# Sieve Search Centroiding Algorithm for Star Sensors

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## Abstract

**:**

## 1. Introduction

## 2. Sieve Search Algorithm

#### 2.1. Characteristics of PSF Gray Value Distribution

- the ROI matrix is nearly symmetric. For an $n\times n$ ROI matrix, let ${G}_{L}$ represent the n-element vector, consisting of gray values in the pixels along the leading diagonal ${d}_{L}$. The off-diagonals ${d}_{LU}$ and ${d}_{LD}$ are a pixels apart from ${d}_{L}$. The gray value vectors corresponding to ${d}_{LU}$ and ${d}_{LD}$ are ${G}_{LU}$ and ${G}_{LD}$, respectively. It can be observed that the value of the ${b}^{th}$ elements in ${G}_{LU}$ and ${G}_{LD}$ are almost similar, where $b\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}[1,n-a]$. In Figure 3, the off-diagonals ${d}_{LU}$ and ${d}_{LD}$ are two pixels apart from ${d}_{L}$. The grey value vectors ${G}_{L}=\left[2\phantom{\rule{3.33333pt}{0ex}}6\phantom{\rule{3.33333pt}{0ex}}29\phantom{\rule{3.33333pt}{0ex}}94\phantom{\rule{3.33333pt}{0ex}}149\phantom{\rule{3.33333pt}{0ex}}115\phantom{\rule{3.33333pt}{0ex}}43\phantom{\rule{3.33333pt}{0ex}}9\phantom{\rule{3.33333pt}{0ex}}3\right]$, ${G}_{LU}=\left[4\phantom{\rule{3.33333pt}{0ex}}19\phantom{\rule{3.33333pt}{0ex}}63\phantom{\rule{3.33333pt}{0ex}}104\phantom{\rule{3.33333pt}{0ex}}83\phantom{\rule{3.33333pt}{0ex}}33\phantom{\rule{3.33333pt}{0ex}}7\right]$, and ${G}_{LD}=\left[5\phantom{\rule{3.33333pt}{0ex}}22\phantom{\rule{3.33333pt}{0ex}}68\phantom{\rule{3.33333pt}{0ex}}104\phantom{\rule{3.33333pt}{0ex}}77\phantom{\rule{3.33333pt}{0ex}}28\phantom{\rule{3.33333pt}{0ex}}6\right]$. The values of ${b}^{th}$ elements in ${G}_{LU}$ and ${G}_{LD}$ are found to be comparable, where $b\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}[1,7]$. In other words, the pixels along the off-diagonals, equidistant from a diagonal element on both sides of the principal diagonals, have comparable gray value accumulation.
- as we transverse along the pixels at the same radial distance from the brightest pixel, the magnitude increases as we move closer to the principal diagonal pixels. Here, magnitude is referred to the gray value of the pixel. Let ${g}_{c}$ represent the gray value of a pixel ${p}_{c}$, along ${d}_{L}$, at some radial distance from the most illuminated pixel in the ROI. At the same radial distance, in the neighborhood of ${p}_{c}$, the gray value of the pixels along ${d}_{LU}$ and ${d}_{LD}$ are ${g}_{bu}$ and ${g}_{bd}$, respectively. Then, ${g}_{c}$ is marginally greater than ${g}_{bu}$ and ${g}_{bd}$. In Figure 3, the brightest pixel has an accumulated gray value of 149. Among the pixels at the same radial distance from it, the gray value at the pixel along the principal diagonal $\left\{43\right\}$ is marginally greater than that along the off-diagonal pixels $\{28,33\}$.

#### 2.2. Sieve Segmentation and Symmetry

#### 2.3. Characteristics of Sieve Magnitude

#### 2.4. Centroiding Based on Sieve Search Algorithm

- (i)
- An ROI of size not less than $5\times 5$ pixel window is created around the star image spot. The accumulated gray value distribution inside the ROI is used to populate the elements of the corresponding ROI matrix.
- (ii)
- The ROI matrix is segregated into square sub-matrices called sieves. The degree of symmetry and magnitude of each sieve ${S}_{i}$ in the ROI is evaluated and stored as ${\varphi}_{i}$ and ${\mu}_{i}$, respectively, where $i\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}1,\dots ,{k}^{2}$. The sieves are rotated by ${90}^{\circ}$ clockwise. The corresponding ${\varphi}_{i[+{90}^{\circ}]}$ and ${\mu}_{i[+{90}^{\circ}]}$ are determined.
- (iii)
- The sieve probability index $P\left({S}_{i}\right)$, evaluated for each sieve ${S}_{i}$ in the ROI according to Equation (7), gives the probability of a sieve to containing the centroid location of the image spot. This is depicted in Figure 6.$${\rho}_{i}=({\varphi}_{i}+{\varphi}_{i[+{90}^{\circ}]})\xb7({\mu}_{i}+{\mu}_{i[+{90}^{\circ}]}),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}P\left({S}_{i}\right)=\frac{{\rho}_{i}}{max\left({\rho}_{i}\right)}$$
- (iv)
- For every sieve ${S}_{i}$ in the ROI, the value of its degree of symmetry–magnitude product ${\rho}_{i}$ is associated with each pixel contained in it. Consequently, for a pixel j, the value of ${\rho}_{i}$, resulting from the various sieves it constitutes, adds up, given by,$${\psi}_{j}=\sum _{i\u03f5U}{\rho}_{i}$$$$P\left(j\right)=\frac{{\psi}_{j}}{max\left({\psi}_{j}\right)}$$
- (v)
- Let ${h}_{j}$ and ${v}_{j}$ correspond to the horizontal and vertical coordinates of pixel j. The value of the centroid is computed as,$${h}_{c}=\frac{{\sum}_{j\u03f5W}{h}_{j}\xb7{\psi}_{j}}{{\sum}_{j\u03f5W}{\psi}_{j}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{v}_{c}=\frac{{\sum}_{j\u03f5W}{v}_{j}\xb7{\psi}_{j}}{{\sum}_{j\u03f5W}{\psi}_{j}}$$

#### 2.5. Optimal Sieve Size

## 3. Materials and Methods

#### 3.1. Simulation of Star Images

#### 3.2. Simulation of Noise Process

#### 3.3. Selection of Attributes

- the radius of the Gaussian spread in both directions, ${\sigma}_{PSF}=0.75$.
- the size of the star image pixel window was $3\times 3$.
- the average level of noise corresponding to a star image spot was restricted to 7% of the accumulated gray value in the most illuminated pixel of that star image.

#### 3.4. Particulars of Experiments

## 4. Results and Analysis

#### 4.1. Sensitivity of Centroiding Accuracy to Various Parameters

#### 4.1.1. Brightness

#### 4.1.2. Location of Star Spot Inside a Pixel

#### 4.1.3. Noise Floor

#### 4.1.4. Gaussian Spread

#### 4.2. Special Cases: Performance Evaluation

#### 4.2.1. Non-Uniform PSF

#### 4.2.2. Stuck Pixel Noise

#### 4.2.3. Optical-Double Stars

#### 4.3. Execution Time

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Pixel intensity of star image spot on a sensor array. The actual centroid of this star image is designated at $(804.35,182.87)$. The pixel closest to the centroid is the brightest and generates 1491 photons. (

**a**) The point spread function is (

**b**) The gray-scale accumulation inside the ROI.

**Figure 4.**The degree of symmetry for all sieves ${S}_{i}$ and ${S}_{i[+{90}^{\circ}]}$ in the ROI. (

**a**) $\varphi $ of all sieves S in the ROI. $\varphi $ peaks for sieves at index numbers $u=[1,\phantom{\rule{3.33333pt}{0ex}}9,\phantom{\rule{3.33333pt}{0ex}}17,\phantom{\rule{3.33333pt}{0ex}}25,\phantom{\rule{3.33333pt}{0ex}}33,\phantom{\rule{3.33333pt}{0ex}}41,\phantom{\rule{3.33333pt}{0ex}}49]$; (

**b**) ${\varphi}_{[+{90}^{\circ}]}$ of all sieves ${S}_{[+{90}^{\circ}]}$ in the ROI. ${\varphi}_{[+{90}^{\circ}]}$ peaks at $v=[7,\phantom{\rule{3.33333pt}{0ex}}13,\phantom{\rule{3.33333pt}{0ex}}19,\phantom{\rule{3.33333pt}{0ex}}25,\phantom{\rule{3.33333pt}{0ex}}31,\phantom{\rule{3.33333pt}{0ex}}37,\phantom{\rule{3.33333pt}{0ex}}43]$.

**Figure 9.**The optimal size of the sieve. (

**a**) Transient and steady-state centroiding errors for sieve size $m=2,3$ and 6; (

**b**) Comparison of various sieve sizes, based on the centroiding error and execution time.

**Figure 11.**The variation of centroiding error with respect to the changes in the brightness level of the star spot. The SSA algorithm was comparatively stable throughout the operation regime.

**Figure 12.**The variation of centroiding errors to changes in the location of the star image spot inside a pixel for various centroiding algorithms. The variation centroiding accuracy, due to S-curve errors, was minimal for SSA, compared to COM and IWCOG. (

**a**) GAC; (

**b**) COM; (

**c**) IWCOG; (

**d**) GBF, G3P and FGF; (

**e**) SSA.

**Figure 13.**The variation of centroiding error with the changes in noise level. The noise robustness of SSA was found to be equivalent to that of FGF and G3P.

**Figure 14.**The variation of centroiding error with the changes in the Gaussian radius. The centroiding error in SSA was high until ${\sigma}_{PSF}\approx 0.9$. Beyond this, as the Gaussian radius increased, the centroiding accuracy of SSA coincided with those of GBF, G3P, and FGF.

**Figure 15.**The special cases encountered by the star sensor camera during its operational lifespan. (

**a**) Non-uniform PSF; (

**b**) Stuck-pixel noise; (

**c**) Optical double stars.

**Figure 16.**The variation of centroiding error with an increase in ${\sigma}_{PSF-Y}$ from 0.5 to 1.5. For ${\sigma}_{PSF-Y}>0.9$, the centroiding accuracy of SSA was comparable with those of GBF, G3P, and FGF.

**Figure 17.**Sensitivity of centroiding accuracy to the location of stuck-pixel noise in the ROI. The magnitude of error in SSA was insignificant compared to the other algorithms, except for GBF.

**Figure 18.**The centroiding error in resolving the visual magnitude barycenter of an optical double star pair. The SSA follows the actual centroid closely, resulting in minimum centroiding error.

**Figure 19.**Comparison of relative real-time computer-simulated execution time for various centroiding algorithms. The gray-scale algorithms COM, IWCOG, and GAC were found to have the lowest execution time. SSA was found to require considerably less execution time in comparison with the curve fitting algorithms.

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**MDPI and ACS Style**

Karaparambil, V.C.; Manjarekar, N.S.; Singru, P.M. Sieve Search Centroiding Algorithm for Star Sensors. *Sensors* **2023**, *23*, 3222.
https://doi.org/10.3390/s23063222

**AMA Style**

Karaparambil VC, Manjarekar NS, Singru PM. Sieve Search Centroiding Algorithm for Star Sensors. *Sensors*. 2023; 23(6):3222.
https://doi.org/10.3390/s23063222

**Chicago/Turabian Style**

Karaparambil, Vivek Chandran, Narayan Suresh Manjarekar, and Pravin Madanrao Singru. 2023. "Sieve Search Centroiding Algorithm for Star Sensors" *Sensors* 23, no. 6: 3222.
https://doi.org/10.3390/s23063222