The navigation measurements of deep space optical navigation systems are generally combined to calculate the LOS direction and spacecraft location [

25]. The accuracy of the apparent diameter and celestial body centroiding is determined by the precision of edge detection. The accuracy of the LOS direction of the navigation sensor is determined by the attitude measurement precision, that is, the centroiding accuracy of the background stars. In this section, the effect of exposure parameters on the accuracy of star centroiding and edge detection is analyzed using the proposed image model, which provides theoretical support for parameter optimization.

#### 4.1. Edge Detection Accuracy Performance Utilizing the WCA Scheme

The edge is the part of the image where brightness changes sharply. The edge points based on the blurred edge model are given by the maxima of the first image derivative or zero crossing point of the second image derivative. Steger proposed a subpixel edge extraction algorithm in his doctoral thesis [

26]. The basic principle of the algorithm is to perform the second-order Taylor expansion about the pixel where the local gradient is maximized in the direction of the edge normal and to determine the subpixel location of the zero crossing point of the second derivative. In this study, the edge of the celestial body is extracted using this algorithm, which is essentially a fitting interpolation algorithm.

Given that the edge detection algorithm is based on image derivative information, it is highly sensitive to noise. Therefore, the image derivatives must be estimated by convolving the image with the derivatives of the Gaussian smoothing kernel. Edges appear as bright lines in an image that contains the absolute value of the gradient. The second-order Taylor expansion about the maximized gradient pixel in the direction of the edge normal is expressed as:

where

${x}_{0}$ is the pixel center. The subpixel location of the edge where

${r}^{\u2033}\left(x\right)=0$ is expressed as:

The 1D celestial body edge model is a piecewise function when the WCA scheme is applied. The gradient of the edge model is derived by convolving the edge model with the first derivative of the Gaussian smoothing kernel and expressed as:

Thus, the second derivative of edge model is derived as:

where

$\mathsf{\Theta}(x)=\frac{{\sigma}_{PSF}^{2}({x}_{c}-x)+{\sigma}^{2}({x}_{c}-l)}{\sqrt{2({\sigma}_{PSF}^{2}+{\sigma}^{2})}{\sigma}_{PSF}\sigma}$. The edge location is the zero crossing point of the second derivative, which indicates that the solution of Equation (23) and

${r}^{\u2034}\left(x\right){r}^{\prime}\left(x\right)<0$ are required. Given that the analytical solution cannot be calculated, the effect of exposure parameters on the edge detection accuracy is analyzed by numerical simulations. The edge localization error is the function of the Gaussian radius

${\sigma}_{PSF}$, well capacity

${Q}_{S}$, total integration time

$T$, AIT

${T}_{S}$, Gaussian smoothing kernel radius

$\sigma $, and edge location

$l$. For a given optical system,

${\sigma}_{PSF}$ is constant,

$\sigma $ is a parameter of the edge detection algorithm, and

$T$ is determined by the limiting detectable star visual magnitude for the navigation sensor. This study focuses on analyzing the edge detection error caused by

${Q}_{S}$ and

${T}_{S}$. Systematic error in edge detection is also introduced by pixelization. However,

$l$ is a random variable in practice that is uniformly distributed over a pixel. The root mean square error is defined as the error of edge detection, which is expressed as:

${\sigma}_{PSF}=0.67$ pixels,

$\sigma \text{\hspace{0.17em}}=\text{}0.55$ pixels,

$T=30\text{\hspace{0.17em}}\mathrm{ms}$ are set at this point. Then, the simulated celestial body images with temporal noise and fixed pattern noise are generated. Sources of noises which are taken into consideration include photon shot noise, dark current noise, readout noise, quantization noise, dark signal non-uniformity and photon response non-uniformity. The full well capacity is set to

${Q}_{MAX}=15,000{e}^{-}$ consist with the image sensor we utilized. The edge detection error simulation results are shown in

Figure 8.

First, the relationship between edge detection error ${\delta}_{E}$ and AIT ${T}_{S}$ is discussed. The black line indicates that ${\delta}_{E}$ is evidently affected by ${T}_{S}$. An interval exists wherein ${\delta}_{E}$ is significantly small. The second segment of the integration time $T-{T}_{S}$ is relatively long when ${T}_{S}$ is short, which leads to the oversaturation of the central region pixels and extension of the apparent diameter of the celestial body. However, $T\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{T}_{S}$ is relatively short when ${T}_{S}$ is excessively long, which leads to a small intensity contrast between the central region and the energy ring. The algorithm will extract the edge of the “rings” instead of the actual edge location. Therefore, ${\delta}_{E}$ initially reaches the minimum number and then increases with the increase in ${T}_{S}$.

Second, the relationship between edge detection error

${\delta}_{E}$ and well capacity

${Q}_{S}$ is discussed. Setting the AIT to an appropriate value leads to a relatively small

${\delta}_{E}$.

${T}_{S}=29.6\text{\hspace{0.17em}}\mathrm{ms}$ is set at this point. The red line indicates that

${\delta}_{E}$ varies slightly and remains nearly constant at the beginning with the increase in

${Q}_{S}$. As

${Q}_{S}$ continues to increase,

${\delta}_{E}$ increases sharply. The reason for this relationship is provided in

Figure 9, which shows the simulated images of the celestial object and the second derivative of the edge model expressed in Equation (23) when the well capacity values are 1000

${e}^{-}$, 5000

${e}^{-}$and 14,000

${e}^{-}$.

In

Figure 9, the purple color represents the saturated pixels, whereas the cyan color represents the pixels with zero intensity. The yellow arc represents the true edge of the celestial body. The red cross symbol indicates the zero-crossing point of the second derivative, which is located at the true edge location, whereas the blue circle symbol indicates the zero-crossing point that deviates from the true edge location.

Figure 9a shows that the intensity of the energy ring is small when the well capacity

${Q}_{S}$ is small and that the zero-crossing points can be extracted at the actual edge location.

Figure 9b shows that the intensity of the energy ring increases with the increase in

${Q}_{S}$ and that the zero crossing points exist at the location of the energy ring. The algorithm extracts double edges, and the false edge can be rejected.

Figure 9c shows that the intensity of the energy ring is higher with a larger

${Q}_{S}$ and that no zero-crossing points are obtained at the actual edge location. The extracted edge deviates from the true location. Thus, a large

${Q}_{S}$ value results in false edge extraction.

Thus, the well capacity and AIT are the main factors that affect the edge detection error. ${T}_{S}$ evidently influences on the accuracy of edge detection. The edge detection error initially reaches the minimum number and then increases with the increase in ${T}_{S}$; ${Q}_{S}$ must not be excessively large.

#### 4.2. Star Centroiding Accuracy Performance Utilizing the WCA Scheme

Star centroiding accuracy is the basis for attitude accuracy. The star centroiding accuracy performance is analyzed when the WCA scheme is employed to ensure attitude accuracy. The total centroiding error is decomposed into the x-and y-component errors. The errors in each case can be proven to be the same. Thus, this study focuses on analyzing the x-component errors as an example.

The centroiding error of the x-component

${\delta}_{x}$ when the WCA scheme is applied is expressed as:

In Equation (25),

${\delta}_{x}$ is the function of the Gaussian radius

${\sigma}_{PSF}$, well capacity

${Q}_{S}$, total integration time

$T$, AIT

${T}_{S}$, incident flux of the star on the image plane

${\varphi}_{S}$, and actual star location

${x}_{0}$.

$T$ is determined by the limiting detectable star visual magnitude for the navigation sensor. If

${x}_{0}$ moves within a pixel, then

${\delta}_{x}$ changes periodically. However,

${x}_{0}$ is a random variable that is uniformly distributed over a pixel within the range [−0.5, 0.5) in practice. The root mean square error is defined as the error of x, as:

After adding temporal noise and fixed pattern noise to the simulated star image, the relationship among

${\delta}_{x,S}$,

${Q}_{S}$, and

${T}_{S}$ is analyzed with different star magnitudes.

Figure 10 shows the simulated results when the star magnitude is 2, 4, 5, and 6. First, the relationship between star centroiding error

${\delta}_{x,S}$ and AIT

${T}_{S}$ is discussed.

Figure 10a shows that the centroiding error of the star magnitude = 2 slowly increases with the increase in

${T}_{S}$.

Figure 10b–d shows that the centroiding error variation caused by

${T}_{S}$ can be neglected. Thus,

${\delta}_{x,S}$ is less affected by

${T}_{S}$. Second, the relationship between star centroiding error

${\delta}_{x,S}$ and well capacity

${Q}_{S}$ is discussed.

Figure 10a–c shows that

${\delta}_{x,S}$ decreases with the increase in

${Q}_{S}$. However, the centroiding error variation of a dim star (

Figure 10d) caused by well capacity can be neglected.

The total star centroiding error is expressed as:

The relationship between total star centroiding error and exposure parameters is consistent with the x-component errors. Thus, the centroiding error of a dim star is unaffected by the WAC scheme. The centroiding error of a bright star decreases with the increase in well capacity, and the AIT effect can be ignored.

In the preceding sections, the centroiding accuracy performance of a single star is analyzed when the WCA scheme is adopted. Many stars in the FOV are required to increase the attitude determination accuracy in practice. More dim stars have been generally recorded than bright stars in the FOV. The overall star centroiding error is defined as the weighted average of centroiding errors for different star magnitudes at this point. The overall centroiding error can directly reflect the attitude determination accuracy of the optical navigation sensor. The star magnitudes range from 0 to 7 at 0.5 intervals. The star magnitudes that range from

${m}_{V}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}0.25$ to

${m}_{V}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}0.25$ are considered the magnitude

${m}_{V}$ to simplify the analysis process. Therefore, the overall star centroiding error is expressed as:

where

${N}_{{M}_{Vi},FOV}$ is the average number of stars brighter than magnitude

${M}_{Vi}$ in the FOV, and is expressed as [

27]:

where

$A$ is the FOV size. The number of stars in the FOV increases exponentially with the increase in star magnitude

${M}_{Vi}$. There are much more dim stars than the bright ones in the FOV. As a result, the star centroiding error of dim star contributes more to the overall centroiding error.

Figure 11 shows the relationship between overall centroiding error and well capacity

${Q}_{S}$ when

${T}_{S}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}29.6\text{\hspace{0.17em}}\mathrm{ms}$. The overall centroiding error fluctuation is relatively stable with the increase in

${Q}_{S}$ because variation of star centroiding error of dim star caused by

${Q}_{S}$ is less affected than bright star Therefore, the variation of overall centroiding error caused by well capacity can be neglected.

In summary, the well capacity ${Q}_{S}$ is mainly responsible for the centroiding error of a single star. However, the attitude accuracy affected by well capacity can be neglected.