# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Motion Degradation Model for Single-Pixel SFDI

_{th}measurement satisfies the phase-shift and frequency-shift properties, as,

_{0}= 0 for the DC image; and N is the pixel resolution of the image. Here, it is assumed that the target moves within the FOV during the entire imaging procedure. Because the Fourier spectrum of a natural image is conjugate symmetric, only the Fourier coefficients for $u\ge 0$ are required to measure. The coefficients for $u<0$ are obtained using the conjugate symmetry.

#### 2.2. Motion Estimation

#### 2.3. Sampling Strategy for Single-Pixel SFDI

## 3. Results

#### 3.1. Simulation Validations

^{−1}, as shown in Figure 3a. The pixel resolution of the image is 128 × 128. The target in piecewise uniform motion along X and Y directions and in circular motion with a constant angular acceleration are simulated, respectively. 500 sampling patterns are used to reconstruct the reflected image, with ${\mathrm{P}}_{x}$ and ${\mathrm{P}}_{y}$ measured every 10 samplings (K = 10) to estimate target displacements. A single frame is measured in the piecewise uniform motion simulation, while two complete frames are measured in the circular motion simulation. Then, 30 dB Gaussian noise is added to the simulated data. The structural similarity index measure (SSIM) is used to evaluate the effects of the method.

#### 3.2. Experimental Validations

^{2}. The spatial frequency of the sinusoidal illumination is 0.2 mm

^{−1}. The number of samplings for image reconstruction is 132, and the ${\mathrm{P}}_{x}$ and ${\mathrm{P}}_{y}$ are measured every 10 samplings (K = 10) for motion estimation. Then, 3-step phase-shifting FSI is adopted to acquire the complex-valued Fourier coefficients, resulting in a total of 474 measurements to obtain a deblurred image.

^{−1}and the reduced scattering coefficient (${{\mu}^{\prime}}_{s}$) of 1.2 mm

^{−1}. A cylindrical target with the diameter of 1 cm and the depth of 0.5 cm is in the middle of the phantom. The ${\mu}_{a}$ and ${{\mu}^{\prime}}_{s}$ of the target are 0.04 mm

^{−1}and 1.2 mm

^{−1}, respectively. In the experiment, the phantom moves in different speeds within the FOV. The speed is controlled by manually operating a linear translation stage. Since manual control is hard to ensure a constant speed, we use the average speed as the true value, which is calculated according to the total measurement time and the total displacement of the target. The average speeds of the phantom are approximately v = 0, 0.5, 1, 1.5, 2 and 2.5 mm/s, respectively. The maximum validated speed is 2.5 mm/s, as the higher speed will force the target to move out of the FOV.

## 4. Discussion

^{−1}, the performance of the method is greatly reduced. In practice, the typically used spatial frequency is between 0.1–0.2 mm

^{−1}, so the method is applicable to most scenarios.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cuccia, D.; Bevilacqua, J.F.; Durkin, A.J.; Tromberg, B.J. Modulated imaging: Quantitative analysis and tomography of turbid media in the spatial-frequency domain. Opt. Lett.
**2005**, 30, 1354–1356. [Google Scholar] [CrossRef] [PubMed] - Cuccia, D.; Bevilacqua, J.F.; Durkin, A.J.; Ayers, F.R.; Tromberg, B.J. Quantitation and mapping of tissue optical properties using modulated imaging. J. Biomed. Opt.
**2009**, 14, 024012. [Google Scholar] [CrossRef] [PubMed] - Giessen, M.V.; Angelo, J.P.; Gioux, S. Real-time, profile corrected single snapshot imaging of optical properties. Biomed. Opt. Express
**2015**, 6, 4051–4062. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ghijsen, M.; Choi, B.; Durkin, A.J.; Gioux, S.; Tromberg, B.J. Real-time simultaneous single snapshot of optical properties and blood flow using coherent spatial frequency domain imaging (cSFDI). Biomed. Opt. Express
**2016**, 7, 870–882. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Horan, S.T.; Gardner, A.R.; Saager, R.; Durkin, A.J.; Venugopalan, V. Recovery of layered tissue optical properties from spatial frequency-domain spectroscopy and a deterministic radiative transport solver. J. Biomed. Opt.
**2018**, 24, 071607. [Google Scholar] - Angelo, J.P.; Chen, S.J.; Ochoa, M.; Sunar, U.; Gioux, S.; Intes, X. Review of structured light in diffuse optical imaging. J. Biomed. Opt.
**2018**, 24, 071602. [Google Scholar] [CrossRef] [Green Version] - Wirth, D.; Sibai, M.; Olson, J.; Wilson, B.C.; Roberts, D.W.; Paulsen, K. Feasibility of using spatial frequency-domain imaging intraoperatively during tumor resection. J. Biomed. Opt.
**2018**, 24, 071608. [Google Scholar] [CrossRef] - Schmidt, M.; Aguenounon, E.; Nahas, A.; Torregrossa, M.; Tromberg, B.J.; Uhring, W.; Gioux, S. Real-time, wide-field, and quantitative oxygenation imaging using spatiotemporal modulation of light. J. Biomed. Opt.
**2019**, 24, 071610. [Google Scholar] [CrossRef] [Green Version] - Kennedy, G.T.; Stone, R., II; Kowalczewski, A.C.; Rowland, R.; Chen, J.H.; Baldado, M.L.; Ponticorvo, A.; Bernal, N.; Christy, R.J.; Durkin, A.J. Spatial frequency domain imaging: A quantitative, noninvasive tool for in vivo monitoring of burn wound and skin graft healing. J. Biomed. Opt.
**2019**, 24, 071615. [Google Scholar] [CrossRef] - Gioux, S.; Mazhar, A.; Cuccia, D.J. Spatial frequency domain imaging in 2019: Principles, applications and perspectives. J. Biomed. Opt.
**2019**, 24, 071613. [Google Scholar] [CrossRef] [Green Version] - Wirth, D.J.; Sibai, M.; Wilson, B.C.; Roberts, D.W.; Paulsen, K. First experience with spatial frequency domain imaging and red-light excitation of protoporphyrin IX fluorescence during tumor resection. Biomed. Opt. Express
**2020**, 11, 4306–4315. [Google Scholar] [CrossRef] [PubMed] - Bounds, A.D.; Girkin, J.M. Early stage dental caries detection using near infrared spatial frequency domain imaging. Sci. Rep.
**2021**, 11, 2433. [Google Scholar] [CrossRef] [PubMed] - Duarte, M.F.; Davenport, M.A.; Takhar, D.; Laska, J.N.; Sun, T.; Kelly, K.F.; Baraniuk, R.G. Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag.
**2008**, 25, 83–91. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.B.; Ma, X.; Zhong, J.G. Single-pixel imaging by means of Fourier spectrum acquisition. Nat. Commun.
**2015**, 6, 6225. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Torabzadeh, M.; Park, Y.; Bartels, R.A.; Durkin, A.J.; Tromberg, B.J. Compressed single pixel imaging in the spatial frequency domain. J. Biomed. Opt.
**2017**, 22, 030501. [Google Scholar] [CrossRef] [Green Version] - Peller, J.; Farahi, F.; Trammell, S.R. Hyperspectral imaging system based on a single-pixel camera design for detecting differences in tissue properties. Appl. Opt.
**2018**, 57, 7651–7658. [Google Scholar] [CrossRef] - Li, T.X.; Qin, Z.P.; Hou, X.; Dan, M.; Li, J.; Zhang, L.M.; Zhou, Z.X.; Gao, F. Multi-wavelength spatial frequency domain diffuse optical tomography using single-pixel imaging based on lock-in photon counting. Opt. Express
**2019**, 27, 23138–23156. [Google Scholar] [CrossRef] - Aguénounon, E.; Dadouche, F.; Uhring, W.; Ducros, N.; Gioux, S. Single snapshot imaging of optical properties using a single-pixel camera: A simulation study. J. Biomed. Opt.
**2019**, 24, 071612. [Google Scholar] [CrossRef] - Mellors, B.O.L.; Bentley, A.; Spear, A.M.; Howle, C.R.; Dehghani, H. Applications of compressive sensing in spatial frequency domain imaging. J. Biomed. Opt.
**2020**, 25, 112904. [Google Scholar] [CrossRef] - Lenz, A.J.M.; Clemente, P.; Climent, V.; Lancis, J.; Tajahuerce, E. Single-pixel spatial frequency domain imaging with integrating detection. In Proceedings of the Diffuse Optical Spectroscopy and Imaging VIII (SPIE Conference 11920), Online, 20–24 June 2021; Volume 11920. [Google Scholar] [CrossRef]
- Xu, Z.H.; Chen, W.; Penuelas, J.; Padgett, M.; Sun, M.J. 1000 fps computational ghost imaging using LED-based structured illumination. Opt. Express
**2018**, 26, 2427–2434. [Google Scholar] [CrossRef] [Green Version] - Huynh, N.; Zhang, E.; Betcke, M.; Arridge, S.; Beard, P.; Cox, B. Single-pixel optical camera for video rate ultrasonic imaging. Optica
**2016**, 3, 26–29. [Google Scholar] [CrossRef] - Higham, C.F.; Murray-Smith, R.; Padgett, M.J.; Edgar, M.P. Deep learning for real-time single-pixel video. Sci. Rep.
**2018**, 8, 2369. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jiao, S.M.; Sun, M.J.; Gao, Y.; Lei, T.; Xie, Z.W.; Yuan, X.C. Motion estimation and quality enhancement for a single image in dynamic single-pixel imaging. Opt. Express
**2019**, 27, 12841–12854. [Google Scholar] [CrossRef] [PubMed] - Li, E.R.; Bo, Z.W.; Chen, M.L.; Gong, W.L.; Han, S.S. Ghost imaging of a moving target with an unknown constant speed. Appl. Phys. Lett.
**2014**, 104, 251120. [Google Scholar] [CrossRef] - Li, X.H.; Deng, C.J.; Chen, M.L.; Gong, W.L.; Han, S.S. Ghost imaging for an axially moving target with an unknown constant speed. Photonics Res.
**2015**, 3, 153–157. [Google Scholar] [CrossRef] [Green Version] - Sun, S.; Gu, J.H.; Lin, H.Z.; Jiang, L.; Liu, W.T. Gradual ghost imaging of moving objects by tracking based on cross correlation. Opt. Lett.
**2019**, 44, 5594–5597. [Google Scholar] [CrossRef] - Yang, D.Y.; Chang, C.; Wu, G.H.; Luo, B.; Yin, L.F. Compressive ghost imaging of the moving object using the low-order moments. Appl. Sci.
**2020**, 10, 7941. [Google Scholar] [CrossRef] - Jiang, W.J.; Li, X.Y.; Peng, X.L.; Sun, B.Q. Imaging high-speed moving targets with a single-pixel detector. Opt. Express
**2020**, 28, 7889–7897. [Google Scholar] [CrossRef] - Wu, J.J.; Hu, L.F.; Wang, J.C. Fast tracking and imaging of a moving object with single-pixel imaging. Opt. Express
**2021**, 29, 42589–42598. [Google Scholar] [CrossRef] - Vervandier, J.; Gioux, S. Single snapshot imaging of optical properties. Biomed. Opt. Express
**2013**, 4, 2938–2944. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.B.; Ye, J.Q.; Deng, Q.W.; Zhong, J.G. Image-free real-time detection and tracking of fast-moving object using a single-pixel detector. Opt. Express
**2019**, 27, 35394–35401. [Google Scholar] [CrossRef] [PubMed] - Deng, Q.W.; Zhang, Z.B.; Zhong, J.G. Image-free real-time 3-D tracking of a fast-moving object using dual-pixel detection. Opt. Lett.
**2020**, 45, 4734–4737. [Google Scholar] [CrossRef] - Zhang, J.Y.; Hu, T.Y.; Shao, X.L.; Xiao, M.X.; Rong, Y.J.; Xiao, Z.L. Multi-target tracking using windowed Fourier single-pixel imaging. Sensors
**2021**, 21, 7934. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.B.; Wang, X.Y.; Zheng, G.A.; Zhong, J.G. Hadamard single-pixel imaging versus Fourier single-pixel imaging. Opt. Express
**2017**, 25, 19619–19639. [Google Scholar] [CrossRef] - Zhang, Z.B.; Wang, X.Y.; Zheng, G.A.; Zhong, J.G. Fast Fourier single-pixel imaging via binary illumination. Sci. Rep.
**2017**, 7, 12029. [Google Scholar] [CrossRef] [PubMed] - Saxena, M.; Eluru, G.; Gorthi, S.S. Structured illumination microscopy. Adv. Opt. Photonics
**2015**, 7, 241–275. [Google Scholar] [CrossRef] - Van der Jeught, S.; Dirckx, J. Real-time structured light profilometry: A review. Opt. Lasers Eng.
**2016**, 87, 18–31. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the sampling patterns for motion estimation and dynamic reconstruction.

**Figure 2.**Sampling strategy for single-pixel SFDI. (

**a**) Normalized Fourier spectrum of a reflected image, (

**b**) modified circular path (white curves) and measuring priority of sampling patterns.

**Figure 3.**Simulation results of the target in piecewise uniform motion. (

**a**) True target images. Red arrows show the moving path and directions of the target. Results obtained using (

**b**) conventional single-pixel SFDI and (

**c**) the proposed method. Target displacements in (

**d**) X and (

**e**) Y directions, calculated from motion estimation.

**Figure 4.**Simulation results of the target in circular motion. (

**a**) True target images. Red line and arrows show the moving path and directions of the target. Results (Frame #30) obtained using (

**b**) conventional single-pixel SFDI and (

**c**) the proposed method. Target displacements in (

**d**) X and (

**e**) Y directions and (

**f**) angular velocities, calculated from motion estimation. The red windows indicate the measurements used for reconstructing Frame #30.

**Figure 5.**Experimental results of moving phantom. Absorption and scattering property maps of (

**a**) the static phantom and (

**b**) the phantom at v = 2 mm/s obtained using the conventional single-pixel SFDI (top) and the proposed method (bottom), respectively. (

**c**) Absorption maps of the phantom at different speeds. The dashed circles and white arrow indicate the true location and moving direction of the target, respectively.

Speed (mm/s) | RMSE (%) | SSIM | ||
---|---|---|---|---|

Conventional | Proposed | Conventional | Proposed | |

v = 0.5 | 0.19 | 0.12 | 0.9960 | 0.9984 |

v = 1.0 | 0.27 | 0.13 | 0.9926 | 0.9982 |

v = 1.5 | 0.31 | 0.14 | 0.9902 | 0.9979 |

v = 2.0 | 0.34 | 0.15 | 0.9878 | 0.9975 |

v = 2.5 | 0.38 | 0.15 | 0.9846 | 0.9975 |

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**MDPI and ACS Style**

Dan, M.; Liu, M.; Gao, F.
Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging. *Appl. Sci.* **2022**, *12*, 7402.
https://doi.org/10.3390/app12157402

**AMA Style**

Dan M, Liu M, Gao F.
Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging. *Applied Sciences*. 2022; 12(15):7402.
https://doi.org/10.3390/app12157402

**Chicago/Turabian Style**

Dan, Mai, Meihui Liu, and Feng Gao.
2022. "Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging" *Applied Sciences* 12, no. 15: 7402.
https://doi.org/10.3390/app12157402