# A Full-Polarization Radar Image Reconstruction Method with Orthogonal Coding Apertures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Signal Model

#### 2.1. Radar Imaging Model

**x-O-y**was established with the target centroid

**O**as the origin. The distance between the radar and the target centroid is R

_{0}. The motion of the target is transformed into a rotation around the centroid of the target with an angular velocity ω. Assume that the translational motion of the target has been accurately compensated, while ignoring the range cell migration (RCM) caused by the target rotation across the azimuth aperture. The classic ‘stop-and-go’ model was adopted for the equivalent rotation of the target, in which the target is considered to rotate uniformly between two pulses and remain stationary within one pulse.

_{0}is signal carrier frequency, k is the frequency modulation slope, T is the pulse width, and the signal bandwidth is B = kT. Besides, $\widehat{t}$ is the fast time variable, and ${t}_{m}$ is the slow-time variable. ${t}_{m}=m{T}_{d}$, where T

_{d}is the pulse repetition interval (PRI) and m is the number of the pulses with $0\le m<M$. M represents the total number of imaging observation pulses. $t={t}_{m}+\widehat{t}$ is the full-time variable, satisfying 0 ≤ t ≤ T

_{A}, where T

_{A}is the total observation time of imaging with T

_{A}= MT

_{d}.

_{T}scattering points. For the i-th scattering point P(x

_{i}, y

_{i}), assume its equivalent rotational momentum in one pulse repetition interval can be ignored. The distance between the target and the radar at time t

_{m}is given by

_{i}represents echo amplitude and c represents the velocity of the electromagnetic wave.

_{s}, then the number of sampling points in one single pulse is N = f

_{s}T. After receiving the target signal echo at time t

_{m}, the high-resolution range profile (HRRP) of the target can be obtained by performing the dechirp process and inverse Fourier transform in range dimension. After obtaining the slow-time range profile sequence of the target, envelope alignment and phase correction are carried out. The processed range profile sequence is recorded as a $M\times N$ matrix $\mathit{S}$, which is represented as

#### 2.2. Orthogonal Coding Apertures for Polarization Radar Imaging

#### 2.3. The Shortcoming of FFT-Based Imaging Method

## 3. Reconstruction Algorithm Based on CS Theory

#### 3.1. Sparse Representation of Orthogonal Coding Aperture

**P**of H polarization channel is given by:

_{H}#### 3.2. Multichannel Joint Reconstruction Algorithm Based on CS Theory

#### 3.3. Procedure of the Proposed Reconstruction Algorithm

**Input:**Slow-time range profile matrix ${\mathit{S}}_{f,k}$, sparse measurement matrix ${\mathit{\Phi}}_{k}$ and ISAR image sparsity L;

**Output:**ISAR images ${\mathit{S}}_{2D,k}$ to be reconstructed;

**Step1:**Initialize residual matrix ${\mathit{R}}_{0,k}={\mathit{S}}_{f,k}$, index set ${\mathit{\Lambda}}_{0}=$ Ø, and set cycle index $l=0$.

**Step2:**Calculate $\sum _{k=1}^{4}\mathrm{abs}({{\mathit{\Phi}}_{k}}^{H}{\mathit{R}}_{l,k})$. Find the coordinates $(p,q)$ of the largest element in $\sum _{k=1}^{4}\mathrm{abs}({{\mathit{\Phi}}_{k}}^{H}{\mathit{R}}_{l,k})$. Then update the index set ${\mathit{\Lambda}}_{l}={\mathit{\Lambda}}_{l-1}{{\displaystyle \cup}}^{}\left\{\left(p,q\right)\right\}$.

**Step3:**Update the ISAR image ${\widehat{\mathit{S}}}_{2D,k}$, where the position of non-zero elements is determined by index set, and the scattering coefficient ${\mathit{\Gamma}}_{k}$ is calculated by the following method:Ŝ

**Step4:**Update residual matrix ${\mathit{R}}_{l,k}={\mathit{S}}_{f,k}-{\mathit{\Phi}}_{k}{\widehat{\mathit{S}}}_{2D,k}$. If $l\le L$,then back to Step2; Otherwise, stop the iteration, and output the ISAR image ${\mathit{S}}_{2D,k}$.

## 4. Simulation and Analysis

#### 4.1. Numerical Simulation

#### 4.1.1. Parameter Setting and Reconstruction Images

#### 4.1.2. Consistency of Reconstruction Images

#### 4.1.3. Reconstructed Image Quality

#### Image Entropy

#### Image Contrast

#### 4.2. Measured Data Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Two different aperture arrangements: (

**a**) uniform alternating aperture; (

**b**) orthogonal coding apertures.

**Figure 3.**Imaging results of Yake-42 simulation model under different aperture extraction ratio: (

**a**) Yake-42 simulation model; (

**b**) Aperture extraction ratio 100%; (

**c**) Aperture extraction ratio 75%; (

**d**) Aperture extraction ratio 50%.

**Figure 8.**Comparison of reconstructed image between RD algorithm and CS algorithm: (

**a**–

**d**) represent the images obtained by RD algorithm in HH, HV, VH and VV polarization channels, respectively; (

**e**–

**h**) represent the images obtained by CS algorithm in HH, HV, VH and VV polarization channels, respectively.

**Figure 9.**Mapping images of single-channel reconstruction and multichannel joint reconstruction: (

**a**–

**d**) represent the single-channel reconstruction images in HH, HV, VH and VV polarization channels, respectively; (

**e**–

**h**) represent the multichannel joint reconstruction images in HH, HV, VH and VV polarization channels, respectively. Notice that the blue boxes mark the scattering points at the same location in the different images. It is clear that the positions of the scattering points differ from each other when each polarization channel is reconstructed separately. In contrast, the positions of the scattering points from the joint reconstruction results remain consistent within each channel.

**Figure 11.**UAV imaging results using RD algorithm: (

**a**) HH polarization channel; (

**b**) HV polarization channel; (

**c**) VH polarization channel; (

**d**) VV polarization channel.

**Figure 12.**Imaging results of UAV with sparse aperture extraction: (

**a**–

**d**) represent the imaging results using RD algorithm in HH, HV, VH and VV polarization channels, respectively; (

**e**–

**h**) represent the imaging results using CAS-OMP algorithm in HH, HV, VH and VV polarization channels, respectively.

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

carrier frequency f_{0} | 10 GHz |

pulse width T_{p} | 100 $\mathsf{\mu}\mathrm{s}$ |

bandwidth B | 500 MHz |

azimuth pulse number | 256 |

distance sampling number | 1024 |

pulse repetition interval | 1 ms |

polarization mode | HH\HV\VH\VV |

Reconstruction Algorithm | Polarization Channel | Experiment Serial Number | Average Image Entropy | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |||

RD algorithm | HH channel | 5.955 | 5.785 | 5.710 | 5.809 | 5.803 | 5.813 |

HV channel | 4.498 | 4.782 | 4.565 | 4.471 | 3.982 | 4.460 | |

VH channel | 4.518 | 4.739 | 4.636 | 4.476 | 4.024 | 4.479 | |

VV channel | 5.850 | 5.983 | 5.876 | 5.866 | 5.758 | 5.867 | |

Full-polarization joint reconstruction algorithm | HH channel | 0.706 | 0.696 | 0.707 | 0.700 | 0.701 | 0.702 |

HV channel | 0.663 | 0.683 | 0.670 | 0.660 | 0.644 | 0.664 | |

VH channel | 0.674 | 0.673 | 0.674 | 0.657 | 0.639 | 0.663 | |

VV channel | 0.696 | 0.697 | 0.704 | 0.704 | 0.705 | 0.701 |

Reconstruction Algorithm | Polarization Channel | Experiment Serial Number | Average Image Contrast | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |||

RD algorithm | HH channel | 347.839 | 307.060 | 273.733 | 295.230 | 303.917 | 305.556 |

HV channel | 153.530 | 168.654 | 164.888 | 151.233 | 136.719 | 155.005 | |

VH channel | 156.150 | 166.363 | 163.592 | 149.898 | 136.994 | 154.600 | |

VV channel | 301.904 | 335.617 | 304.306 | 312.230 | 287.988 | 308.409 | |

Full-polarization joint reconstruction algorithm | HH channel | 499.981 | 435.572 | 411.414 | 444.382 | 423.317 | 442.933 |

HV channel | 188.784 | 201.384 | 180.080 | 162.159 | 148.806 | 176.242 | |

VH channel | 190.817 | 196.223 | 181.416 | 166.137 | 146.995 | 176.317 | |

VV channel | 452.815 | 510.040 | 510.289 | 467.005 | 493.793 | 486.788 |

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

initial frequency | 8 GHz |

terminal frequency | 12 GHz |

bandwidth B | 4 GHz |

frequency interval | 20 MHz |

number of sub pulses | 200 |

pitch angle | 0° |

azimuth angle | −180°~180° |

azimuth interval | 0.2° |

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## Share and Cite

**MDPI and ACS Style**

Zhao, T.; Wu, Q.; Zhao, F.; Xu, Z.; Xiao, S.
A Full-Polarization Radar Image Reconstruction Method with Orthogonal Coding Apertures. *Remote Sens.* **2021**, *13*, 4626.
https://doi.org/10.3390/rs13224626

**AMA Style**

Zhao T, Wu Q, Zhao F, Xu Z, Xiao S.
A Full-Polarization Radar Image Reconstruction Method with Orthogonal Coding Apertures. *Remote Sensing*. 2021; 13(22):4626.
https://doi.org/10.3390/rs13224626

**Chicago/Turabian Style**

Zhao, Tiehua, Qihua Wu, Feng Zhao, Zhiming Xu, and Shunping Xiao.
2021. "A Full-Polarization Radar Image Reconstruction Method with Orthogonal Coding Apertures" *Remote Sensing* 13, no. 22: 4626.
https://doi.org/10.3390/rs13224626