# Application of Compressive Sensing in the Presence of Noise for Transient Photometric Events

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CS Architecture

- Generate a spatial sky image of size n × n using uniform random distribution, in the range of 50 and 5000 pixel value. The radius of these star sources are generated using an airy point spread function with an aperture radius between 1 and 5 pixel units, which is randomly chosen.
- Magnification for the source star experiencing a single-lens microlensing event is determined by the microlensing equations [4]. The center pixel value, $P[{x}_{0},{x}_{1}]$, at any time, t, is given by$$Amp[{x}_{0},{x}_{1}]=M\left(t\right)\times P[{x}_{0},{x}_{1}]$$
- If adding source noise or background noise, generate a noise image of the same size, n × n. Add this image to the image generated in step 1.
- Generate a CS-based projection matrix of size m × n, where $m=q\%\times n$. In our simulations, we use $q=25$.
- Create CS-based measurements by$$\begin{array}{c}\hfill {y}_{o}=A{x}_{o}+{n}_{t}\end{array}$$
- Create CS-based measurements from a reference image, ${x}_{r}$, and the same measurement matrix, A:$$\begin{array}{c}\hfill {y}_{r}=A{x}_{r}\end{array}$$
- Obtain the difference, ${y}_{diff}={y}_{o}-{y}_{r}$.
- Reconstruct ${x}_{diff}$ using CS reconstruction algorithms, given ${y}_{diff}$ and A. Reconstruction algorithms such as orthogonal matching pursuit (OMP) or optimization algorithms can be used. We use OMP in our work. For OMP alogirthms, we set the sparsity level to be $10\%$ of n. Hence, once $10\%$ of n non-zero elements are obtained, the algorithm successfully exits. This value was used based on prior knowledge about transient sources in spatial sky images.

## 3. Noise in Compressive Sensing Measurements

#### 3.1. Source Noise

#### 3.2. Measurement Noise

#### 3.2.1. Shot Noise

#### 3.2.2. Thermal Noise

#### 3.3. Total Noise in Detectors

#### 3.4. SNR for CS Applications

#### 3.5. Expected Value and Variance for CS Applications

#### 3.6. Mutual Coherence of a Matrix

## 4. Numerical Results

#### 4.1. Gravitational Microlensing Setup

#### 4.2. CS Analysis with Source Noise

#### 4.3. CS with Added Measurement Noise

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CS | Compressive sensing |

SNR | Signal-to-noise ratio |

## References

- Eldar, Y.C.; Kutyniok, G. (Eds.) Compressed Sensing: Theory and Applications; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Bobin, J.; Starck, J.; Ottensamer, R. Compressed Sensing in Astronomy. IEEE J. Sel. Top. Signal Process.
**2008**, 2, 718–726. [Google Scholar] [CrossRef] [Green Version] - Candès, E.J. Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, 22–30 August 2006. [Google Scholar]
- Seager, S. (Ed.) Exoplanets; University of Arizona Press: Tucson, AZ, USA, 2010. [Google Scholar]
- Pope, G. Compressive Sensing: A Summary of Reconstruction Algorithms. Master’s Thesis, ETH, Swiss Federal Institute of Technology Zurich, Department of Computer Science, Zurich, Switzerland, 2009. [Google Scholar]
- Diamond, S.; Boyd, S. CVXPY: A Python-embedded modeling language for convex optimization. J. Mach. Learn. Res.
**2016**, 17, 2909–2913. [Google Scholar] - Korde-Patel, A.; Barry, R.K.; Mohsenin, T. Application of Compressive Sensing to Gravitational Microlensing Experiments. In Proceedings of the International Astronomical Union, Sorento, Italy, 19–25 October 2016; pp. 67–70. [Google Scholar]
- Korde-Patel, A.; Barry, R.K.; Mohsenin, T. Compressive Sensing Based Data Acquisition Architecture for Transient Stellar Events in Crowded Star Fields. In Proceedings of the IEEE International Instrumentation and Measurement Technology Conference (I2MTC), Dubrovnik, Croatia, 25–28 May 2020; pp. 1–6. [Google Scholar]
- Arias-Castro, E.; Eldar, Y.C. Noise folding in compressed sensing. IEEE Signal Process. Lett.
**2011**, 18, 478–481. [Google Scholar] [CrossRef] [Green Version] - Jauregui-Sánchez, Y.; Clemente, P.; Latorre-Carmona, P.; Tajahuerce, E.; Lancis, J. Signal-to-noise ratio of single-pixel cameras based on photodiodes. Appl. Ortics
**2018**, 57, B67–B73. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hasinoff, S.W. Photon, Poisson Noise. Comput. Vision Ref. Guide
**2014**, 14, 608–610. [Google Scholar] - Zmuidzinas, J. Thermal noise and correlations in photon detection. Appl. Opt.
**2003**, 42, 4989–5008. [Google Scholar] [CrossRef] [PubMed]

**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.

**Figure 7.**Rate of decay of coefficients, zoomed in towards the higher magnification coefficients to view the difference between ${\mu}_{0}=0.1$ and ${\mu}_{0}=0.01$.

**Figure 8.**Summary of the % error for an image with added Gaussian source noise for a single-lensed microlensing event with ${\mu}_{0}=0.1$ and ${\mu}_{0}=0.01$ for varying levels of Gaussian noise addition to the spatial region of interest.

**Figure 9.**Average % error for an image with added Gaussian source noise for a single-lensed microlensing event with ${\mu}_{0}=0.1$ and ${\mu}_{0}=0.01$.

**Figure 10.**Graph of % error for an image with added Gaussian source noise for a single-lensed microlensing event with ${\mu}_{0}=0.01$. Binomial and Gaussian measurement matrices, with the given variance, are used for comparison.

**Figure 11.**Graph of % error for an image with added Gaussian source noise for a single-lensed microlensing event with ${\mu}_{0}=0.1$. Binomial and Gaussian measurement matricies, with the given variance, are used for comparison.

**Figure 12.**Average % error for an image with applied Poisson noise to CS measurements for a single-lensed microlensing event with ${\mu}_{0}=0.1$ using binomial measurement matrix with $p=0.5$ and $p=0.25$.

**Figure 13.**Average % error for an image with applied Poisson noise to CS measurements for a single-lensed microlensing event with ${\mu}_{0}=0.01$ using binomial measurement matrix with $p=0.5$ and $p=0.25$.

**Table 1.**Total noise variance and mutual coherence of A, $\mu \left(A\right)$, with the given properties of A.

Measurement Matrix, A | Total Noise Variance | Average $\mathit{\mu}\left(\mathit{A}\right)$ |
---|---|---|

Gaussian with ${\sigma}^{2}=0.25$ | $0.25\times {\sigma}_{n}^{2}$ | 0.616 |

Binomial with ${\sigma}^{2}=0.25$ | $0.25\times {\sigma}_{n}^{2}$ | 0.841 |

**Table 2.**Total noise variance and mutual coherence for A with the given properties of a binomial distribution.

Measurement Matrix, A | Expected Value of Total Noise Variance | Average $\mathit{\mu}\left(\mathit{A}\right)$ |
---|---|---|

Binomial with $p=0.5$ | $0.5E\left[x\right]$ | 0.841 |

Binomial with $p=0.25$ | $0.25E\left[x\right]$ | 0.789 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Korde-Patel, A.; Barry, R.K.; Mohsenin, T.
Application of Compressive Sensing in the Presence of Noise for Transient Photometric Events. *Signals* **2022**, *3*, 794-806.
https://doi.org/10.3390/signals3040047

**AMA Style**

Korde-Patel A, Barry RK, Mohsenin T.
Application of Compressive Sensing in the Presence of Noise for Transient Photometric Events. *Signals*. 2022; 3(4):794-806.
https://doi.org/10.3390/signals3040047

**Chicago/Turabian Style**

Korde-Patel, Asmita, Richard K. Barry, and Tinoosh Mohsenin.
2022. "Application of Compressive Sensing in the Presence of Noise for Transient Photometric Events" *Signals* 3, no. 4: 794-806.
https://doi.org/10.3390/signals3040047