# Squint Model InISAR Imaging Method Based on Reference Interferometric Phase Construction and Coordinate Transformation

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## Abstract

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## 1. Introduction

## 2. Principle of 3D InISAR Imaging

#### 2.1. Normal Model

_{0}, Y

_{0}, Z

_{0}) is the initial location of the target center, and P(x

_{p}, y

_{p}, z

_{p}) is one of the scattering centers of the target. The initial distances between antennas A, B, C, and P are denoted as R

_{AP0}, R

_{BP0}, R

_{CP0}, respectively.

_{m}are the fast and slow times, f

_{0}is the carrier frequency, γ is the frequency-modulation rate, and T

_{P}is the time width of the transmitted pulse. Then, the echo from a single scatterer P can be expressed as,

_{K}represents the scattering intensity of scatterer P received by antenna K, and τ

_{K}(t

_{m}) is the echo delay relative to antenna K which can be expressed as,

_{KP}(t

_{m}) represents the instantaneous distance from P to the radar K. During the short imaging time, the R

_{KP}(t

_{m}) can be treated as linearly varied, that is, R

_{KP}(t

_{m}) = R

_{KP0}+ V

_{KP}t

_{m}. After the pulse compression, motion compensation, and ISAR imaging processing, the ISAR image of each antenna can be expressed as,

_{m}denotes the imaging time, f

_{m}denotes the Doppler frequency, and λ denotes the wavelength of the carrier frequency of the transmitted signal. We should stress that if the aspect angle range (AAR) of observation is small, i.e., less than 1–2 degrees, Equation (4) can be obtained easily by azimuthal FFT after range compression. If the AAR is larger than several degrees and in a high-range resolution situation, then the 2D interpolation is required, i.e., Equation (4) can be obtained after complex interpolation processing, which is out of the scope of this paper.

_{AB}and ΔR

_{AC}represent the wave path differences between antennas A and B and between A and C, respectively. According to the imaging geometry of Figure 1, ΔR

_{AB}and ΔR

_{AC}can be estimated by,

_{P}Z

_{P}) along the baselines. However, the real InPha may be wrapped due to the measured InPha being unable to exceed (−π, π) and thus it suffers from InPhA. In other words, only when X

_{P}, Z

_{P}are small enough to meet the following conditions, can InPhaA can be avoided.

_{P}and Z

_{P}are all determined by two parts, i.e., X

_{P}= X

_{O}+ x

_{p}and Z

_{P}= Z

_{O}+ z

_{p}, where X

_{O}and Z

_{O}respectively represent the X and Y coordinates of the origin of the target coordinate system in the coordinate system of InISAR, as introduced above, and x

_{p}and z

_{p}depend on the target size. As previously mentioned, the InPhaA caused by X

_{O}is called the squint effect which is focused on in this research. For the convenience of derivation, we temporarily ignore the InPhaA caused by x

_{p}, that is, we assume there is no InPhaA when X

_{O}= 0.

_{O}is so small that X

_{P}is within the constraints of Equation (10) so X

_{P}can be directly estimated by

_{P}can be approximated by the LOS range R

_{AP}.

#### 2.2. Squint Model

_{AB}in Equation (8) can be rewritten as,

_{AB}is not only a function of x

_{p}, but also a function of X

_{O}as well, even if x

_{p}= 0, φ

_{AB}will also change along with the variation of X

_{O}. Figure 2 demonstrates the influence of the squint effect on the InPha with Y

_{O}= 10 km, Z

_{O}= 0, λ = 0.03 m, and L = 3 m assumed, where Figure 2a shows the real InPhas variation as X

_{O}changes from 5 km to 10 km, while Figure 2b shows the measured InPhas variation, which is wrapped. As a result, the measured InPhas will not satisfy Equation (10), which results in difficulties for subsequent InPha unwrapping and 3D target reconstruction.

_{AP}of the ISAR image is not consistent with the Y-coordinate of the actual target under the squint model. If it is directly used for 3D reconstruction, distortion will be introduced, however, it can be corrected through coordinate transformation as addressed later in Section 3.2.

## 3. InISAR Imaging Based on Reference InPhas Construction

#### 3.1. Reference InPhas Construction

_{AQ}assumed,

_{AQ}can be directly obtained by measuring the echo delay, ΔR

_{AB}and ΔR

_{AC}represent the wave path differences between antennas A and B and between A and C, respectively, which can be obtained by calculating the cross-correlation function (CCF) between two ISAR images of different channels as follows,

_{Q}, Y

_{Q}, Z

_{Q}) can be calculated by substituting R

_{AQ}, ΔR

_{AB}, and ΔR

_{AC}into Equation (14), and then Q (X

_{Q}, Y

_{Q}, Z

_{Q}) is taken as the reference scatterer used to construct the reference InPhas by,

_{AQ}, R

_{BQ}, and R

_{CQ}are the radial distances from the reference scatterer Q to the antennas A, B, and C, respectively.

#### 3.2. InPhas Restoration

_{AP}− R

_{BP}− (R

_{AQ}− R

_{BQ}) and R

_{AP}− R

_{CP}− (R

_{AQ}− R

_{CQ}) can be very small, φ

_{AB_focus}and φ

_{AC_focus}will not exceed (−π, π), i.e., the InPhA is eliminated. Then, in order to make the InPha correctly reflect the echo difference between two antennas, we must restore the real InPhas by,

#### 3.3. 3D Coordinates Estimation

_{P}= X

_{O}+ x

_{p}and Z

_{P}= Z

_{O}+ z

_{p}, here X

_{O}and Z

_{O}are replaced by X

_{Q}and Z

_{Q}.

_{AB_real}and φ

_{AC_freal}) can be very large in the far-field case, so R

_{AP}, R

_{BP}, and R

_{CP}in Equation (24) can no longer be approximated by R

_{AO}as the imaging method under the normal model does. Therefore, in order to guarantee the estimation accuracy, the LOS ranges of all scatterers are directly obtained by the echo delays of different receiving antennas.

**M**is the rotation matrix as follows,

_{p}can be obtained from (28)

_{P}= R

_{AP}− R

_{AQ}. It should be noted that the Q does not need to be in the exact target center, because even if it is situated away from the target center by some distance (which depends on the accuracy of 3D locating of the target according to (14)), introducing some error on the y-coordinate estimation by coordinate transformation, the error is acceptable in consideration of the far-field condition.

## 4. Simulation Results

_{AB}and φ

_{AC}calculated by substituting X

_{P}= Y

_{P}= Z

_{P}= 10 km into Equations (8) and (9) are both about 120.91 radians, i.e., the InPhaA caused by the squint effect will appear inevitably. Moreover, the unambiguity ranges of x

_{p}and z

_{p}calculated by Equation (24) are ∈ (−260, 260) m, while the maximum dimension of the target is smaller than 50 m, so Equation (10) is satisfied, i.e., there is no InPhaA that can be caused by x

_{p}and z

_{p}. Therefore, we need to remove the influence of InPhaA caused by the squint effect.

#### 4.1. Reference InPhas Construction

#### 4.2. InPhas Restoration

_{AB_dir}cannot exceed (−π, π), the real InPhas φ

_{AB_real}should be the sum of φ

_{AB_dir}and 2 kπ, where k is an integer to be decided. Figure 7a shows the measured InPhas directly obtained by the traditional InISAR imaging method, as it is shown, the φ

_{AB}and φ

_{AB}vary from 1.3 to 1.7 radians. They are ambiguous with 2 kπ radians absent according to the geometry. We used the constructed reference InPhas by Equations (20) and (21) to remove the InPhaA, and plot the real InPhas obtained by Equation (23) in Figure 7b. As can be clearly seen from Figure 7a,b, the InPhaA caused by the squint effect is eliminated, and the real InPhas of the all scatterers was successfully reconstructed.

#### 4.3. 3D Coordinates Estimation

#### 4.4. Robustness of the Proposed Method

## 5. Experiment Results

_{X1}, T

_{X2}, T

_{X3}) and four receiving antennas (R

_{X1}, R

_{X2}, R

_{X3}, R

_{X4}). Figure 14a presents the simplified observation geometry for describing the experiment clearly. The antennas appear to not be positioned in the L-shape or cross-shape that the traditional 3D InISAR imaging methods usually rely on, however they can be configured to an L-type as follows. First, a 3D coordinate system is established by taking the transmitting antenna T

_{X1}as the origin, so the coordinates of T

_{X2}, R

_{X4}, R

_{X1}, and R

_{X2}can be obtained as (−λ, 0, λ/2), (d, 0, 0), (d + 3λ/2, 0, 0) and (d + λ, 0, 0), respectively. This configuration guarantees that a moderate bistatic acquisition geometry can be obtained, which can then be approximated by an equivalent monostatic acquisition geometry with virtual self-transmitting and self-receiving antennas positioned in the middle of the original transmitting antennas and receiving antennas [28]. Therefore, the echo from T

_{X1}transmitting and R

_{X4}receiving (1T4R) can be equivalent to the echo from a virtual self-transmitting and self-receiving antenna at A(d/2, 0, 0). Similarly, the echoes from 1T1R and 2T2R can be equivalent to the echoes from self-transmitting and self-receiving antennas at B(d/2 + 3λ/4, 0, 0) and C(d/2, 0, λ/4), respectively. Obviously, the three virtual antennas A, B, and C form an L-type InISAR system as shown in Figure 14b, which is then used to verify the proposed method. Table 4 lists some typical parameters of the experiment system. The length and height of the UAV are about 0.6 m and 0.2 m, respectively, and the target center is approximately positioned at (0.5, 1.1, 0.1) m as shown in Figure 13b. Suppose P is an arbitrarily selected scattering center from the UAV, then X

_{P}∈ (0.2, 0.8) and Z

_{P}∈ (0, 0.2). Then, the range of InPhas can be roughly estimated according to Equations (4) and (5), i.e., φ

_{AB}

_{_real}∈ (π/2, 2π), and φ

_{AC}

_{_real}∈ (0, π/2), and it is apparent that the InPhas of the all scattering centers from the Z-direction interferometry are within (−π, π), while the InPhas of some scattering centers from the X-direction interferometry exceed (−π, π), which is caused by the squint effect as mentioned in Section 2, that is to say, the InPhaA occurs, which must be removed before imaging processing. The above analysis is validated by the results shown in Figure 15.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The influence of the squint effect on InPhas: (

**a**) the real InPha; (

**b**) the measured InPhas.

**Figure 5.**Target model in (

**a**) 3D; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 8.**3D imaging results by the traditional method without removing the squint effect, in (

**a**) 3D; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 9.**3D imaging results obtained by the proposed method with the Q scatterer at position (9.980, 9.939, 9.980) km taken as the reference scatterer in (

**a**) 3D; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 10.**3D imaging results obtained by the method in [25] with the Q scatterer at position (9.980, 9.939, 9.980) km taken as the reference scatterer in (

**a**) 3D; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 11.**3D imaging results obtained by the method in [25] with the scatterer at position (9.995, 9.995, 9.995) km taken as the reference scatterer in (

**a**) 3D; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 14.**Experiment observation geometry: (

**a**) real observation geometry; (

**b**) equivalent observation geometry.

**Figure 15.**InPha images of measured InPhas: (

**a**) from

**X**-direction interferometry, the InPha is discontinuous as indicated by the abrupt change of colors between yellow and dark blue; (

**b**) from

**Z**-direction interferometry, the InPha is continuous as indicated by the gradual change of colors between yellow and light blue.

**Figure 16.**InPha images of real InPhas: (

**a**) from X-direction interferometry, the InPha is continuous as indicated by the gradual change of colors between yellow and light blue; (

**b**) from Z-direction interferometry, the InPha is continuous as indicated by the gradual change of colors between yellow and light blue.

**Figure 17.**Three-dimensional imaging results without InPhaA removed: (

**a**) 3D reconstruction; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

**Figure 18.**Three-dimensional imaging results with InPhaA removed (

**a**) 3D reconstruction; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (d) projection on y-z plane.

**Figure 19.**Three-dimensional imaging results with InPhaA removed and image distortion corrected: (

**a**) 3D reconstruction; (

**b**) projection on x-y plane; (

**c**) projection on x-z plane; (

**d**) projection on y-z plane.

Parameters | Value |
---|---|

Carrier frequency | 10 GHz |

Bandwidth | 500 MHz |

PRF | 500 Hz |

Location of antenna A | (0,0,0) |

Location of antenna B | (1,0,0) m |

Location of antenna C | (0,0,1) m |

Target velocity | 50 m/s |

Roll/pitch/yaw | 0/0/0.03 rad/s |

SNR | 10 dB |

Target location | (10,10,10) km |

θ | 45 deg |

φ | 45 deg |

**Table 3.**Comparison of RMSEs and runtimes of our method and the method in [25].

Method | RMSE of x | RMSE of y | RMSE of z | Runtime |
---|---|---|---|---|

Our method | 0.2063 | 0.3389 | 0.1914 | 0.01 s |

Ref. [25] | 0.2125 | 0.3734 | 0.2195 | 0.26 s |

Parameters | Value |
---|---|

Carrier frequency | 77 GHz |

Chirp rate | 39.976 MHz |

Pulse width | 100 μs |

PRF | 200 Hz |

Rotational velocity | 0.2618 rad/s |

Imaging time | 0.6 s |

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**MDPI and ACS Style**

Li, Y.; Zhang, Y.; Dong, X.
Squint Model InISAR Imaging Method Based on Reference Interferometric Phase Construction and Coordinate Transformation. *Remote Sens.* **2021**, *13*, 2224.
https://doi.org/10.3390/rs13112224

**AMA Style**

Li Y, Zhang Y, Dong X.
Squint Model InISAR Imaging Method Based on Reference Interferometric Phase Construction and Coordinate Transformation. *Remote Sensing*. 2021; 13(11):2224.
https://doi.org/10.3390/rs13112224

**Chicago/Turabian Style**

Li, Yu, Yunhua Zhang, and Xiao Dong.
2021. "Squint Model InISAR Imaging Method Based on Reference Interferometric Phase Construction and Coordinate Transformation" *Remote Sensing* 13, no. 11: 2224.
https://doi.org/10.3390/rs13112224