# Fast and Accurate Refocusing for Moving Ships in SAR Imagery Based on FrFT

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- To avoid the computationally expensive problem caused by 2D peak search, this paper proposes to calculate the azimuth line’s optimal rotation order by searching for the minimum entropy using the advance and retreat method according to the variation law of the signal entropy with the rotation order in FrFT domain.
- In order to accelerate the calculation of the optimal rotation order for each azimuth line in the SAR image, this scheme proposes a fast refocusing approach and a fine refocusing approach according to the optimal rotation order distribution of each azimuth line on a linear moving ship. The fast refocusing approach only requires calculating the optimal rotation order of the best azimuth line in the SAR image. The fine refocusing approach further refines the optimal rotation order for other azimuth lines based on the optimal rotation order of the best azimuth line.
- Extensive experiments have been carried out on Gaofen-3 SAR images in different modes. In particular, the effectiveness on high resolution SAR images is verified. The experimental results validate that the proposed fast refocusing approach can achieve the fastest speed, while the proposed fine refocusing approach can achieve the best focusing performance compared to existing refocusing algorithms.

## 2. Review of the Signal Model and FrFT

## 3. Proposed Scheme

#### 3.1. Removing Sea Backgrounds

#### 3.2. Finding the Best Azimuth Line

#### 3.3. Calculating the Optimal Rotation Order

#### 3.4. Refocusing Module

Algorithm 1: Fast refocusing approach |

Input: Best azimuth line’s optimal rotation order ${a}_{best\_opt}$, azimuth line set $S$. |

Output: Refoucsed SAR image ${S}_{refocused}$ |

for $g(:,{n}_{k})$ in $S$ |

$g{(:,{n}_{k})}_{refocused}=FrFT(g(:,{n}_{k}),{a}_{best\_opt})$ |

end for |

${S}_{refocused}=[g{(:,{n}_{1})}_{refocused},g{(:,{n}_{2})}_{refocused},\cdots ,g{(:,{n}_{k})}_{refocused}]$ |

Algorithm 2: Fine refocusing approach |

Input: Best azimuth line’s optimal rotation order ${a}_{best\_opt}$, azimuth line set $S$. |

Output: Refoucsed SAR image ${S}_{refocused}$ |

for $g(:,{n}_{k})$ in $S$ |

${\alpha}_{opt}=MinimumEntropySearch(g(:,{n}_{k}),{a}_{best\_opt})$ |

$g{(:,{n}_{k})}_{refocused}=FrFT(g(:,{n}_{k}),{a}_{opt})$ |

end for |

${S}_{refocused}=[g{(:,{n}_{1})}_{refocused},g{(:,{n}_{2})}_{refocused},\cdots ,g{(:,{n}_{k})}_{refocused}]$ |

#### 3.5. Numerical Analysis of Computional Burden

## 4. Experiments

#### 4.1. Refocusing Experiments on Simulated Moving Point Target

^{2}, and azimuth acceleration of 15 m/s

^{2}are shown in Figure 4, respectively. It can be observed that the target azimuth velocity and range acceleration will cause the SAR imaging results to broaden in the azimuth direction, and the target azimuth acceleration will cause the SAR imaging results to be asymmetrically distorted in the azimuth direction.

#### 4.2. Refocusing Experiments on Moving Ships of Gaofen-3 UFS SAR Images

#### 4.3. Refocusing Experiments on Moving Ships of Gaofen-3 SL SAR Images

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Curlander, J.C.; McDonough, R.N. Synthetic Aperture Radar; Wiley: New York, NY, USA, 1991; Volume 11. [Google Scholar]
- Cumming, I.G.; Wong, F.H. Digital Processing of Synthetic Aperture Radar Data; Artech House: Boston, MA, USA, 2005; pp. 108–110. [Google Scholar]
- Kuang, G.; Gao, G.; Jiang, Y.; Lu, J.; Jia, C. Theory, Algorithm and Application for Target Detection in Synthetic Aperture Radar; Press of National University of Defense Technology: Changsha, China, 2007. [Google Scholar]
- Ouchi, K. Recent trend and advance of synthetic aperture radar with selected topics. Remote Sens.
**2013**, 5, 716–807. [Google Scholar] [CrossRef] [Green Version] - Zhu, X.X.; Wang, Y.; Montazeri, S.; Ge, N. A review of ten-year advances of multi-baseline SAR interferometry using TerraSAR-X data. Remote Sens.
**2018**, 10, 1374. [Google Scholar] [CrossRef] [Green Version] - Yamaguchi, Y. Disaster monitoring by fully polarimetric SAR data acquired with ALOS-PALSAR. Proc. IEEE
**2012**, 100, 2851–2860. [Google Scholar] [CrossRef] - Lopez-Sanchez, J.M.; Ballester-Berman, J.D. Potentials of polarimetric SAR interferometry for agriculture monitoring. Radio Sci.
**2009**, 44, 1–20. [Google Scholar] [CrossRef] [Green Version] - Pelich, R.; Longépé, N.; Mercier, G.; Hajduch, G.; Garello, R. AIS-based evaluation of target detectors and SAR sensors characteristics for maritime surveillance. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2014**, 8, 3892–3901. [Google Scholar] [CrossRef] [Green Version] - Leng, X.; Ji, K.; Kuang, G. Ship Detection from Raw SAR Echo Data. IEEE Trans. Geosci. Remote Sens.
**2023**, 61, 5207811. [Google Scholar] [CrossRef] - Zhan, R.; Cui, Z. Ship Recognition for SAR Scene Images under Imbalance Data. Remote Sens.
**2022**, 14, 6294. [Google Scholar] [CrossRef] - Zhang, L.; Leng, X.; Feng, S.; Ma, X.; Ji, K.; Kuang, G.; Liu, L. Azimuth-Aware Discriminative Representation Learning for Semi-Supervised Few-Shot SAR Vehicle Recognition. Remote Sens.
**2023**, 15, 331. [Google Scholar] [CrossRef] - Shao, Z.; Zhang, T.; Ke, X. A Dual-Polarization Information-Guided Network for SAR Ship Classification. Remote Sens.
**2023**, 15, 2138. [Google Scholar] [CrossRef] - Kang, M.; Leng, X.; Lin, Z.; Ji, K. A modified faster R-CNN based on CFAR algorithm for SAR ship detection. In Proceedings of the 2017 International Workshop on Remote Sensing with Intelligent Processing (RSIP), Shanghai, China, 18–21 May 2017; pp. 1–4. [Google Scholar]
- Lin, Z.; Ji, K.; Leng, X.; Kuang, G. Squeeze and excitation rank faster R-CNN for ship detection in SAR images. IEEE Geosci. Remote Sens. Lett.
**2018**, 16, 751–755. [Google Scholar] [CrossRef] - Sun, Z.; Dai, M.; Leng, X.; Lei, Y.; Xiong, B.; Ji, K.; Kuang, G. An Anchor-Free Detection Method for Ship Targets in High-Resolution SAR Images. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2021**, 14, 7788–7816. [Google Scholar] [CrossRef] - Sun, Z.; Leng, X.; Lei, Y.; Xiong, B.; Ji, K.; Kuang, G. BiFA-YOLO: A novel YOLO-based method for arbitrary-oriented ship detection in high-resolution SAR images. Remote Sens.
**2021**, 13, 4209. [Google Scholar] [CrossRef] - Leng, X.; Ji, K.; Xiong, B.; Kuang, G. Complex Signal Kurtosis—Indicator of Ship Target Signature in SAR Images. IEEE Trans. Geosci. Remote Sens.
**2022**, 13, 4209. [Google Scholar] [CrossRef] - Xiong, B.; Sun, Z.; Wang, J.; Leng, X.; Ji, K. A Lightweight Model for Ship Detection and Recognition in Complex-Scene SAR Images. Remote Sens.
**2022**, 14, 6053. [Google Scholar] [CrossRef] - Zhang, T.; Quan, S.; Yang, Z.; Guo, W.; Zhang, Z.; Gan, H. A two-stage method for ship detection using PolSAR image. IEEE Trans. Geosci. Remote Sens.
**2022**, 60, 5236918. [Google Scholar] [CrossRef] - Zhang, Y.; Lu, D.; Qiu, X.; Li, F. Scattering-Point-Guided RPN for Oriented Ship Detection in SAR Images. Remote Sens.
**2023**, 15, 1411. [Google Scholar] [CrossRef] - Yang, Q.; Li, Z.; Li, J.; An, H.; Wu, J.; Pi, Y.; Yang, J. A Novel Bistatic SAR Maritime Ship Target Imaging Algorithm Based on Cubic Phase Time-Scaled Transformation. Remote Sens.
**2023**, 15, 1330. [Google Scholar] [CrossRef] - Chen, V.C.; Li, F.; Ho, S.S.; Wechsler, H. Micro-Doppler effect in radar: Phenomenon, model, and simulation study. IEEE Trans. Aerosp. Electron. Syst.
**2006**, 42, 2–21. [Google Scholar] [CrossRef] - Li, X.; Deng, B.; Qin, Y.; Wang, H.; Li, Y. The influence of target micromotion on SAR and GMTI. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 2738–2751. [Google Scholar] [CrossRef] - Raney, R.K. Synthetic Aperture Imaging Radar and Moving Targets. IEEE Trans. Aerosp. Electron. Syst.
**1971**, 7, 499–505. [Google Scholar] [CrossRef] - Sharma, J.J.; Gierull, C.H.; Collins, M.J. Compensating the effects of target acceleration in dual-channel SAR–GMTI. IEE Proc.-Radar Sonar Navig.
**2006**, 153, 53–62. [Google Scholar] [CrossRef] - Ruegg, M.; Meier, E.; Nuesch, D. Capabilities of dual-frequency millimeter wave SAR with monopulse processing for ground moving target indication. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 539–553. [Google Scholar] [CrossRef] - Bethke, K.H.; Baumgartner, S.; Gabele, M.; Hounam, D.; Kemptner, E.; Klement, D.; Krieger, G.; Erxleben, R. Air and spaceborne monitoring of road traffic using SAR moving target indica-tion—Project TRAMRAD. ISPRS J. Photogramm. Remote Sens.
**2006**, 61, 243–259. [Google Scholar] [CrossRef] - Baumgartner, S.; Gabele, M.; Krieger, G.; Bethke, K.-H. Traffic monitoring with SAR: Implications of target acceleration. In Proceedings of the European Conference on Synthetic Aperture Radar (EUSAR), Dresden, Germany, 16–18 May 2006; VDE Verlag GmbH: Berlin, Germany, 2006; pp. 1–4. [Google Scholar]
- Djurović, I.; Ioana, C.; Thayaparan, T.; Stanković, L.; Wang, P.; Popović, V.; Simeunović, M. Cubic-phase function evaluation for multicomponent signals with application to SAR imaging. IET Signal Process.
**2010**, 4, 371–381. [Google Scholar] [CrossRef] [Green Version] - Liu, P.; Jin, Y.Q. A study of ship rotation effects on SAR image. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 3132–3144. [Google Scholar] [CrossRef] - Zhou, B.; Qi, X.; Zhang, J.; Zhang, H. Effect of 6-DOF Oscillation of Ship Target on SAR Imaging. Remote Sens.
**2021**, 13, 1821. [Google Scholar] [CrossRef] - Huang, L.M.; Duan, W.Y.; Han, Y.; Chen, Y.S. A review of short-term prediction techniques for ship motions in seaway. J. Ship Mech.
**2014**, 18, 1534–1542. [Google Scholar] - Doerry, A.W. Ship Dynamics for Maritime ISAR Imaging; Sandia National Laboratories (SNL): Albuquerque, NM, USA; Livermore, CA, USA, 2008. [Google Scholar]
- Chen, X.L.; Dong, Y.L.; Li, X.Y.; Guan, J. Modeling of Micromotion and A-nalysis of Properties of Rigid Marine Targets. J. Radars
**2015**, 4, 630–638. [Google Scholar] [CrossRef] - Chen, J.; Xing, M.; Yu, H.; Liang, B.; Peng, J.; Sun, G.-C. Motion compensation/autofocus in airborne synthetic aperture radar: A review. IEEE Geosci. Remote Sens. Mag.
**2021**, 10, 185–206. [Google Scholar] [CrossRef] - Chen, J.; Yu, H. Wide-beam SAR autofocus based on blind resampling. Sci. China Inf. Sci.
**2023**, 66, 140304. [Google Scholar] [CrossRef] - Bao, Z.; Ye, W. Improvements of Autofocusing Technique for ISAR Motion Compensation. Acta Electron. Sin.
**1996**, 24, 74–79. [Google Scholar] - Itoh, T.; Sueda, H.; Watanabe, Y. Motion compensation for ISAR via centroid tracking. IEEE Trans. Aerosp. Electron. Syst.
**1996**, 32, 1191–1197. [Google Scholar] [CrossRef] - Zhu, Z.; Qiu, X.; She, Z. ISAR motion compensation using modified Doppler centroid tracking method. In Proceedings of the IEEE 1996 National Aerospace and Electronics Conference NAECON 1996, Dayton, OH, USA, 20–22 May 1996; Volume 1, pp. 359–363. [Google Scholar]
- Wahl, D.E.; Eichel, P.H.; Ghiglia, D.C.; Jakowatz, C. Phase gradient autofocus—A robust tool for high resolution SAR phase correction. IEEE Trans. Aerosp. Electron. Syst.
**1994**, 30, 827–835. [Google Scholar] [CrossRef] [Green Version] - Zhu, D.; Jiang, R.; Mao, X.; Zhu, Z. Multi-subaperture PGA for SAR autofocusing. IEEE Trans. Aerosp. Electron. Syst.
**2013**, 49, 468–488. [Google Scholar] [CrossRef] - Morrison, R.L.; Do, M.N.; Munson, D.C. SAR image autofocus by sharpness optimization: A theoretical study. IEEE Trans. Image Process.
**2007**, 16, 2309–2321. [Google Scholar] [CrossRef] [Green Version] - Zeng, T.; Wang, R.; Li, F. SAR image autofocus utilizing minimum-entropy criterion. IEEE Geosci. Remote Sens. Lett.
**2013**, 10, 1552–1556. [Google Scholar] [CrossRef] - Schulz, T.J. Optimal sharpness function for SAR autofocus. IEEE Signal Process. Lett.
**2006**, 14, 27–30. [Google Scholar] [CrossRef] - Martorella, M.; Berizzi, F.; Haywood, B. Contrast maximisation based technique for 2-D ISAR autofocusing. IEE Proc.-Radar Sonar Navig.
**2005**, 152, 253–262. [Google Scholar] [CrossRef] [Green Version] - Huang, X.; Ji, K.; Leng, X.; Dong, G.; Xing, X. Refocusing moving ship targets in SAR images based on fast minimum entropy phase compensation. Sensors
**2019**, 19, 1154. [Google Scholar] [CrossRef] [Green Version] - Ozaktas, H.M.; Arikan, O.; Kutay, M.A.; Bozdagt, G. Digital computation of the fractional Fourier transform. IEEE Trans. Signal Process.
**1996**, 44, 2141–2150. [Google Scholar] [CrossRef] [Green Version] - Sejdić, E.; Djurović, I.; Stanković, L.J. Fractional Fourier transform as a signal processing tool: An overview of recent developments. Signal Process.
**2011**, 91, 1351–1369. [Google Scholar] [CrossRef] [Green Version] - Sun, H.B.; Liu, G.S.; Gu, H.; Su, W.-M. Application of the fractional Fourier transform to moving target detection in airborne SAR. IEEE Trans. Aerosp. Electron. Syst.
**2002**, 38, 1416–1424. [Google Scholar] - Amein, A.S.; Soraghan, J.J. The fractional Fourier transform and its application to high resolution SAR imaging. In Proceedings of the 2007 IEEE International Geoscience and Remote Sensing Symposium, Barcelona, Spain, 23–28 July 2007; pp. 5174–5177. [Google Scholar]
- Li, Z.; Zhang, X.; Yang, Q.; Xiao, Y.; An, H.; Yang, H.; Wu, J.; Ya, J. Hybrid SAR-ISAR image formation via joint FrFT-WVD processing for BFSAR ship target high-resolution imaging. IEEE Trans. Geosci. Remote Sens.
**2021**, 60, 5215713. [Google Scholar] [CrossRef] - Pelich, R.; Longépé, N.; Mercier, G.; Hajduch, G.; Garello, R. Vessel refocusing and velocity estimation on SAR imagery using the fractional Fourier transform. IEEE Trans. Geosci. Remote Sens.
**2015**, 54, 1670–1684. [Google Scholar] [CrossRef] - Wang, J.; Leng, X.; Sun, Z.; Zhang, X.; Ji, K. Refocusing Swing Ships in SAR Imagery Based on Spatial-Variant Defocusing Property. Remote Sens.
**2023**, 15, 3159. [Google Scholar] [CrossRef]

**Figure 4.**Azimuth profiles of SAR imaging results for different moving states of a point target. (

**a**) Azimuth velocity of 20 m/s. (

**b**) Range acceleration of 10 m/s

^{2}. (

**c**) Azimuth acceleration of 15 m/s

^{2}.

**Figure 5.**Time–frequency distribution of signals in different motion states. (

**a**) Azimuth velocity of 20 m/s. (

**b**) Range acceleration of 10 m/s

^{2}. (

**c**) Azimuth acceleration of 15 m/s

^{2}.

**Figure 6.**Distribution of signals with different motion states in the FrFT domain. (

**a**) Azimuth velocity of 20 m/s. (

**b**) Range acceleration of 10 m/s

^{2}. (

**c**) Azimuth acceleration of 15 m/s

^{2}.

**Figure 7.**The results of the defocused signals with different motion states after FrFT at optimal rotation order. (

**a**) Azimuth velocity of 20 m/s. (

**b**) Range acceleration of 10 m/s

^{2}. (

**c**) Azimuth acceleration of 15 m/s

^{2}.

**Figure 13.**Distribution of optimal rotation order for ship’s each azimuth line. (

**a**) Ship1 (

**b**) Ship2.

**Figure 14.**Distribution in FrFT domain of non-dominant scattering point range cells. (

**a**) Ship1 (

**b**) Ship2.

**Figure 16.**The optimal rotation order of each range cell on the ship obtained by different methods. (

**a**) Ship1. (

**b**) Ship2.

**Figure 17.**The relationship between the peak and entropy of the signal after FrFT and the rotation order. (

**a**) Ship1. (

**b**) Ship2.

**Figure 19.**The refocused image of ship1 by different methods. (

**a**) Origin image. (

**b**) PGA. (

**c**) FMEPC. (

**d**) Pelich’s method. (

**e**) Proposed fast refocusing approach. (

**f**) Proposed fine refocusing approach.

**Figure 20.**The refocused image of ship2 by different methods. (

**a**) Origin image. (

**b**) PGA. (

**c**) FMEPC. (

**d**) Pelich’s method. (

**e**) Proposed fast refocusing approach. (

**f**) Proposed fine refocusing approach.

**Figure 22.**Time–frequency distribution of each ship’s best azimuth line after STFT. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 23.**Distribution of each ship’s best azimuth line in the FrFT domain. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 24.**The result of each best azimuth line after FrFT at the optimal rotation order. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 25.**Distribution of the optimal rotation order for each range cell. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 26.**Distributions of non-dominant scattering point range cells in FrFT domain. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3.

**Figure 27.**Image entropy at different range cell’s optimal rotation order. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 28.**The optimal rotation order of each range cell on the ship obtained by different methods. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 29.**Relationship between the entropy and the processing time. (

**a**) Ship1. (

**b**) Ship2. (

**c**) Ship3. (

**d**) Ship4. (

**e**) Ship5. (

**f**) Ship6.

**Figure 30.**The refocused image of Ship1–Ship6 obtained by different methods which are shown from the first row to the sixth row, respectively. (

**a**) Origin image. (

**b**) Refocused by PGA. (

**c**) Refocused by FMEPC. (

**d**) Refocused by Pelich’s method using FrFT. (

**e**) Refocused by the proposed fast refocusing approach. (

**f**) Refocused by the proposed fine refocusing approach.

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

Carrier Frequency | 3 GHZ |

Pulse Repetition Frequency | 188 HZ |

Band Width | 150 MHZ |

Platform Height | 3000 m |

Antenna Length | 2 m |

Platform Velocity | 150 m/s |

Pulse Width | 1.5 μs |

Proposed Algorithm | 2D Peak Search Method | |
---|---|---|

Uniform motion in azimuth | 12 | 60 |

Acceleration in range | 15 | 60 |

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

Carrier frequency (GHZ) | 5.4 |

Platform velocity (m/s) | 7568 |

Band width (MHZ) | 80 |

Pulse Width (μs) | 55 |

Pulse repetition frequency (Hz) | 2179 |

Original | PGA | FMEPC | Pelich’s Method | Fast Refocusing | Fine Refocusing | |
---|---|---|---|---|---|---|

Ship1 | 7.91 | 7.61 | 7.61 | 7.6 | 7.66 | 7.54 |

Ship2 | 5.88 | 5.09 | 5.08 | 5.05 | 5.1 | 5 |

PGA | FMEPC | Pelich’s Method | Fast Refocusing | Fine Refocusing | |
---|---|---|---|---|---|

Ship1 | 0.53 s | 3.14 s | 4.06 s | 0.085 s | 0.43 s |

Ship2 | 0.48 s | 1.16 s | 1.2 s | 0.03 s | 0.15 s |

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

Carrier frequency (GHZ) | 5.4 |

Platform velocity (m/s) | 7567 |

Band width (MHZ) | 240 |

Pulse Width (μs) | 45 |

Pulse repetition frequency (Hz) | 3738 |

Sub-Image | Ship1 | Ship2 | Ship3 | Ship4 | Ship5 | Ship6 | Mean |
---|---|---|---|---|---|---|---|

Original Image | 9.16 | 7.92 | 8.35 | 9.98 | 8.54 | 9.55 | 8.92 |

PGA | 7.51 | 6.72 | 6.7 | 8.58 | 7.15 | 8.1 | 7.46 |

FMEPC | 7.5 | 6.64 | 6.66 | 8.57 | 7.12 | 8.07 | 7.43 |

Pelich’s method using FrFT | 7.34 | 6.56 | 6.5 | 8.51 | 6.58 | 7.71 | 7.20 |

Proposed fast refocusing approach | 7.46 | 6.56 | 6.62 | 8.56 | 6.62 | 7.96 | 7.30 |

Proposedfine refocusing approach | 7.35 | 6.51 | 6.49 | 8.52 | 6.57 | 7.67 | 7.18 |

Sub-Image | Ship1 | Ship2 | Ship3 | Ship4 | Ship5 | Ship6 | Mean |
---|---|---|---|---|---|---|---|

PGA | 2.05 | 0.75 | 0.67 | 1.62 | 3.01 | 2.25 | 1.73 |

FMEPC | 4.46 | 1.23 | 1.49 | 16 | 10.18 | 11.65 | 7.50 |

Pelich’s method using FrFT | 8.7 | 2.6 | 3.78 | 9.08 | 7.05 | 4.85 | 6.01 |

Proposed fast refocusing approach | 0.17 | 0.06 | 0.11 | 0.2 | 0.15 | 0.1 | 0.13 |

Proposedfine refocusing approach | 0.9 | 0.39 | 0.48 | 0.85 | 0.63 | 0.57 | 0.64 |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Leng, X.; Sun, Z.; Zhang, X.; Ji, K.
Fast and Accurate Refocusing for Moving Ships in SAR Imagery Based on FrFT. *Remote Sens.* **2023**, *15*, 3656.
https://doi.org/10.3390/rs15143656

**AMA Style**

Wang J, Leng X, Sun Z, Zhang X, Ji K.
Fast and Accurate Refocusing for Moving Ships in SAR Imagery Based on FrFT. *Remote Sensing*. 2023; 15(14):3656.
https://doi.org/10.3390/rs15143656

**Chicago/Turabian Style**

Wang, Jin, Xiangguang Leng, Zhongzhen Sun, Xi Zhang, and Kefeng Ji.
2023. "Fast and Accurate Refocusing for Moving Ships in SAR Imagery Based on FrFT" *Remote Sensing* 15, no. 14: 3656.
https://doi.org/10.3390/rs15143656