ axis and $y$ axis are not subject to being estimated separately; thus, the functional relationship between these two directions is preserved.

In our follow-up work, we will conduct research in examining the following areas: (1) whether the physical model used in this paper can serve to suppress star position measurement errors caused by atmospheric refraction; (2) whether our algorithm can also be pertinent for other telescopes; and (3) whether the algorithm can correct the error of the assembled CCDs.

## Acknowledgment

I thank my supervisor for his guidance during this experiment. At the same time, I would like to thank my classmates and family for their support and care in this process. Also, thanks to Software Routines from the IAU SOFA; its work provides us with reliable intermediate data.

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