# Compressive Sensing Based Space Flight Instrument Constellation for Measuring Gravitational Microlensing Parallax

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Goals

#### 1.3. Organization

- In Section 2, we provide background on the theory of gravitational microlensing parallax measurements.
- In Section 3, we provide background on CS theory and its application for gravitational microlensing measurements.
- In Section 4, we show a potential CS detector implementation architecture for a telescope constellation.
- In Section 5, we describe our simulation and modelling setup.
- In Section 6, we provide our simulation results and analysis.
- In Section 7, we discuss data volume and on-board resources required for space instrument implementation.
- Finally, in Section 8, we provide conclusions.

## 2. Microlensing Parallax

- Motion of the Earth around the sun causing an annual parallax;
- Two or more space based observatories, separated by a significant baseline;
- Terrestrial parallax measured using a ground and space based observatory.

**o**= $<{o}_{1},{o}_{2}>$, where:

## 3. CS Application for Gravitational Microlensing

#### 3.1. CS Theory

#### 3.2. CS Application

- 1
- Obtain CS based measurements, ${y}_{o}$, for a spatial image.CS can be applied by projecting a matrix, A, onto the region of interest, ${x}_{o}$. This can be done on a column-by-column basis for a n × n spatial region, ${x}_{o}$. Thus, for 2D images, ${y}_{0}$ and A are of size m × n, where $m<<n$.
- 2
- Given A and a clean reference image, ${x}_{r}$, construct measurements matrix ${y}_{r}$, where ${y}_{r}=A{x}_{r}$.
- 3
- Apply a 2D differencing algorithm on ${y}_{o}$ and ${y}_{r}$ to obtain a differenced image, ${y}_{\mathrm{diff}}$, and the corresponding convolution kernel, M, which is used to match the observed and reference CS measurement vectors, ${y}_{o}$ and ${y}_{r}$ [11]. In our modelling, we use ${y}_{\mathrm{diff}}={y}_{o}-{y}_{r}$, by using the assumption that the PSF of the reference and observed image is the same. This assumption is valid for space flight instruments when both observed and reference images are obtained on-board the instrument.
- 4
- Reconstruct the differenced image, ${x}_{\mathrm{diff}}^{\prime}$ using CS reconstruction algorithms, given A and ${y}_{\mathrm{diff}}$.

## 4. CS Detector Architecture

#### State-of-the-Art Space Flight Instrumentation

## 5. Simulation Setup

#### 5.1. Parallax Measurement Setup

- Source is in the galactic bulge: ${D}_{s}$ = 8 kpc
- ${D}_{L}$ = 4 kpc
- ${\mu}_{rel}=200{\displaystyle \frac{Km}{s}}$

#### 5.2. Compressive Sensing Setup

## 6. Results

## 7. Data Volume and Resources

- 1
- Data volume storageUsing a $n\times n$ image, with a 14-bit ADC resolution, we would expect the total data volume to be:
Traditional Detector CS detector 14 bits × $n\times n$ 14 bits × $m\times n$ We would make the assumption that the photodetector is not saturated with the ADC bit resolution needed to sample. Without data compression, we will need to transfer $14{n}^{2}$ bits/ FOV using a traditional detector. Using CS approach for $25\%$ measurements, we can will need to transmit $14\times 0.25\times n\times n=3.5{n}^{2}$ bits/ FOV. - 2
- Computational resourcesOn-board computation will consist of programming the spatial modulator and storing the $m\times n$ size acquired data for each $n\times n$ spatial image. To compare this with a traditional detector system, we would require computational resources for compressing data on-board, in order to be accommodated in the data down-link bandwidth.In terms of on-board Field Programmable Gate Array (FPGA) resources for each of the modules listed in Table 13, we would expect a similar amount of logic gates, except for item 3. There are different methods for implementing data compression, including compression algorithms and pixel averaging [24,25]. For CS detectors, spatial modulation implementation will depend on the spatial modulator used. In addition, we would require either storage or generation and transmission of the spatial modulation matrix (CS measurement matrix) on-board. The on-board storage needed for traditional detectors would be significantly higher than storage needed for CS architecture modules.
- 3
- OpticsA traditional detector consists of a telescope and a detector, typically a CCD camera. In the case for CS, we would need a telescope, as well as lenses, to focus the light on a spatial modulator device, such as a DMD array, followed by a photodetector. However, lensless cameras for CS applications have been implemented [16,17,18] and would need to be studied for a SmallSat type instrument. The optical path required to implement the detector system will be further studied in future work.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CS | Compressive Sensing |

SmallSat | Small Satellite |

FOV | Field-of-View |

ASTERIA | Arcsecond Space Telescope Enabling Research in Astrophysics |

FPGA | Field Programmable Gate Array |

DMD | Digital Micro-mirror Device |

ADC | Analog-to-Digital Converter |

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**Figure 1.**CS Architecture. The blue block represents CS data acquisition which can be performed on-board a spaceflight instrument, while the orange blocks represent computations which can be performed on the ground. Image differencing can also be performed on-board to further reduce data volume.

**Figure 5.**A diagram of satellite constellations observing the same spatial region in order to capture a microlensing parallax of any microlensing event occurring the given field-of-view. X represents a satellite with a CS detector system.

**Figure 6.**A potential CS implementation of the detector system using a telescope to acquire the light from the spatial region, a set of micro-mirror arrays to reflect light using CS projection methods, and a photodiode to capture a single measurement of the total reflected light.

**Figure 7.**Photometric curves generated by different parallax values, shown with its corresponding CS reconstructed curve for R = 7000 km.

**Figure 8.**Legend for Figure 7.

**Figure 9.**Photometric curves generated by different parallax values, shown with its corresponding CS reconstructed curve for R = 42,000 km. The original photometric curve without any microlensing effects is shown in red for comparison.

**Figure 10.**Legend for Figure 9.

**Figure 11.**Photometric curves generated by different parallax values, shown with its corresponding CS reconstructed curve for R = 1 AU. The magnification is significantly lower because the differenced image is reconstructed using our CS technique, and the $\mathsf{\Delta}$ in both ${u}_{0}$ and ${t}_{0}$ is significantly high.

**Figure 12.**Legend for Figure 11.

Parameter | Definition |
---|---|

${t}_{0}$ | Time of peak magnification |

${t}_{E}$ | Einstein ring crossing time: $\frac{{\theta}_{E}}{{\mu}_{rel}}$ |

${\mu}_{0}$ | Impact parameter in units of ${\theta}_{E}$ |

${F}_{s}$ | Microlensing source star flux |

${F}_{b}$ | Microlensing source star blended flux |

Traditional Detectors | CS Detectors |
---|---|

CCD Detectors | Typically designed with spatial light modulators and photodiode |

Pixel by pixel readout of the image | Total power reflected from the matrix projected onto the image is measured |

Digitization of each pixel readout | Digitization of the total power read |

R | ${\mathit{t}}_{\mathit{E}}$ | Cadence | Observation Time |
---|---|---|---|

7000 km | 1 day | 48 min | 1 day |

42,000 km | 1 day | 48 min | 1 day |

1 AU | 1 day | 5.02 days | 150.5 days |

**Table 4.**Percent Error for CS reconstruction for each $\mathsf{\Phi}$ for R = 7000 km. The second row shows average % error over all time samples, the third row shows average % error at peak magnification, and the last row shows the standard deviation of the % error at peak magnification.

$\mathbf{\Phi}$ | 0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ |
---|---|---|---|---|---|---|---|---|

Avg % Err | 0.175 | 0.230 | 0.088 | 0.109 | 0.163 | 0.161 | 0.240 | 0.098 |

Avg % Err at peak | 0.075 | 0.06 | 1.07 | 0.068 | 1.09 | 0.076 | 0.081 | 0.073 |

Std dev. % Err at peak | 0.057 | 0.064 | 9.94 | 0.056 | 9.94 | 0.086 | 0.070 | 0.068 |

**Table 5.**Percent error at peak magnification over 100 Monte Carlo simulations, between a microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. Error values for R = 7000 km. Values in bold underline show where % error between the two curves is less than $10\%$.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 12.0 | 25.3 | 40.0 | 50.2 | 56.5 | 59.8 | 60.1 |

$\frac{\pi}{8}$ | 13.6 | - | 15.2 | 31.8 | 43.4 | 50.6 | 54.4 | 54.7 |

$\frac{\pi}{4}$ | 33.9 | 17.9 | - | 19.6 | 33.3 | 41.7 | 46.2 | 46.6 |

$\frac{3\pi}{8}$ | 66.7 | 46.7 | 24.5 | - | 16.9 | 27.5 | 33.1 | 33.5 |

$\frac{\pi}{2}$ | 101 | 76.6 | 49.8 | 20.4 | - | 12.7 | 19.4 | 20.0 |

$\frac{5\pi}{8}$ | 130 | 102 | 71.6 | 37.9 | 14.5 | - | 7.72 | 8.34 |

$\frac{3\pi}{4}$ | 149 | 119 | 85.9 | 49.4 | 24.1 | 8.36 | - | 0.674 |

$\frac{7\pi}{8}$ | 151 | 121 | 87.2 | 50.4 | 24.9 | 9.10 | 0.679 | - |

**Table 6.**Time difference in

**Hours**at peak magnification between a microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. R = 7000 km.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 0 | 0.828 | 0.828 | 1.66 | 1.66 | 0.828 | 0.828 |

$\frac{\pi}{8}$ | 0 | - | 0.828 | 0.828 | 1.66 | 1.66 | 0.828 | 0.828 |

$\frac{\pi}{4}$ | 0.828 | 0.828 | - | 0 | 0.828 | 0.828 | 0. | 0 |

$\frac{3\pi}{8}$ | 0.828 | 0.828 | 0 | - | 0.828 | 0.828 | 0 | 0 |

$\frac{\pi}{2}$ | 1.66 | 1.66 | 0.828 | 0.828 | - | 0 | 0.828 | 0.828 |

$\frac{5\pi}{8}$ | 1.66 | 1.66 | 0.828 | 0.828 | 0 | - | 0.828 | 0.828 |

$\frac{3\pi}{4}$ | 0.828 | 0.828 | 0 | 0 | 0.828 | 0.828 | - | 0 |

$\frac{7\pi}{8}$ | 0.828 | 0.828 | 0 | 0 | 0.828 | 0.828 | 0 | - |

**Table 7.**Percent Error for CS reconstruction for each $\mathsf{\Phi}$ for R = 42,000 km. The second row shows average % error over all time samples, the third row shows average % error at ${t}_{0}$, and the last row shows the standard deviation of the % error at ${t}_{0}$.

$\mathbf{\Phi}$ | 0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ |
---|---|---|---|---|---|---|---|---|

Avg % Err | 0.108 | 0.110 | 0.182 | 0.110 | 0.191 | 0.208 | 0.173 | 0.201 |

Avg % Err at peak | 1.07 | 0.059 | 0.091 | 1.07 | 0.086 | 0.063 | 0.062 | 0.071 |

Std dev. % Err at peak | 9.94 | 0.041 | 0.205 | 9.94 | 0.094 | 0.049 | 0.062 | 0.057 |

**Table 8.**Percent error at peak magnification between a microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. Error values for R = 42,000 km. Values in bold underline show where % error between the two curves is less than $10\%$.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 22.1 | 7.50 | 109 | 215 | 25.1 | 59.7 | 70.9 |

$\frac{\pi}{8}$ | 28.3 | - | 18.7 | 168 | 305 | 3.85 | 48.3 | 62.6 |

$\frac{\pi}{4}$ | 8.11 | 15.8 | - | 126 | 241 | 19.0 | 56.5 | 68.5 |

$\frac{3\pi}{8}$ | 52.1 | 62.7 | 55.7 | - | 51.0 | 64.1 | 80.7 | 86.1 |

$\frac{\pi}{2}$ | 68.3 | 75.3 | 70.7 | 33.8 | - | 76.2 | 87.2 | 90.8 |

$\frac{5\pi}{8}$ | 33.5 | 4.00 | 23.5 | 179 | 321 | - | 46.2 | 61.1 |

$\frac{3\pi}{4}$ | 148 | 93.4 | 130 | 418 | 682 | 86.0 | - | 27.7 |

$\frac{7\pi}{8}$ | 243 | 168 | 218 | 617 | 982 | 157 | 38.3 | - |

**Table 9.**Time difference in

**Hours**at peak magnification between microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. Error values for R = 42,000 km.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 1.66 | 4.14 | 6.62 | 9.10 | 10.8 | 10.8 | 10.8 |

$\frac{\pi}{8}$ | 1.66 | - | 2.48 | 4.97 | 7.45 | 9.10 | 9.10 | 9.10 |

$\frac{\pi}{4}$ | 4.14 | 2.48 | - | 2.48 | 4.97 | 6.62 | 6.62 | 6.62 |

$\frac{3\pi}{8}$ | 6.62 | 4.97 | 2.48 | - | 2.48 | 4.14 | 4.14 | 4.14 |

$\frac{\pi}{2}$ | 9.10 | 7.45 | 4.97 | 2.48 | - | 1.66 | 1.66 | 1.66 |

$\frac{5\pi}{8}$ | 10.8 | 9.10 | 6.62 | 4.14 | 1.66 | - | 0 | 0 |

$\frac{3\pi}{4}$ | 10.8 | 9.10 | 6.62 | 4.14 | 1.66 | 0 | - | 0 |

$\frac{7\pi}{8}$ | 10.8 | 9.10 | 6.62 | 4.14 | 1.66 | 0 | 0 | - |

**Table 10.**Percent Error for CS reconstruction for each $\mathsf{\Phi}$ for R = 1 AU. The second row shows average % error over all time samples, the third row shows average % error at the peak of each curve, and the last row shows the standard deviation of the % error at the peak.

$\mathbf{\Phi}$ | 0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ |
---|---|---|---|---|---|---|---|---|

Avg % Err | 0.437 | 0.441 | 0.633 | 0.743 | 0.621 | 0.348 | 0.616 | 0.582 |

Avg % Err at peak | 0.186 | 0.192 | 0.192 | 0.194 | 0.183 | 0.119 | 0.106 | 0.117 |

Std dev. % Err at peak | 0.146 | 0.190 | 0.178 | 0.183 | 0.137 | 0.287 | 0.092 | 0.083 |

**Table 11.**Percent error at peak between a microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. Error values for R = 1 AU. Values in bold underline show where % error between the two curves is less than $5\%$.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 12.7 | 27.2 | 43.7 | 18.7 | 276 | 217 | 167 |

$\frac{\pi}{8}$ | 11.3 | - | 12.8 | 27.5 | 5.34 | 234 | 181 | 137 |

$\frac{\pi}{4}$ | 21.4 | 11.4 | - | 13.0 | 6.65 | 196 | 149 | 110 |

$\frac{3\pi}{8}$ | 30.4 | 21.5 | 11.5 | - | 17.4 | 162 | 121 | 85.6 |

$\frac{\pi}{2}$ | 15.8 | 5.07 | 7.12 | 21.0 | - | 217 | 167 | 125 |

$\frac{5\pi}{8}$ | 73.4 | 70.0 | 66.2 | 61.8 | 68.4 | - | 15.7 | 29.1 |

$\frac{3\pi}{4}$ | 68.5 | 64.4 | 59.9 | 54.7 | 62.5 | 18.6 | - | 15.9 |

$\frac{7\pi}{8}$ | 62.5 | 57.7 | 52.3 | 46.1 | 55.5 | 41.0 | 18.9 | - |

**Table 12.**Time difference in

**Days**at peak between a microlensing photometric curve with $\mathsf{\Phi}$ shown in the first row, compared to the photometric curve with $\mathsf{\Phi}$ in the first column. Error values for R = 1 AU. Values in bold underline show where % error between the two curves is less than $5\%$.

0 | $\frac{\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{\mathit{\pi}}{2}$ | $\frac{5\mathit{\pi}}{8}$ | $\frac{3\mathit{\pi}}{4}$ | $\frac{7\mathit{\pi}}{8}$ | |
---|---|---|---|---|---|---|---|---|

0 | - | 25.9 | 51.9 | 77.8 | 88.2 | 62.3 | 36.3 | 10.4 |

$\frac{\pi}{8}$ | 25.9 | - | 25.9 | 51.9 | 62.3 | 88.2 | 62.3 | 36.3 |

$\frac{\pi}{4}$ | 51.9 | 25.9 | - | 25.9 | 36.3 | 114 | 88.2 | 62.3 |

$\frac{3\pi}{8}$ | 77.8 | 51.9 | 25.9 | - | 10.4 | 140 | 114 | 88.2 |

$\frac{\pi}{2}$ | 88.2 | 62.3 | 36.3 | 10.4 | - | 151 | 125 | 98.6 |

$\frac{5\pi}{8}$ | 62.3 | 88.2 | 114 | 140 | 151 | - | 25.9 | 51.9 |

$\frac{3\pi}{4}$ | 36.3 | 62.3 | 88.2 | 114 | 125 | 25.9 | - | 25.9 |

$\frac{7\pi}{8}$ | 10.4 | 36.3 | 62.3 | 88.2 | 98.6 | 51.9 | 25.9 | - |

Traditional Detector | CS Detector | |
---|---|---|

1 | Data acquisition (ADC) interface | Data acquisition (ADC) interface |

2 | Data storage module | Data storage module |

3 | Data compression | Spatial modulation implementation |

4 | Data packetization and transmission | Data packetization and transmission |

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**MDPI and ACS Style**

Korde-Patel, A.; Barry, R.K.; Mohsenin, T.
Compressive Sensing Based Space Flight Instrument Constellation for Measuring Gravitational Microlensing Parallax. *Signals* **2022**, *3*, 559-576.
https://doi.org/10.3390/signals3030034

**AMA Style**

Korde-Patel A, Barry RK, Mohsenin T.
Compressive Sensing Based Space Flight Instrument Constellation for Measuring Gravitational Microlensing Parallax. *Signals*. 2022; 3(3):559-576.
https://doi.org/10.3390/signals3030034

**Chicago/Turabian Style**

Korde-Patel, Asmita, Richard K. Barry, and Tinoosh Mohsenin.
2022. "Compressive Sensing Based Space Flight Instrument Constellation for Measuring Gravitational Microlensing Parallax" *Signals* 3, no. 3: 559-576.
https://doi.org/10.3390/signals3030034