# Sparsity-Based Recovery of Three-Dimensional Photoacoustic Images from Compressed Single-Shot Optical Detection

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Optical Setup

#### 2.2. Continuous Model

#### 2.3. Discrete Model

#### 2.4. Image Reconstruction via Compressed Sensing

#### 2.5. Simulation Setup and Analysis

## 3. Results

#### 3.1. Baseline Test

#### 3.2. Simulated Experiments

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PA | photoacoustic |

FPE | Fabry–Pérot etalon |

DMD | digital micro-mirror device |

IPD | initial pressure distribution |

TwIST | Two-Step Iterative Shrinkage/Thresholding |

IST | iterative shrinkage/thresholding |

IRS | iterative reweighted shrinkage |

MSE | mean square error |

MS-SSIM | multi-scale structural similarity |

SNR | signal-to-noise ratio |

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**Figure 1.**The proposed optical system. SMF = single-mode fiber, OI = optical isolator, COL = collimator, PBS = polarized beam splitter, $\lambda $/4 = quarter wave plate, L = lens, CAM = camera, DMD = digital micromirror device, LP = linear polarizer, FPE = Fabry–Pérot etalon.

**Figure 3.**(

**a**) A cross-section of the ground truth impulse IPD and (

**b**–

**d**) cross-sections of a no-noise reconstruction of the impulse IPD. Here we use regularization parameter $\lambda =2.5\times {10}^{-4}$ in (6).

**Figure 5.**Average MS-SSIM (

**a**) and average MSE (

**b**) of a cylinder IPD with a radius of six perpendicular to the direction of shearing. Average MS-SSIM (

**c**) and average MSE (

**d**) of a cylinder IPD with a radius of 6 h parallel to the direction of shearing. Average MS-SSIM (

**e**) and average MSE (

**f**) of a vessel IPD with ten vessels present. Each point is averaged over five trials for each SNR value considered and is plotted against the SNR value used to calculate the additive Gaussian noise.

**Figure 6.**Average MSE (

**a**) and average MS-SSIM (

**b**) of the reconstructed cylinder IPD with varying radius, averaged over five trials for each radius value considered. A cross section of the ground truth cylinder with radius of five (

**c**), ten (

**d**) and fifteen (

**e**) voxel widths and a cross section of the reconstruction of the same cylinder, respectively, (

**f**–

**h**). Each cylinder considered is orthogonal to the $xz$-plane.

**Figure 7.**Average MSE (

**a**) and average MS-SSIM (

**b**) of the reconstructed vessel IPDs with a varying number of vessels present, averaged over twenty IPDs for each number of vessels considered. Ground truth projection onto the xy-plane of a four vessel IPD (

**c**), eight vessel IPD (

**d**), and twelve vessel IPD (

**e**), and the reconstruction of the same IPDs, respectively, (

**f**–

**h**). In images (

**c**–

**h**), hue represents depth in the z-dimension, with the colorbar indicating pixel lengths away from the FPE, while intensity is proportional the value of the voxels after being thresholded at 0.15.

**Figure 8.**Average MSE (

**a**) and average MS-SSIM (

**b**) of the reconstructed cylinder IPD as well as average MSE (

**c**) and average MS-SSIM (

**d**) of the reconstructed vessel-like IPD, averaged over five trials for each value of the regularization parameter considered. Low, medium, and high values of SNR are considered for comparison.

Parameter | Description | Value |
---|---|---|

N | length of IPD grid [pixels] | 64 |

${N}_{c}$ | length of computational grid [pixels] | 96 |

${N}_{L}$ | width of boundary condition layer [pixels] | 15 |

${c}_{s}$ | speed of sound in the medium [m/s] | 1540 |

h | pixel width [$\mu $m] | 122 |

f | center frequency [MHz] | 5 |

T | total time steps | 400 |

$\Delta t$ | time step size [ns] | 24 |

$\alpha $ | [pixels-widths/s] | 12 |

s | downsize factor | 4 |

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**MDPI and ACS Style**

Green, D.; Gelb, A.; Luke, G.P.
Sparsity-Based Recovery of Three-Dimensional Photoacoustic Images from Compressed Single-Shot Optical Detection. *J. Imaging* **2021**, *7*, 201.
https://doi.org/10.3390/jimaging7100201

**AMA Style**

Green D, Gelb A, Luke GP.
Sparsity-Based Recovery of Three-Dimensional Photoacoustic Images from Compressed Single-Shot Optical Detection. *Journal of Imaging*. 2021; 7(10):201.
https://doi.org/10.3390/jimaging7100201

**Chicago/Turabian Style**

Green, Dylan, Anne Gelb, and Geoffrey P. Luke.
2021. "Sparsity-Based Recovery of Three-Dimensional Photoacoustic Images from Compressed Single-Shot Optical Detection" *Journal of Imaging* 7, no. 10: 201.
https://doi.org/10.3390/jimaging7100201