Attitude-correlated frames adding approach to improve signal-to-noise ratio of star image for star tracker

Yuanman Ni; Dongkai Dai; Wenfeng Tan; Xingshu Wang; Shiqiao Qin

doi:10.1364/OE.27.015548

1. Introduction

A star tracker is an avionics instrument used to provide the absolute 3-axis attitude of a spacecraft utilizing star observations [1, 2]. First-generation star trackers are characterized by outputting positions of a few bright stars in sensor-referenced coordinates and requires external processing [3]. Its star observation for navigation purposes is performed by aiming the Stellar INS optoelectronic tools at the brightest stars and long-term holding them using gyrostabilized platforms [4].

The present second-generation star tracker, also called star imagers, performs this task by pattern recognition of the star constellations in the field of view (FOV) and by means of a star catalog covering the entire firmament [5]. It has a smoother and more robust operation, lower cost and higher accuracy compared to first-generation star trackers and becomes the preferred autonomous navigator onboard most spacecraft [6–9]. However, when used inside the atmosphere, it is susceptible to sky background produced by atmospheric scattering and stray light and the star image has an extremely low SNR. The dramatically SNR decline of star image reduces the detection capability and makes it out of operation during daytime [10–13]. Therefore, it is crucial to improve the SNR of star images and to overcome the obstacle of the daytime star detection under strong noise in order to adapt the star tracker to the near-ground condition [14–16].

Selecting an image sensor whose spectral response lies in the short-wave infrared band is the mainstream solution for daytime star trackers [17]. It has higher SNR compared with the camera detecting visible light because the background radiation in infrared spectrum region is much weaker than that in visible spectrum region. To further enhance the star detection ability, there are three ways to improve the SNR of infrared star images. The first way is to improve the optical system by using lens that has a large aperture and a long focal length, which definitely increases the cost and size of the system. For example, the daytime star tracker developed by Trex Enterprise Corp. adopts three optical lens of 200mm aperture and 0.5° × 0.4° FOV to observe stars on the H-band, enabling the detection of the 6.8 magnitude stars at sea level during daytime, and the total weight of the assembly is up to 100 pounds [18,19]. Extending the integration time is the second way, because the star signal increases faster than the noise with integration time [20]. The main problem is that the pixels are easy to saturate under strong background radiation, and the long exposure time will induce the blurring of the star image under dynamic conditions. Thirdly, Mark O’Malley proposed a frame adding approach as an alternative to long integration time to improve the SNR for low light detection [21]. By shortening the exposure time of each single-frame image and adding the frames directly, it effectively prevents the saturation of the electric detector under strong noise and has a good effect on image enhancement of static targets. However, under dynamic conditions, the star is shifting on the image with the angular motion of the vehicle [22,23]. Adding the star images directly results in motion-blurring of the image and the deterioration in star centroiding accuracy [24,25].

In order to solve the problems of the third way and make it adapted to the dynamic condition, a star images adding method based on the attitude-correlated frame (ACF) is proposed in this paper. In the ACF method [26], the star image frames measured in different time are correlated by the angular motions of the star tracker and calculated with the angular rates sensed by the strap-down GUs. Then the translation and rotation transformations between different frames (namely correlated transformation) are conducted by using the gyro-measured attitude changes. By correlating and adding successive image frames, the SNR of the star image is enhanced while the background noise is suppressed under dynamic conditions, and the star detection capacity can be improved.

The paper is organized as follows. The ACFA approach is depicted in detail in Section 2. Simulation results are presented in Section 3. In Section 4, the experimental result is given to verify the approach. Finally, the conclusion is drawn in Section 5.

2. Basic theory

2.1. Frame adding to improve SNR under static conditions

For optical sensors, shot noise occurs due to the random fluctuations (described by the Gaussian distribution) of the number of photons detected, and its standard deviation is proportional to the square-root of the average intensity. During the day time, the performance of the focal plane array (FPA) is limited by the shot noise induced by strong sky background radiation. The SNR of the image can be enhanced by increasing the integration time. However, the pixels are easy to saturate under strong radiation condition. Frame adding is a viable way to improve the SNR with successive short integration time images and to keep them from saturation.

Considering a single frame obtained for low light detection, the SNR is given by [21]:

SNR = \frac{{\bar{s}}_{i}}{\sqrt{N_{τ}^{2} + 2 N_{0}^{2}}},

where s̄_i is an estimate of the input signal, N_τ is the integration dependent noise including shot noise, dark current noise, and transfer noise, and N₀ is the output noise, such as readout, amplifier, and quantization noise. It should be noted that the shot noise is the main source in strong sky background.

Consider n frames taken one after another and added together. The mean input signal increases to ns̄_i. Assuming mean square fluctuations add, then the noise for n frames added together becomes $\sqrt{n (N_{τ}^{2} + 2 N_{0}^{2})}$ . Therefore, the SNR for frame adding is given by [21]:

{SNR}_{add} = \frac{n {\bar{s}}_{i}}{\sqrt{n (N_{τ}^{2} + 2 N_{0}^{2})}} = \sqrt{n} SNR .

There is an obvious improvement of SNR ( $\sqrt{n}$ ) for n correlated frames over a single frame. Theoretically, adding successive star images can improve the signal-to-noise ratio of the image, but in fact, the attitude of the vehicle is constantly changing, causing a certain star to shift on different images. Consequently, directly adding the images, the energy of a certain star is no longer distributed within a small area. The method of frame adding to improve SNR is restricted to static target signal on different images. So this method needs to be modified before applied to dynamic conditions.

2.2. Attitude-correlated transformation process

Under dynamic conditions, the attitude of the camera is changing with the vehicle, therefore, the position of a certain star on Focal Plane Array (FPA) is shifting with time as shown in Fig. 1. Adding dynamic frames directly results in motion blurring of star image and deterioration in star centroiding accuracy. The Attitude Correlated Frames Adding (ACFA) Approach is proposed to solve the problem. Considering n successive frames, to make the input signal after addition increase to n times of a single frame, the pixels of a certain star on different images should be added. As shown in Fig. 1(b), for a certain point in the celestial sphere, its coordinate on the FPA of star tracker is (u_j, v_j) at t_j. While its coordinate is (u_k, v_k) at t_k. Because of the attitude change of the star tracker from t_k to t_j, there is a correlated transformation between (u_j, v_j) and (u_k, v_k). The information of GUs is used to calculate the attitude change between two moments and to eliminate its impact on the star coordinate on FPA. By transforming the coordinate at t_k to find its position at other moments, then the corresponding pixels can be added. In this way, star radiation can be regarded as a static signal on the image, and adding successive frames after attitude-correlated transformation can improve SNR of star image and star detection capability under dynamic conditions.

Fig. 1 Two successive frames taken by the star tracker at different moments. (a) The attitude of the star tracker is changing with time in the inertial coordinate system. (b) There are 4 same stars on two different images taken at t_k and t_j respectively. ST refers to star tracker, and GUs refers to gyroscope units.

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The coordinate systems used in this paper are defined as follows:

The Earth-centered inertial coordinate system, which is fixed in inertial space, is represented by i-frame. Its X_i axis is in the equatorial plane and points to the vernal equinox. The Z_i axis is aligned with the Earth’s rotation axis and vertical to the equatorial plane. The Y_i axis completes a right-handed frame with the other two axes. The star tracker coordinate system (s-frame) has its origin at the detector center. The X_s and Y_s axes are parallel to a row and a column of the detector, respectively. The three axes satisfy the right-hand rule. The strap-down GUs system, which is represented by b-frame, has its X_b, Y_b and Z_b axes consistent with the three mutually orthogonal sensitive axes of the GUs. And it is rigidly mounted to the star tracker platform.

The vector of a certain star in i-frame is rⁱ, and it can be transformed to s-frame when given the attitude matrix of the star tracker at the time of t_k and t_j respectively:

{\begin{cases} r_{k}^{s} = {[\begin{matrix} x_{k}^{s} & y_{k}^{s} & z_{k}^{s} \end{matrix}]}^{T} = C_{i}^{s (k)} \cdot r^{i} \\ r_{j}^{s} = {[\begin{matrix} x_{j}^{s} & y_{j}^{s} & z_{j}^{s} \end{matrix}]}^{T} = C_{i}^{s (j)} \cdot r^{i} \end{cases},

where

C_{i}^{s}

is calculated from the following equation with installation matrix (the transformation matrix from b-frame to s-frame)

C_{b}^{s}

and attitude matrix of the GUs

C_{i}^{b}

:

{\begin{cases} C_{i}^{s (k)} = C_{b}^{s} C_{i}^{b (k)} \\ C_{i}^{s (j)} = C_{b}^{s} C_{i}^{b (j)} \end{cases} .

It can be deduced that:

C_{s (j)}^{s (k)} = C_{b}^{s} C_{b (j)}^{b (k)} {(C_{b}^{s})}^{T},

where

{(C_{b}^{s})}^{T}

is the trasposed matrix of

C_{b}^{s}

, and

C_{b (j)}^{b (k)}

can be calculated from the information of GUs [27]:

C_{b (j)}^{b (k)} = (I + [Φ_{j} \times]) \dots (I + [Φ_{k - 2} \times]) (I + [Φ_{k - 1} \times]),

where [Φ×] is the skew-symmetric matrix of the angle increment:

[Φ \times] = [\begin{matrix} 0 & - ϕ_{z} & ϕ_{y} \\ ϕ_{z} & 0 & - ϕ_{x} \\ - ϕ_{y} & ϕ_{x} & 0 \end{matrix}] .

For the sampling time of Δt, the angle increment can be calculated using the angular rate of GUs in the b-frame $ω_{i b}^{b}$ :

Φ = ω_{i b}^{b} Δ t = [\begin{matrix} ϕ_{x} & ϕ_{y} & ϕ_{z} \end{matrix}] .

The vectors of a certain star in i-frame are fixed at different time. Thus, the vectors of the star in s-frame between t_k and t_j have the following linear transformation relationship:

r_{k}^{s} = C_{s (j)}^{s (k)} r_{j}^{s} .

At two different moments, the vector of a certain star $r_{k}^{s}$ and $r_{j}^{s}$ become correlated because of $C_{s (j)}^{s (k)}$ . So $C_{s (j)}^{s (k)}$ is defined as the attitude-correlated matrix (ACM) from t_k to t_j. T_mn is the value in m row and n column of $C_{s (j)}^{s (k)}$ , Eq. (9) can be rewritten as:

{\begin{cases} x_{k}^{s} = T_{11} x_{j}^{s} + T_{12} y_{j}^{s} + T_{13} z_{j}^{s} \\ y_{k}^{s} = T_{21} x_{j}^{s} + T_{22} y_{j}^{s} + T_{23} z_{j}^{s} \\ z_{k}^{s} = T_{31} x_{j}^{s} + T_{32} y_{j}^{s} + T_{33} z_{j}^{s} \end{cases} .

The ideal imaging of a star tracker can be simplified to a pinhole model [28]. If the principal point coordinates (u₀, v₀), the focal length f and the aberration of the camera δu, δv of the star tracker are known, the relationship between the vector of a star in s-frame is given by:

{\begin{cases} u = u_{0} + f \frac{x^{s}}{z^{s}} + δ u \\ v = v_{0} + f \frac{y^{s}}{z^{s}} + δ v \end{cases} .

(u_j, v_j) is the coordinate on FPA in frame # j, and (u_k, v_k) is the coordinate on FPA in frame # k:

{\begin{cases} u_{j} = u_{0} + f \frac{x_{j}^{s}}{z_{j}^{s}} + δ u \\ v_{j} = v_{0} + f \frac{y_{j}^{s}}{z_{j}^{s}} + δ v \end{cases},

{\begin{cases} u_{k} = u_{0} + f \frac{x_{k}^{s}}{z_{k}^{s}} + δ u \\ v_{k} = v_{0} + f \frac{y_{k}^{s}}{z_{k}^{s}} + δ v \end{cases},

while (u_k, v_k) is the corresponding coordinate of (u_j, v_j) at t_k under dynamic conditions, Eq. (9) can be rewritten as:

{\begin{cases} u_{k} = \frac{T_{11} (u_{j} - u_{0} - δ u) + T_{12} (v_{j} - v_{0} - δ v) + T_{13}}{T_{31} (u_{j} - u_{0} - δ u) + T_{32} (v_{j} - v_{0} - δ v) + T_{33}} + δ u \\ v_{k} = \frac{T_{21} (u_{j} - u_{0} - δ u) + T_{22} (v_{j} - v_{0} - δ v) + T_{23}}{T_{31} (u_{j} - u_{0} - δ u) + T_{32} (v_{j} - v_{0} - δ v) + T_{33}} + δ v \end{cases} .

We transform each pixel of frame # j through Eq. (14), finding the coordinate on FPA of corresponding pixels in frame # k.

For a moment t_n when the attitude is to be measured, the former successive n − 1 star images are correlated to the nth image by the attitude-correlated matrices before added. The exact number of correlated images should be chosen considering some other factors as the dynamic condition of the vehicle, integration time, the frequency of star tracker, SNR threshold for star detection and etc.

2.3. Area weighed averaging adding method

For static star images, there is a one-to-one correspondence of the pixels between different frames. However, in reality, the integer coordinate becomes fractional after transformation. Consequently, the pixels between the correlated images are not perfectly overlapping. Considering that each row and column has hundreds of pixels and we add the entire image, the transformed pixels located at the edge of the image can be ignored. A certain transformed pixel in the jth image (j = 1, 2, ..., n − 1) is located in a window in the nth image. The transformed pixel of the jth image centered at A₀ has overlapping areas with four pixels of the nth image with the center of A₁, A₂, A₃ and A₄ respectively, which is shown in Fig. 2.

Fig. 2 Coordinates of a transformed pixel: the coordinate of A₀ is an integer between 1 and 512, in the jth image. But it becomes fractional after transformation, which has overlapping areas with four pixels of the nth image.

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The gray value of pixel A₀ is divided into four parts according to the overlapping areas S and then added to the four pixels of nth image A₁, A₂, A₃ and A₄ by area weighed averaging adding method which is given by:

{\begin{cases} I (A_{1}) = I_{n} (A_{1}) + S_{1} \times I_{j} (A_{0}) \\ I (A_{2}) = I_{n} (A_{2}) + S_{2} \times I_{j} (A_{0}) \\ I (A_{3}) = I_{n} (A_{3}) + S_{3} \times I_{j} (A_{0}) \\ I (A_{4}) = I_{n} (A_{4}) + S_{4} \times I_{j} (A_{0}) \end{cases},

where I_n, I_j, I is the gray value function of the image at t_n, t_j(j = 1, 2, ..., n − 1) and after addition respectively.

Due to the area weighed averaging adding method, the original pixel is divided into four parts, the standard deviation is linearly split as well. While the images are added after the division, it is the variance of each frame that is linearly superimposed. The standard deviation of the image noise in a single frame is defined as σ. And the noise standard deviation of the correlated image is $\sqrt{n} / 2$ to $\sqrt{n}$ times that of a single frame (The specific derivation process is elaborated in the appendix):

\frac{\sqrt{n}}{2} σ ⩽ σ_{add} ⩽ \sqrt{n} σ .

The star signal is still proportional to the number of correlated frames n. Therefore, the added SNR satisfies the following inequality:

\sqrt{n} SNR ⩽ {SNR}_{add} ⩽ 2 \sqrt{n} SNR .

The area weighted averaging method can solve the problem that the pixels between correlated images are not perfectly overlapping and meanwhile make the noise standard deviation after addition lower than $\sqrt{n}$ times of a single frame.

The flowchart shown in Fig. 3 depicts the procedure of the ACFA approach. The system consists of a star tracker and GUs which are rigidly fixed together. Firstly, the star tracker takes successive star images while the GUs outputs its angular rate to calculate the ACM. Then the mathematical relationship between the image coordinates at different time and the ACM are constructed. Using the ACT, the image regions in different frames which are corresponding to the same star can be correlated to the current frame. Next the area weighed averaging method is adopted to add the correlated frames. Finally the added frame with increased SNR is used for star extraction and identification.

Fig. 3 Flow chart of ACFA approach.

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3. Simulation

3.1. Simulation parameters

The star images are generated by the star tracker simulator when given intrinsic parameters and attitude of the star tracker. The performance of a typical star tracker and GUs used for this simulation are listed in Table 1:

Table 1. Simulation parameters

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The three-axis attitude of the star tracker in i-frame is shown in Fig. 4. The GUs data is generated based on the given attitude of the star tracker at a sample rate of 50 Hz.

Fig. 4 12 seconds’ three-axis attitude of the vehicle in i-frame.

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The feasibility of the proposed method is validated by the calculated SNR of star images and star point centroiding error. The gray values of the image pixels reflect the level of star signal and noise charges, which are used to calculate the SNR. Firstly, a sub-image of 50 × 50 pixels containing the navigation star is selected. And the mean of the gray values is subtracted from each pixel in the sub-image. Then a window of 5 × 5 pixels containing the star is extracted, and their mean gray value is calculated as the signal intensity. Next, the standard deviation of the remaining pixels (exclude the 5 × 5 window) in the sub-image is calculated as the noise intensity. Finally, the SNR is equal to the signal intensity divided by noise intensity.

3.2. Adding result of correlated images

Using the approach proposed in Section 2, the simulated successive star images are correlated and added. It can be seen in Fig. 5(a) that the star signal is overwhelmed by strong noise in a single frame. When 60 frames are correlated and added in Fig. 5(b), the star signal has an obvious improvement over the noise and is easier to be detected. Besides, the star point after addition still keeps its shape.

Fig. 5 The same region of a single image and correlated image: (a) The SNR of a single frame is too low to see the dim star in the image. (b) The star signal becomes obvious when 60 frames are correlated and added.

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Then the star signal intensity and the noise intensity of correlated image are calculated. The signal of a certain star and noise growth with respect to the number of correlated frames are shown in Fig. 6. The star signal intensity in the window is proportional to the number of correlated frames, which indicates that the star point energy is still concentrated within a small region that is dominated by a certain star. The noise after the correlation and addition is between $\sqrt{n} / 2$ and $\sqrt{n}$ of a single frame, which verifies the result of Eq. (16). The SNR of the star image by ACFA approach as shown in Fig. 7(a) is between $\sqrt{n}$ and $2 \sqrt{n}$ times of a single frame, which agrees with Eq. (17).

Fig. 6 The star signal growth and noise growth of of correlated images. (a) Star signal is proportional to number of added frames. (b) Noise is between $\sqrt{n} / 2$ and $\sqrt{n}$ of a single frame.

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Fig. 7 SNR growth and centroiding accuracy estimation: (a) SNR is between $\sqrt{n}$ and $2 \sqrt{n}$ of a single frame. (b) The centroiding error of the navigation star with respect to the number of correlated frames. δx and δy represents the centroiding error in the x-axis and y-axis direction of the image coordinate system, δr represents the centroiding error in radical derenction.

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The star centroids are subsequently determined and compared with the present true value. As shown in Fig. 7(b), The centroiding error is always less than 0.1 pixel and declines with the number of correlated frames, and gradually approaches to the limit of systematic errors. Therefore the ACFA approach improves the SNR of star image making dim stars detectable and meanwhile keeps the shape of stars which guarantees the centroiding accuracy.

3.3. Influence of attitude-correlated matrix error

The attitude-correlated matrix (calculated by Eq. (5)) accuracy is one of the main factors affecting the adding result of the ACFA approach. Since the GUs information is used to calculate the ACM, the gyro error is the leading source of error. In practice, the GUs have two dominant errors including bias and noise. And typical performances of gyroscopes in different grades are listed in Table 2 [29]. GUs errors of three different grades are added (navigation, tactical and automotive respectively) while correlating the images, and the SNR of star image is analyzed. The simulation result is shown in Fig. 8(a). Compared with the error-free condition, the effect of navigation and tactical gyroscope errors on the SNR after the correlation is rather small, which can be ignored. As for the automotive grade, when the number of correlated frames is small, the simulation result agrees with theory. But when more frames are added, the SNR slightly decreases.

Fig. 8 Adding result with errors: (a) GUs of four different error grades; (b) Different fixed angle errors of three perpendicular axes.

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Table 2. GUs error

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As indicated in Eq. (5), the installation matrix $C_{b}^{s}$ is used to calculate the attitude-correlated matrix. As the fixed angle is used to calculate the installation matrix, if it deviates from the actual value, the calculated attitude-correlated matrix is inaccurate. And the correlating result of the ACFA approach is affected as well. Different levels of fixed angle error are added while correlating the images, and the SNR of the star image is analyzed. The simulation result is shown in Fig. 8(b). It can be seen that for fixed angle error less than [500, 500, 1000]^″, its effect on the SNR growth is rather small, which can be ignored. As for higher-level fixed angle error, the SNR shows a decreasing trend when more frames are correlated. Therefore, the ACFA approach performs well with middle-level fixed angle errors.

The effects of two kinds of errors are then analyzed in detail. When the GUs error is automotive-grade, 10 frames with only star signal are correlated and added by ACFA approach, the result is shown in Fig. 9(a). The added star signal is no longer concentrated within a small region, and with increased noise, the SNR is reduced.

Fig. 9 Analyze the reason of declined SNR: For automotive-graded GUs, star signal distribution of 10 correlated frames is dispersed.

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The result of ACFA depends on the accuracy of attitude-correlated matrix, so we further analyze the effects of GUs error and fixed angle error on attitude-correlated matrix. The error of attitude-correlated matrix can be calculated as follows [27]:

E = I - C_{s (j)}^{s (k)} {({\bar{C}}_{s (j)}^{s (k)})}^{T}

Where $C_{s (j)}^{s (k)}$ is the true value, ${\bar{C}}_{s (j)}^{s (k)}$ is the calculated matrix with errors, and I is the unit matrix. Therefore, the corresponding attitude error angle can be obtained by transforming E to three-axis Euler angles. The error of attitude-correlated matrix caused by automotive Gus as shown in Fig. 10(a) accumulates with navigation time. For a fixed angle error of [500, 500, 1000]^″, the error of attitude-correlated matrix is shown in Fig. 10(b). It can be seen that the error fluctuates around zero and is less than 0.02 arc-seconds. So it will not accumulate with time and is too small to affect the ACFA approach.

Fig. 10 Analyze the error of attitude-correlated matrix: (a) The output attitude error of gyro is accumulating with time. (b) The output attitude error of fixed angle is small.

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4. Experiment

The experiment was conducted to verify the proposed method at the laboratory building (Changsha, China) on 28th Nov. at 2 am. The equipment used in experiments are shown in Fig. 11. The star tracker and GUs are rigidly fixed together and mounted on a rotary table, and their performance specifications are listed in Table 3. The rotary table provided a dynamic condition for the system. The experiment last for about 40 minutes, and the three-axis attitude of the star tracker in i-frame is shown in Fig. 12.

Fig. 11 Experimental system.

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Table 3. Experimental parameters

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Fig. 12 Attitude change of the vehicle.

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During the process, 4556 successive star images were taken by star tracker. The three-axis angular rate sensed by the GUs can be transformed to the s frame through the pre-calibrated installation matrix $C_{b}^{s}$ . And then the attitude-correlated matrix between different frames can be calculated. Since the attitude-correlated matrix is calculated, the star images can be correlated and added by the ACFA approach.

The contrast of the star images before and after the correlation is shown in Fig. 13. The star on the image after addition still keeps its shape as shown in Fig. 13(b), so it guarantees the subsequent star centroiding accuracy. There is a dim star on the original image, and it can be detected after adding 5 frames. The star identification result is then analyzed. For the single image in Fig. 13(a), 9 stars can be identified. When 5 successive frames are correlated and added as shown in Fig. 13(b), the number of identified stars increases to 24. Since the dim stars become “brighter”, the number of detectable stars increases as well.

Fig. 13 Contrast before and after addition: red circle is the identified stars. (a) A single frame; (b) 5 correlated frames.

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The noise and SNR growth with the number of correlated frames is shown in Fig. 14. The noise after the correlation and addition is between $\sqrt{n} / 2$ and $\sqrt{n}$ of a single frame, which verifies the simulation result. The SNR of n correlated frames is $\sqrt{n} ~ 2 \sqrt{n}$ times that of a single frame, which means the experimental results are consistent with the theory.

Fig. 14 Experimental result of ACFA approach: (a) The noise of n correlated frames is between $\sqrt{n} / 2$ and $\sqrt{n}$ of a single image. (b) The SNR of n correlated frames is between $\sqrt{n}$ and $2 \sqrt{n}$ of a single image.

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The 4556 images under dynamic conditions are divided into more than 400 parts, every 10 successive star images is a group and then correlated by ACFA approach. The statistical result is shown in Fig. 15. Stars with magnitude lower than 5 are detectable for a single frame. As the SNR of star signals is increasing with the number of correlated frames, the maximum magnitude of detectable stars rises as well and reaches 5.64 for 10 correlated frames. Therefore, the number of identified stars increases sharply from 1 to 9 when 7 frames are added. The star identification failure rate shows an opposite trend decreasing from 80% to 20%.

Fig. 15 The influence of ACFA on the star detection and star identification: (a) Maximum of identified star magnitude. (b) The mean number of identified stars. (c) Failure rate. (d) RMS value of angular distance error.

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In the case where the number of bright stars is less than 3, the star identification is likely to fail, and the star tracker cannot output the attitude of the vehicle normally. So the ACFA approach can improve the threshold of detectable star magnitude and the star identification numbers. When the failure rate is accordingly lower, the application of star tracker can be expanded to strong background radiation condition. Besides, with more identified stars on the image, the output attitude accuracy of the star tracker is improved as well [9].

Since the true coordinates of the stars on the focal plane are unknown, the angular distance (the angular separation between two stars as observed from the detector) is used to estimate the star centroid error. In different coordinate systems, the angular distance between two certain stars is constant. In i-frame (taken for reference), it can be calculated using the right ascension and declination of the identified stars. The corresponding angular distance in s-frame (measured value) can be obtained through the coordinates on FPA and focal length. And their difference is the angular distance error. The root mean square(RMS) value of star angular distance error is shown in Fig. 15(d). The RMS of a single frame is up to 13.1 arc-seconds. It declines to approximately 7.5 arc-seconds as the number of correlated frames increases. But it cannot keep decreasing to zero due to the limit of systematic errors, which is consistent with the simulation result.

5. Conclusions

The star tracker is sensitive to strong background noise and can be completely out of operation. In this paper, an attitude-correlated frames adding (ACFA) approach is proposed to break the bottleneck. The attitude-correlated matrix at different time is calculated using the information of GUs. And then successive star images are correlated by transforming the pixels to find the regions of a certain star on different images. Next, add the correlated images to enhance star signal intensity and to improve the SNR of star images. Finally, the method of area weighed averaging is adopted to add the transformed pixels which are not perfectly overlapping. Simulations under dynamic conditions and experimental results under oscillating conditions indicate that the star signal of n frames correlated by ACFA approach is n times that of a single frame and the SNR increases to $\sqrt{n} ~ 2 \sqrt{n}$ times. It also improves the star centroid accuracy. And the impact of GUs error (including bias and noise) and fixed angle can be ignored at low angular rate of the vehicle, which verifies the effectiveness of the proposed approach.

For the hardware implementation, the ACFA method, which uses sequent frames for attitude determination, requires larger memory space to store the image data and more computation resource for image adding. That will cost more hardware resource and time for image processing. The increment of image processing time will induce more time delay of the attitude output. To solve this problem, we can use a high-speed central processing unit (CPU), of which the clock speed is up to several GHz, for image processing. Moreover, the computation cost can be reduced by processing the sub-images containing the signal of navigation stars. Because the integration of star tracker and GUs can provide the attitude information, we can predict the coordinates of the navigation stars on each frame. Then the sub-image (for example, a window of 15 pixels×15 pixels) containing a star can be extracted, with its center located at the predicted star coordinates. The ACFA method can only store and process the sequent sub-images. As a consequence, the hardware cost will be greatly reduced.

The method can be applied to star tracker and INS integrated navigation systems. The GUs of INS can provide accurate attitude change for this approach. In return, the accumulating error of the inertial navigation system can be compensated with more accurate attitude information of the star tracker. Using a short-wave infrared camera to detect stars during the daytime is another trend but the noise of infrared image is also high. The feasibility of this method for improving the SNR of infrared star images should be further assessed.

Appendix

After the correlated images are added by the area weighted averaging method, the SNR of the star image is analyzed.

Assuming that the noise of an image (M × N pixels) is Gaussian distribution, its mean value is Ī and standard deviation is σ:

{\begin{matrix} \bar{I} = \frac{1}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} I_{i, j} \\ σ = \sqrt{\frac{1}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {(I_{i, j} - \bar{I})}^{2}} \end{matrix},

Considering that the pixels of correlated images after ACT are not perfectly overlapping as shown in the figure below, a pixel is divided into four parts: S₁, S₂, S₃, and S₄. Before addition each pixel of a single frame is divided into the same four parts which have the ratio of k₁, k₂, k₃ and k₄ respectively. So for each sub-pixel, we have:

{\begin{matrix} I_{i, j}^{S_{1}} = k_{1} \cdot I_{i, j}, I_{i, j}^{S_{2}} = k_{2} \cdot I_{i j}, I_{i, j}^{S_{3}} = k_{3} \cdot I_{i, j} I_{i, j}^{S_{4}} = k_{4} \cdot I_{i, j} \\ k_{1} + k_{2} + k_{3} + k_{4} = 1 \end{matrix},

where

I_{i, j}^{S_{1}}

,

I_{i, j}^{S_{2}}

,

I_{i, j}^{S_{3}}

,

I_{i, j}^{S_{4}}

denotes the gray value of the S₁, S₂, S₃, S₄ potion of the pixel I_i,j and 0 ⩽ k₁, k₂, k₃, k₄ ⩽ 1. The mean value and standard deviation of all the sub-pixels whose area are S₁ can be calculated:

{\begin{matrix} {\bar{I}}^{S_{1}} = \frac{1}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} I_{i, j}^{S_{1}} = \frac{k_{1}}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} I_{i, j} = k_{1} \cdot \bar{I} \\ σ_{1} = \sqrt{\frac{1}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {(I_{i, j}^{S_{1}} - {\bar{I}}^{S_{1}})}^{2}} = \sqrt{\frac{1}{M \cdot N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {(k_{1} I_{i, j} - k_{1} \bar{I})}^{2}} = k_{1} σ \end{matrix},

Similarly, the standard deviation of S2, S3, and S4 sub-pixels are:

σ_{2} = k_{2} \cdot σ, σ_{3} = k_{3} \cdot σ, σ_{4} = k_{4} \cdot σ .

Therefore, the standard deviations of the four arrays has the equation:

σ = σ_{1} + σ_{2} + σ_{3} + σ_{4} .

When these divided parts of n frames with the same noise intensity (standard deviation σ) are added together, it is their variances that add linearly. And the noise intensity of the added frame is σ_add:

σ_{add}^{2} = \sum_{f = 1}^{n} σ_{f 1}^{2} + σ_{f 2}^{2} + σ_{f 3}^{2} + σ_{f 4}^{2},

where σ_f1, σ_f2, σ_f3, σ_f4 are the standard deviation of the sub-pixels with the area of S₁, S₂, S₃ and S₄ in the fth frame respectively. We have:

σ = σ_{f 1} + σ_{f 2} + σ_{f 3} + σ_{f 4} .

For any real number, there exists an inequality:

\frac{1}{4} {(a + b + c + d)}^{2} ⩽ a^{2} + b^{2} + c_{i 3}^{2} + d^{2} ⩽ {(a + b + c + d)}^{2},

hence, for f = 1, 2, ..., n,

\frac{1}{4} {(σ_{f 1} + σ_{f 2} + σ_{f 3} + σ_{f 4})}^{2} ⩽ σ_{f 1}^{2} + σ_{f 2}^{2} + σ_{f 3}^{2} + σ_{f 4}^{2} ⩽ {(σ_{f 1} + σ_{f 2} + σ_{f 3} + σ_{f 4})}^{2},

It can be deduced that:

\sum_{f = 1}^{n} \frac{1}{4} {(σ_{f 1} + σ_{f 2} + σ_{f 3} + σ_{f 4})}^{2} ⩽ \sum_{f = 1}^{n} σ_{f 1}^{2} + σ_{f 2}^{2} + σ_{f 3}^{2} + σ_{f 4}^{2} ⩽ \sum_{f = 1}^{n} {(σ_{f 1} + σ_{f 2} + σ_{f 3} + σ_{f 4})}^{2},

that is:

\frac{n}{4} σ^{2} ⩽ σ_{add}^{2} ⩽ n σ^{2} .

Thus,

\frac{\sqrt{n}}{2} σ ⩽ σ_{add} ⩽ \sqrt{n} σ .

The star signal is proportional to the number of correlated frames n. Therefore, the SNR of the correlated frame after addition satisfies the following inequality:

\sqrt{n} SNR ⩽ {SNR}_{add} ⩽ 2 \sqrt{n} SNR .

Funding

National Natural Science Foundation of China (NSFC) (61803378, 61573368).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Parameter	Value	Parameter	Value
Image size (pixel)	512 × 512	Star tracker update frequency (Hz)	5
Focal length (mm)	100	Field of view (degree)	6
Integration time (ms)	50	Gyroscope sampling frequency (Hz)	50
Gyro Bias (deg/hr)	0.01	Gyro Noise ( $\deg / \sqrt{hr}$ )	0.002

Sensor Quality	Gyro Bias (deg/hr)	Gyro Noise ( $\deg / \sqrt{hr}$ )
Navigation Grade	0.01	0.005
Tactical Grade	1	0.125
Automotive Grade	100	0.3

Parameter	Value	Parameter	Value
Image size (pixel)	1024 × 1024	Star tracker update frequency (Hz)	2
Focal length (mm)	25.6	Gyroscope sampling frequency (Hz)	1000
Integration time (ms)	50	Field of view (degree)	14.2
Gyro Bias (deg/hr)	0.01	Gyro Noise ( $\deg / \sqrt{hr}$ )	0.002

Parameter	Value	Parameter	Value
Image size (pixel)	512 × 512	Star tracker update frequency (Hz)	5
Focal length (mm)	100	Field of view (degree)	6
Integration time (ms)	50	Gyroscope sampling frequency (Hz)	50
Gyro Bias (deg/hr)	0.01	Gyro Noise ( $\deg / \sqrt{hr}$ )	0.002

Sensor Quality	Gyro Bias (deg/hr)	Gyro Noise ( $\deg / \sqrt{hr}$ )
Navigation Grade	0.01	0.005
Tactical Grade	1	0.125
Automotive Grade	100	0.3

Attitude-correlated frames adding approach to improve signal-to-noise ratio of star image for star tracker

Abstract

1. Introduction

2. Basic theory

2.1. Frame adding to improve SNR under static conditions

2.2. Attitude-correlated transformation process

2.3. Area weighed averaging adding method

3. Simulation

3.1. Simulation parameters

3.2. Adding result of correlated images

3.3. Influence of attitude-correlated matrix error

4. Experiment

5. Conclusions

Appendix

Funding

Disclosures

References

Cited By

Figures (15)

Tables (3)

Equations (31)

Optics Express