# ISAR Imaging for Maneuvering Targets with Complex Motion Based on Generalized Radon-Fourier Transform and Gradient-Based Descent under Low SNR

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ISAR Imaging for Ship Targets

#### 2.1. Signal Model for Ship Targets

#### 2.2. Signal Analysis for Targets with Complex Motion

#### 2.3. Proposed Approach Description

#### 2.3.1. Coarse Parameters Estimation with GRFT

#### 2.3.2. Fine Parameters Estimation with Gradient-Based Optimal

#### Image Entropy Combined with Subarray Averaging Operation

#### Parameters Estimation Based on Gradient Descent Method

- Set the initial parameter ${x}^{0}$, initial matrix ${H}_{0}$, and the precision of error $\epsilon $ as $\left(\widehat{\alpha},\widehat{\beta}\right)$, unit matrix, and $1{\mathrm{e}}^{-5}\ge 0$, respectively.
- Calculate the gradient $\nabla E\left({x}^{0}\right)$. If $\Vert \nabla E\left({x}^{0}\right)\Vert <\epsilon $, then stop the calculation and the optimal parameter is ${x}^{*}={x}^{0}$. Otherwise, conduct the next step.
- Set ${p}_{0}=-{H}_{0}\cdot \nabla E\left({x}^{0}\right),\text{}k\leftarrow 0$, and conduct the next step.
- Perform a one-dimensional search to obtain ${t}_{k}$ such that $E\left({x}^{k}+{t}_{k}\cdot {p}_{k}\right)=\mathrm{min}E\left({x}^{k}+t\cdot {p}_{k}\right)$ is satisfied. Set ${x}^{k+1}={x}^{k}+t\cdot {p}_{k}$ and conduct the next step.
- Calculate $\nabla E\left({x}^{k+1}\right)$, if $\Vert \nabla E\left({x}^{k+1}\right)\Vert \le \epsilon $, then stop the calculation, and set the optimal parameter as ${x}^{*}={x}^{k+1}$. Otherwise, conduct the next step.
- If $k+1=n$, then ${x}^{0}={x}^{n}$ and conduct Step 3. Otherwise, conduct the next step.
- Calculate$$\begin{array}{ll}{H}_{k+1}& ={H}_{k}+\frac{\u2206{x}_{k}\xb7\u2206{x}_{k}^{T}}{\u2206{x}_{k}^{T}\cdot \u2206{g}_{k}}\left(1+\frac{\u2206{g}_{k}^{T}\cdot {H}_{k}\cdot \u2206{g}_{k}}{\u2206{x}_{k}^{T}\cdot \u2206{g}_{k}}\right)\\ & \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-\frac{1}{\u2206{x}_{k}^{T}\cdot \u2206{g}_{k}}\left(\u2206{x}_{k}\cdot \u2206{g}_{k}^{T}\cdot {H}_{k}+{H}_{k}\cdot \u2206{g}_{k}\cdot \u2206{x}_{k}^{T}\right)\end{array}$$$${p}_{k+1}=-{H}_{k+1}\cdot \nabla E\left({x}^{k+1}\right)$$

- Obtain the raw echoes, and conduct preprocessing part, e.g., range alignment, phase adjustment, and RCM correction.
- Coarsely search the range of true parameters via detecting the coherent peak with GRFT.
- Finely estimate the optimal parameters by using the gradient descent method.
- Finishing the compensation of 2D spatial-variant phase errors and obtain the well-focused ISAR image.

#### 2.4. Some Considerations for the Proposed Method in Applications

#### 2.4.1. Computational Complexity Analysis

#### 2.4.2. Doppler Frequency Spectrum Analysis

#### 2.4.3. Sampling Rate and PRF

#### 2.4.4. Phase Error Analysis

## 3. Experimental Results and Analysis

#### 3.1. Simulation Results

#### 3.2. Imaging Result under Different SNRs

#### 3.3. Electromagnetic Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Fu, J.; Xing, M.; Sun, G. Time-Frequency Reversion-Based Spectrum Analysis Method and Its Applications in Radar Imaging. Remote Sens.
**2021**, 13, 600. [Google Scholar] [CrossRef] - Prickect, M.J.; Chen, C.C. Principle of inverse synthetic aperture radar (ISAR) imaging. In Proceedings of the EASCON’80; Electronics and Aerospace Systems Conference, Arlington, VA, USA, 29 September–1 October 1980; pp. 340–345. [Google Scholar]
- Tan, X.; Yang, Z.; Li, D.; Liu, H.; Liao, G.; Wu, Y.; Liu, Y. An Efficient Range-Doppler Domain ISAR Imaging Approach for Rapidly Spinning Targets. IEEE Trans. Geosci. Remote Sens.
**2020**, 58, 2670–2681. [Google Scholar] [CrossRef] - Li, D.; Zhan, M.; Liu, H.; Liao, Y.; Liao, G. A Robust Translational Motion Compensation Method for ISAR Imaging Based on Keystone Transform and Fractional Fourier Transform Under Low SNR Environment. IEEE Trans. Aerosp. Electron. Syst.
**2017**, 53, 2140–2156. [Google Scholar] [CrossRef] - Berizzi, F.; Diani, M. ISAR imaging of rolling, pitching and yawing targets. In Proceedings of the International Radar Conference, Edinburgh, UK, 15–17 October 2002; pp. 346–349. [Google Scholar]
- Liu, Z.; Jiang, Y. A Novel Doppler Frequency Model and Imaging Procedure Analysis for Bistatic ISAR Configuration with Shorebase Transmitter and Shipborne Receiver. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2016**, 9, 2989–3000. [Google Scholar] [CrossRef] - Li, D.; Zhan, M.; Su, J.; Liu, H.; Liao, G. Performances analysis of coherently integrated cubic phase function for LFM signal and its application to ground moving target imaging. IEEE Trans. Geosci. Remote Sens.
**2017**, 11, 6402–6419. [Google Scholar] [CrossRef] - O’Shea, P.J. A new technique for instantaneous frequency rate estimation. IEEE Signal Process. Lett.
**2002**, 9, 251–252. [Google Scholar] [CrossRef] [Green Version] - O’Shea, P.J. A Fast Algorithm for Estimating the Parameters of a Quadratic FM Signal. IEEE Trans. Signal Process.
**2004**, 52, 385–393. [Google Scholar] [CrossRef] [Green Version] - Simeunovi´c, M.; Djurovi´c, I. Non-uniform sampled cubic phase function. Signal Process.
**2014**, 101, 99–103. [Google Scholar] [CrossRef] - Wang, Y.; Lin, Y. ISAR Imaging of Non-Uniformly Rotating Target via Range-Instantaneous-Doppler-Derivatives Algorithm. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2013**, 7, 167–176. [Google Scholar] [CrossRef] - Bai, X.; Tao, R.; Wang, Z.; Wang, Y. ISAR Imaging of a Ship Target Based on Parameter Estimation of Multicomponent Quadratic Frequency-Modulated Signals. IEEE Trans. Geosci. Remote Sens.
**2013**, 52, 1418–1429. [Google Scholar] [CrossRef] - Zheng, J.; Liu, H.; Liao, G.; Su, T.; Liu, Z.; Liu, Q.H. ISAR Imaging of Nonuniformly Rotating Targets Based on Generalized Decoupling Technique. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2016**, 9, 520–532. [Google Scholar] [CrossRef] - Chen, V.; Ling, H. Time-Frequency Transform for Radar Imaging and Signal Analysis; Artech House: Boston, MA, USA, 2002. [Google Scholar]
- Peleg, S.; Friedlander, B. The discrete polynomial-phase transform. IEEE Trans. Signal Process.
**1995**, 43, 1901–1914. [Google Scholar] [CrossRef] - Cohen, L. Time-Frequency Analysis; Englewood Cliffs, Prentice-Hall: Bergen, NJ, USA, 1996. [Google Scholar]
- Kim, K.-T.; Choi, I.-S.; Kim, H.-T. Efficient radar target classification using adaptive joint time-frequency processing. IEEE Trans. Antennas Propag.
**2000**, 48, 1789–1801. [Google Scholar] [CrossRef] [Green Version] - Berizzi, F.; Corsini, G. Autofocusing of inverse synthetic aperture radar images using contrast optimization. IEEE Trans. Aerosp. Electron. Syst.
**1996**, 32, 1185–1191. [Google Scholar] [CrossRef] - Eichel, P.H.; Ghiglia, D.C.; Jakowatz, C.V. Speckle processing method for synthetic-aperture-radar phase correction. Opt. Lett.
**1989**, 14, 1–3. [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, Dayton, OH, USA, 20–22 May 1966. [Google Scholar]
- Ding, Z.; Zhang, T.; Li, Y.; Li, G.; Dong, X.; Zeng, T.; Ke, M. A Ship ISAR Imaging Algorithm Based on Generalized Radon-Fourier Transform with Low SNR. IEEE Trans. Geosci. Remote Sens.
**2019**, 57, 6385–6396. [Google Scholar] [CrossRef] - Martorella, M.; Berizzi, F. Time windowing for highly focused ISAR image reconstruction. IEEE Trans. Aerosp. Electron. Syst.
**2005**, 41, 992–1007. [Google Scholar] [CrossRef] - Zhou, P.; Zhang, X.; Dai, Y.S.; Sun, W.F.; Wan, Y. Time window selection algorithm for ISAR ship imaging based on in-stantaneous Doppler frequency estimation of multiple scatterers. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2019**, 10, 3799–3812. [Google Scholar] [CrossRef] - Kang, B.-S.; Ryu, B.-H.; Kim, K.-T. Efficient Determination of Frame Time and Length for ISAR Imaging of Targets in Complex 3-D Motion Using Phase Nonlinearity and Discrete Polynomial Phase Transform. IEEE Sensors J.
**2018**, 18, 5739–5752. [Google Scholar] [CrossRef] - Zhang, S.; Sun, S.; Zhang, W.; Zong, Z.; Yeo, T.S. High-Resolution Bistatic ISAR Image Formation for High-Speed and Complex-Motion Targets. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2015**, 8, 1–12. [Google Scholar] [CrossRef] - Chen, V.C.; Miceli, W.J. Time-varying spectral analysis for radar imaging of maneuvering targets. IEEE Proc.-Radar Sonar Navigat.
**1998**, 145, 262–268. [Google Scholar] [CrossRef] - Li, J.; Ling, H.; Chen, V. An Algorithm to Detect the Presence of 3D Target Motion from ISAR Data. Multidimens. Syst. Signal Process.
**2003**, 14, 223–240. [Google Scholar] [CrossRef] - Feng, C.Q.; Tong, N.N.; Huang, D.R.; Guo, Y.D. 2D spatial-variant phase errors compensation for ISAR imagery based on contrast maximization. Electron. Lett.
**2016**, 52, 1480–1482. [Google Scholar] - Xu, J.; Yu, J.; Peng, Y.-N.; Xia, X.-G. Radon-Fourier Transform for Radar Target Detection, I: Generalized Doppler Filter Bank. IEEE Trans. Aerosp. Electron. Syst.
**2011**, 47, 1186–1202. [Google Scholar] [CrossRef] - Zhang, S.; Liu, Y.; Li, X. Fast Entropy Minimization Based Autofocusing Technique for ISAR Imaging. IEEE Trans. Signal Process.
**2015**, 63, 3425–3434. [Google Scholar] [CrossRef] - Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in Fortran: The Art of Scientific Computing. Math. Comput.
**1994**, 62, 433. [Google Scholar] [CrossRef] - Kim, K.T. Reconstruction of high range resolution profiles of moving targets using stepped frequency waveforms. IET Radar Sonar Navig.
**2010**, 4, 564–575. [Google Scholar] [CrossRef] - Jeong, H.-R.; Kim, H.-T.; Kim, K.-T. Application of Subarray Averaging and Entropy Minimization Algorithm to Stepped-Frequency ISAR Autofocus. IEEE Trans. Antennas Propag.
**2008**, 56, 1144–1154. [Google Scholar] [CrossRef] - Mokhtari, A.; Ribeiro, A. Stochastic Quasi-Newton Methods. Proc. IEEE
**2020**, 108, 1906–1922. [Google Scholar] [CrossRef] - Nocedal, J.; Wright, S.J. Numerical Optimization; Springer Science and Business Media LLC: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Marston, T.M.; Plotnick, D.S. Semiparametric statistical strip-map synthetic aperture autofocusing. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 2086–2095. [Google Scholar] [CrossRef] - Wahl, D.E.; Eichel, P.H.; Ghiglia, D.C.; Jakowatz, C.V. 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] - Kouyoumjian, R. Asymptotic high-frequency methods. Proc. IEEE
**1965**, 53, 864–876. [Google Scholar] [CrossRef]

**Figure 4.**Cost surface of image entropy. (

**a**) Cost surface via conventional image entropy. (

**b**) Cost surface via image entropy combined with subarray averaging.

**Figure 10.**ISAR imaging results with different approaches. (

**a**) PGA imaging results. (

**b**) STFT imaging results. (

**c**) Our proposed method.

**Figure 11.**ISAR imaging results with different complex white Gaussian noise. (

**a**–

**c**) Imaging results of PGA, STFT, and proposed method under SNR is 0 dB. (

**d**–

**f**) Imaging results of PGA, STFT, and proposed method under SNR is −7 dB. (

**g**–

**i**) Imaging results of PGA, STFT, and proposed method under SNR is −14 dB.

**Figure 14.**Imaging results for the destroyer. (

**a**) RCS model of the destroyer. (

**b**) Imaging results with RD algorithm. (

**c**) Imaging results with the PGA method. (

**d**) Imaging results with our proposed method.

Target motion parameters (rad/s) | |||||||

${\omega}_{y}$ | ${\omega}_{p}$ | ${\omega}_{r}$ | ${\omega}_{r}^{\prime}$ | ${\omega}_{y}^{\prime}$ | ${\omega}_{p}^{\prime}$ | ||

−0.121 | 0.1 | −0.01 | 3.5 × 10^{−2} | 5.5 × 10^{−2} | −0.025 | ||

LOS motion parameter (rad/s) | |||||||

${\phi}_{0}$ | ${\eta}_{0}$ | $\kappa $ | $\gamma $ | ||||

1.0472 | 0.7854 | 0.35 | 0.5 |

Approach | Image Entropy |
---|---|

PGA algorithm | 9.3565 |

STFT method | 11.3896 |

Proposed method | 8.0824 |

Approach | Image Entropy |
---|---|

PGA algorithm | 8.7717 |

STFT method | 10.1647 |

Proposed method | 8.3391 |

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

© 2021 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**

Yang, Z.; Li, D.; Tan, X.; Liu, H.; Liu, Y.; Liao, G.
ISAR Imaging for Maneuvering Targets with Complex Motion Based on Generalized Radon-Fourier Transform and Gradient-Based Descent under Low SNR. *Remote Sens.* **2021**, *13*, 2198.
https://doi.org/10.3390/rs13112198

**AMA Style**

Yang Z, Li D, Tan X, Liu H, Liu Y, Liao G.
ISAR Imaging for Maneuvering Targets with Complex Motion Based on Generalized Radon-Fourier Transform and Gradient-Based Descent under Low SNR. *Remote Sensing*. 2021; 13(11):2198.
https://doi.org/10.3390/rs13112198

**Chicago/Turabian Style**

Yang, Zhijun, Dong Li, Xiaoheng Tan, Hongqing Liu, Yuchuan Liu, and Guisheng Liao.
2021. "ISAR Imaging for Maneuvering Targets with Complex Motion Based on Generalized Radon-Fourier Transform and Gradient-Based Descent under Low SNR" *Remote Sensing* 13, no. 11: 2198.
https://doi.org/10.3390/rs13112198