# Joint Sparsity Constraint Interferometric ISAR Imaging for 3-D Geometry of Near-Field Targets with Sub-Apertures

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## Abstract

**:**

## 1. Introduction

## 2. Signal Model of Inisar Near-Field Imaging

_{2}is located at origin O’. Antennae TR

_{1}and TR

_{2}form the vertical baseline along axis Z’. Define two coordinate systems, where T’(x’,y’,z’) is the radar coordinate system, axis y’ is the line of sight of radar, x’ and y’ represent the horizontal and vertical directions, respectively. T(x,y,z) is the target coordinate system, in which axis y is coincident with axis y’, and axes x and y represent the azimuth direction and range direction of ISAR, respectively, and distance between origins of the two coordinate systems is R

_{0}(R

_{0}< 4D

^{2}/λ, D is the maximum size of the target and λ is the wavelength of the incident wave). The target is moving at a constant speed in plane (x,y) at an angular velocity ω, and plane (x’,y’) is parallel to plane (x,y). Assuming that coordinate of any point P on the target is (x,y,z), and the coordinate in the cylindrical coordinate system is (r

_{0},θ

_{0},z). Then, at the moment t, the distance from the antenna I (i∈ TR

_{1}, TR

_{2}}) to the point P is:

_{i}refers to the pitch angle from the antenna i to the origin of target coordinate system. Assuming that the antenna transmits a step frequency wideband signal [26]:

## 3. Joint Sparsity Constraint Near-Field 3-D Imaging of Inisar Based on CS

#### 3.1. Algorithm Flow Description

- Step 1:
- Full apertures of the two channels are divided into several sub-apertures by the same criteria, and each sub-aperture has a very small azimuth angle range.
- Step 2:
- Global sparsity constraint and improved OMP algorithm are applied to obtain the 2-D complex images ${I}_{1}$ and ${I}_{2}$ of each sub-aperture in the two channels.
- Step 3:
- Two images of each sub-aperture are performed with interferometric processing to obtain projection coordinates of the scattering points along the baseline.
- Step 4:
- 3-D images of each sub-aperture target are constructed by synthesizing the 2-D ISAR images of the interferometric processing results.
- Step 5:
- 3-D images of all sub-apertures are processed synthetically to obtain the final 3-D images.
- Step 6:
- The imaging flow is shown in Figure 2.

#### 3.2. Joint Sparse Constrained Optimization Model

- (1)
- when $p=1$, ${\Vert g\Vert}_{1,0}={\Vert \left(\left|{g}_{1}\right|+\left|{g}_{2}\right|\right)\Vert}_{0}$, is termed as mixed sum norm, i.e., sparsity of amplitude sum of the two ISAR images is taken as the global sparsity constraint;
- (2)
- when $p=2$, ${\Vert g\Vert}_{2,0}={\Vert {\left({\left|{g}_{1}\right|}^{2}+{\left|{g}_{2}\right|}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\Vert}_{0}$ is termed as mixed Euclidean norm;
- (3)
- when $p=\infty $, ${\Vert g\Vert}_{\infty ,0}={\Vert \mathrm{max}\left(\left|{g}_{1}\right|,\left|{g}_{2}\right|\right)\Vert}_{0}$ is termed as mixed infinite norm, i.e., the sparsity of one of the two ISAR images (with larger amplitude) is taken as the overall sparsity constraint.

#### 3.3. Optimal Solution Algorithm Constrained by Joint Sparse

- (1)
- Initialization: number of iterations $t=1$, and support set ${\mathrm{\Lambda}}_{0}=0$. For the $i\mathrm{th}$ channel, its initialized target vector ${\mathrm{g}}_{i,0}=0$, and incremental matrix ${\mathrm{\Phi}}_{i,0}=0$, which is composed of column vectors in the support set. Make ${\mathrm{r}}_{i,t}$ the residual signal after $t$ iterations, and initialize ${\mathrm{r}}_{i,0}=\mathrm{s}{\prime}_{i}$.
- (2)
- Obtain index ${\lambda}_{t}$ by solving the following formulas:$$\mathrm{Mixed}\mathrm{sum}\mathrm{norm}\text{:}{\lambda}_{t}=\underset{k\in \{1,\cdots ,PQ\}}{\mathrm{arg}\mathrm{max}}{\displaystyle \sum _{i=1}^{2}\left(\left|\langle {r}_{i,t-1}^{}{}^{\ast},{A}_{i,k}^{}\prime \rangle \right|\right)},$$$$\mathrm{Mixed}\mathrm{Euclidean}\mathrm{norm}\text{:}{\lambda}_{t}=\underset{k\in \{1,\cdots ,PQ\}}{\mathrm{arg}\mathrm{max}}{\displaystyle \sum _{i=1}^{2}{\left({\left|\langle {r}_{i,t-1}^{}{}^{\ast},{A}_{i,k}^{}\prime \rangle \right|}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$$$\mathrm{Mixed}\mathrm{infinite}\mathrm{norm}\text{:}{\lambda}_{t}=\underset{k\in \{1,\cdots ,PQ\}}{\mathrm{arg}\mathrm{max}}\left(\underset{i\in \left\{1,2\right\}}{\mathrm{max}}\left|\langle {r}_{i,t-1}^{}{}^{\ast},{A}_{i,k}^{}\prime \rangle \right|\right),$$
- (3)
- Record the obtained index ${\lambda}_{t}$ to the support set and its corresponding vector in ${A}_{i}\prime $ to the incremental matrix:$$\begin{array}{l}{\mathrm{\Lambda}}_{t}={\mathrm{\Lambda}}_{t}\cup \left\{{\lambda}_{t}\right\};\\ {\mathrm{\Phi}}_{i,t}=\left[{\mathrm{\Phi}}_{i,t-1}{\mathrm{A}}_{i,{\lambda}_{t}}^{}\prime \right]\end{array}$$
- (4)
- Adopt the least square method to calculate the projection coefficient of each channel:$${\mathrm{g}}_{i,t}=\underset{{\mathrm{g}}_{i}}{\mathrm{arg}\mathrm{min}}\Vert {\mathrm{s}}_{i}\prime -{\mathrm{\Phi}}_{i,t}{\mathrm{g}}_{i}\Vert \left(i=1,2\right)$$
- (5)
- Update residual signal ${r}_{i,t}^{}$:$${r}_{i,t}^{}={s}_{i}^{}\prime -{\mathrm{\Phi}}_{i,t}{g}_{i,t}\left(i=1,2\right)$$
- (6)
- For the number of iteration $t=t+1$, repeat step (2) to (4) until the energy of the residual signal is lower than the preset threshold Thres or the number of iterations reaches the preset sparsity K.

#### 3.4. Extraction of Target Scattering Information

## 4. Experiments and Analysis

#### 4.1. Numerical Simulations

#### 4.1.1. Precision Analysis of Scattering Point Coordinate Estimation

#### 4.1.2. Precision Analysis of Interferometric Phase

#### 4.1.3. Noise Suppression Performance

#### 4.1.4. Performance of Sparsity Sampling Imaging

#### 4.1.5. Computational Complexity

#### 4.2. Experiments and Analysis of Backhoe

#### 4.3. Actual Measurement Experiment in Anechoic Chamber

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Geometric distribution for scattering model of plane point. (

**a**) 2D distribution diagram of scattering point; (

**b**) 3-D distribution diagram of scattering point.

**Figure 4.**3-D Imaging results of near-field InISAR. (

**a**) Traditional imaging approach, (

**b**) Global sparsity mixed sum norm processing; (

**c**) Global sparsity mixed Euclidean norm processing (

**d**) Global sparsity mixed infinite norm processing.

**Figure 5.**Interferometric phase image of near-field InISAR 3-D imaging. (

**a**) Traditional imaging approach, (

**b**) Global sparsity mixed sum norm processing; (

**c**) Global sparsity mixed Euclidean norm processing (

**d**) Global sparsity mixed infinite norm processing.

**Figure 6.**Imaging results obtained by adopting traditional approach and proposed approach respectively under 5 dB Noise. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); interferometric phase images (top layer); complex images of channels 1 and 2 (second and third layers); final 3-D imaging results (last layer).

**Figure 8.**Imaging results of traditional approach and proposed approach under 20% of effective data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); interferometric phase images (top layer); complex images of channels 1 and 2 (second and third layers); final 3-D imaging results (last layer).

**Figure 11.**InISAR imaging results of Backhoe with complete data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

**Figure 12.**InISAR imaging results of Backhoe with 25% data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

**Figure 15.**Target model of five metal balls. (

**a**) Scanning frame and probe; (

**b**) optical picture of five balls; (

**c**) distribution of target spatial position.

**Figure 16.**InISAR imaging results of five metal balls with complete data. (

**a**) Traditional imaging processing; (

**b**) imaging process of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

**Figure 17.**InISAR imaging results of five metal balls with 20% data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

**Figure 18.**Imaging test for closed chamber. (

**a**) Imaging system and target scene; (

**b**) distribution of targets.

**Figure 19.**InISAR imaging results of anechoic chamber target with complete data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

**Figure 20.**InISAR imaging results of anechoic chamber target with 20% data. (

**a**) Traditional imaging processing; (

**b**) imaging processing of proposed approach (mixed Euclidean norm); complex images of channels 1 and 2 (first and second layers); interferometric phase images (third layer); final 3-D imaging results (last layer).

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

Carrier frequency | 10 GHz |

Bandwidth | 4 GHz |

Frequency step interval | 40 MHz |

Azimuth accumulation angle | 20° |

Azimuth sampling interval | 0.2° |

Distance between antenna and target | 2 m |

Baseline length | 0.02 m |

Scattering Point 1 (x, y, z) | Scattering Points 2 (x, y, z) | Scattering Points 3 (x, y, z) | |
---|---|---|---|

Theoretical coordinate | (0.0125, 0.2688, 0) | (0.1438, 0.2438, 0) | (0.1188, 0.1688, 0.0467) |

Traditional imaging | (0, 0.24, 0.6438) | (0.12, 0.23, 0.14) | (0.1, 0.1, 0.13) |

Mixed sum norm | (0.009, 0.26, 0) | (0.1387, 0.24, 0.046) | (0.11, 0.1, 0.046) |

Mixed infinite norm | (0.008, 0.26, 0) | (0.1387, 0.24, 0.046) | (0.1063, 0.1, 0.046) |

Mixed Euclidean | (0.01, 0.265, 0) | (0.139, 0.241, 0.03) | (0.1163, 0.15, 0.046) |

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

Carrier frequency | 10 GHz |

Bandwidth | 6 GHz |

Sampling point number of frequency | 512 |

Azimuth accumulation angle | 51° |

Sampling point number of direction | 71 * 17 |

Pitch angle | 0.07° |

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

Carrier frequency | 10 GHz |

Bandwidth | 4 GHz |

Frequency step interval | 40 MHz |

Azimuth accumulation angle | 20° |

Azimuth sampling interval | 0.2° |

Distance between antenna and target | 0.02 m |

Baseline length | 2 m |

Approach | Channel | Ball 1 | Ball 2 | Ball 3 | Ball 4 | Ball 5 |
---|---|---|---|---|---|---|

Initial Traditional approach | -- | (−0.20, −0.20) | (0.20, −0.20) | (0.00, 0.00) | (−0.20, 0.20) | (0.20, 0.20) |

1 | (−0.22, −0.19) | (0.18, −0.18) | (−0.05, 0.02) | (−0.22, 0.19) | (0.22, 0.22) | |

2 | (−0.21, −0.18) | (0.18, −0.19) | (−0.03, 0.02) | (−0.22, 0.18) | (0.23, 0.23) | |

Proposed approach | 1 | (−0.20, −0.20) | (0.19, −0.19) | (−0.01, 0.01) | (−0.20, 0.19) | (0.21, 0.21) |

2 | (−0.20, −0.20) | (0.19, −0.19) | (−0.01, 0.01) | (−0.20, 0.19) | (0.21, 0.21) |

Approach | Channel | Ball 1 | Ball 2 | Ball 3 | Ball 4 | Ball 5 |
---|---|---|---|---|---|---|

Initial Traditional approach | -- | (−0.20, −0.20) | (0.20, −0.20) | (0.00, 0.00) | (−0.20, 0.20) | (0.20, 0.20) |

1 | (−0.20, −0.20) | (0.19, −0.19) | (−0.04, 0.01) | (−0.22, 0.19) | (−0.14, 0.20) | |

2 | (−0.21, −0.20) | (0.18, −0.19) | (−0.04, 0.00) | (−0.20, 0.17) | (Null, Null) | |

Proposed approach | 1 | (−0.20, −0.20) | (0.19, −0.19) | (−0.01, 0.01) | (−0.20, 0.19) | (0.20, 0.21) |

2 | (−0.20, −0.20) | (0.19, −0.19) | (−0.01, 0.01) | (−0.20, 0.19) | (0.20, 0.21) |

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## Share and Cite

**MDPI and ACS Style**

Fang, Y.; Wang, B.; Sun, C.; Wang, S.; Hu, J.; Song, Z. Joint Sparsity Constraint Interferometric ISAR Imaging for 3-D Geometry of Near-Field Targets with Sub-Apertures. *Sensors* **2018**, *18*, 3750.
https://doi.org/10.3390/s18113750

**AMA Style**

Fang Y, Wang B, Sun C, Wang S, Hu J, Song Z. Joint Sparsity Constraint Interferometric ISAR Imaging for 3-D Geometry of Near-Field Targets with Sub-Apertures. *Sensors*. 2018; 18(11):3750.
https://doi.org/10.3390/s18113750

**Chicago/Turabian Style**

Fang, Yang, Baoping Wang, Chao Sun, Shuzhen Wang, Jiansheng Hu, and Zuxun Song. 2018. "Joint Sparsity Constraint Interferometric ISAR Imaging for 3-D Geometry of Near-Field Targets with Sub-Apertures" *Sensors* 18, no. 11: 3750.
https://doi.org/10.3390/s18113750