# Non-Uniform Synthetic Aperture Radiometer Image Reconstruction Based on Deep Convolutional Neural Network

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## Abstract

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## 1. Introduction

- The G-matrix method measures the G-matrix of the system response, and then uses the measured G-matrix to reconstruct the brightness temperature image of the measured data by using the regularization algorithm. There are many very small singular values in the G-matrix of the impulse response of a non-uniform sampling synthetic aperture radiometer system. These small singular values are caused by the non-uniform arrangement of element antennas. When the G-matrix reconstruction method is used for image reconstruction, a stable solution cannot be obtained.
- Based on the ideal situation, the FFT method can meet the Fourier transformation relationship between the visibility function and the brightness temperature image. Through the measured visibility function, the anti-Fourier transform is transformed and rebuilds the brightness temperature image. Although a stable solution can be obtained by using the FFT method, the FFT method requires that the sampling points in the frequency domain are evenly distributed, while the sampling points of the non-uniform sampling synthetic aperture radiometer on the UV plane are non-uniform, which will introduce large errors into the inversion image.

## 2. Related Works

#### 2.1. Non-Uniform Synthetic Aperture Radiometer Model

#### 2.2. Definition of the Related Neural Network

## 3. Dataset and Learning for ISAR-CNN

#### 3.1. Dataset Generation

- From the UC Merced (UCM) dataset (created by researchers at the University of California, Merced, CA, USA) of 21 different categories of remote sensing images, an image was selected and converted to grayscale. In order to correspond to the scale of the conventional microwave brightness temperature image, we mapped the scale of the image from 0–255 to 2.7–300 (in the practical application of Earth remote sensing, the scene brightness temperature is mostly distributed between 2.7 K and 300 K, and the simulated scene covers the dynamic range of the whole scene brightness temperature distribution). Then, we used this image as the original scene brightness temperature image ($T$).
- $T$ is then input into the ideal non-uniform synthetic aperture radiometer simulation program to generate the simulated non-uniform synthetic aperture radiometer visibilities function $V\left(u,v\right)$ and its corresponding UV plane coordinates $\left(u,v\right)$. In the non-uniform synthetic aperture radiometer simulation program, the antenna array is set to a randomly distributed 51-element antenna array, and the antenna array is shown in Figure 2 (the X-axes and Y-axes represent the relative position of the antenna distribution, and the unit is the multiple of the wavelength). The receiving frequency was set to 33.5 GHz.
- $V\left(u,v\right)$ (size of 1 $\times $ 269) and $\left(u,v\right)$ (size of 1 $\times $ 269) were combined as one input data $\left[V\left(u,v\right)\left(u,v\right)\right]$, with a size of 3 $\times $ 269.
- $\left[V\left(u,v\right)\left(u,v\right)\right]$ and $T$ (size of 79 $\times $ 79) were combined as one sample.
- After each image in the UCM dataset undergoes steps 1–4, we can obtain multiple samples to form dataset ($\mathrm{D}$).
- Randomly select 90% of the data from dataset ($\mathrm{D}$) as the training dataset (${\mathrm{D}}_{\mathrm{train}}$), and the remaining 10% of the data from dataset ($\mathrm{D}$) is then used for network testing as the testing dataset (${\mathrm{D}}_{\mathrm{test}}$)

#### 3.2. Learning for IASR-CNN

_{c}) is obtained. At the same time, the visibility function part (size of 2 $\times $ 269 $\times $ 269) is input to the convolutional layer (C4), and the output is the convolution features between channels (O

_{4}). The convolutional layer (C4) contains 64 filters, each with a size of 3 $\times $ 3. The stride of the filters is 1 and the pooling size is 0.

_{4}and S

_{c}are multiplied to obtain the weighted channel information (O

_{se}) for subsequent processing.

_{se}and C2 are added and input into the convolution layer (C5). The convolutional layers (C5, C6, C7, C8, and C9) are connected as shown in Figure 3. The convolutional layer (C5) contains 64 filters, each with a size of 5 × 5. The stride of the filters is 1 and the pooling size is 2 × 2. The convolutional layer (C6) contains 64 filters, each with a size of 3 × 3. The stride of the filters is 1 and the pooling size is 2 × 2. The convolutional layer (C7) contains 32 filters, each with a size of 5 × 5. The stride of the filters is 2 and the pooling size is 1 × 1. The convolutional layer (C8) contains 16 filters, each with a size of 5 × 5. The stride of the filters is 2 and the pooling size is 1 × 1. The convolutional layer (C9) contains one filter, each with a size of 3 × 3. The stride of the filters is 1 and the pooling size is 1 × 1.

^{−}

^{m}), respectively.

_{train}contains multiple samples. Each sample is composed of a non-uniform synthetic aperture radiometer visibilities function $V\left(u,v\right)$ as well as its corresponding UV plane coordinates $\left(u,v\right)$ and corresponding original scene brightness temperature image (

**T**). The generation of the dataset is described in detail in the previous section.

**T**

_{cnn}). When all the samples in the training dataset D

_{train}are input into the network, the network completes an epoch training. Through multiple epoch training, the parameters in the IASR-CNN are adjusted iteratively to reduce the root mean square error (RMSE) between the

**T**

_{cnn}and

**T**(in the training dataset D

_{train}). When the number of training cycles reaches the set epoch or the RMSE tends to be stable, it indicates that the network has completed the training. In this paper, by comparing the final convergence results of different gradient descent algorithms (such as the SGD, Adadelta, and Adam algorithms), the standard backpropagation stochastic gradient descent algorithm (SGD) was selected to minimize the RMSE and update the IASR-CNN parameters (the learning rate attenuation strategy will be adopted in the actual use, so the learning rate decreases gradually). The formula for calculating the RMSE between

**T**

_{cnn}and

**T**(in the training dataset) is as follows.

**T**and

**T**

_{cnn}is X × Y; k is the sample sequence number; L is the total number of samples in the training dataset.

## 4. Experiments and Result Analysis

#### 4.1. Ideal Simulation Results

**T**) as well as reconstruct the brightness temperature image. The effect of the network reconstruction of brightness temperature image was evaluated qualitatively and quantitatively.

**T**), which was randomly selected from the test datasets, Figure 5b shows the image

**T**

_{cnn}reconstructed by the network, Figure 5c shows the image

**T**

_{grid}reconstructed by the grid method, and Figure 5d is the image

**T**

_{aff}reconstructed by the array factor forming method. A comparison with the original scene brightness temperature showed that the effect of the IASR–CNN method for the brightness temperature reconstruction was better, and visually, the

**T**

_{cnn}reconstructed by the network was not much different from the original scene image

**T**.

^{−}³, 2.15 × 10

^{−4}and 1.77 × 10

^{−4}(after spectrum normalization). This quantitatively shows that the spectrum of the network reconstructed image was closer to the original scene spectrum.

#### 4.2. Simulation Results with Errors

## 5. Conclusions

^{−4}, 2.21 × 10

^{−4}and 1.76 × 10

^{−4}, which quantitatively showed that the image spectrum reconstructed by the IASR-CNN method was closer to the original scene spectrum compared to the traditional method (the grid method and the AFF method). The quantitative results showed that the reconstructed image quality of the IASR-CNN method was the best.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Randomly generated antenna array (after wavelength normalization) and UV plane sampling point.

**Figure 5.**The IASR images reconstructed by different methods at the same scene. (

**a**) Original scene. (

**b**) IASR-CNN method. (

**c**) Grid method. (

**d**) AFF method.

**Figure 6.**The spectrum of images reconstructed by different methods at the same scene. (

**a**) Spectrum in direction u. (

**b**) Spectrum in direction v. (

**c**) High-frequency spectrum in direction u. (

**d**) High-frequency spectrum in direction v.

**Figure 7.**The brightness temperature images reconstructed by the different methods at variable noise intensities. (

**a**) Original scene. (

**b**) The IASR-CNN method at a 0.1 noise intensity. (

**c**) The grid method at a 0.1 noise intensity. (

**d**) The AFF method at a 0.1 noise intensity. (

**e**) The IASR-CNN method at a 0.2 noise intensity. (

**f**) The grid method at a 0.2 noise intensity. (

**g**) The AFF method at a 0.2 noise intensity. (

**h**) The IASR-CNN method at a 0.3 noise intensity. (

**i**) The grid method at a 0.3 noise intensity. (

**j**) The AFF method at a 0.3 noise intensity.

**Figure 8.**The evaluation metrics of the brightness temperature image reconstructed by different methods at variable noise intensities. (

**a**) PSNR. (

**b**) RMSE.

**Table 1.**The evaluation metrics of the brightness temperature images reconstructed by three methods.

Method | RMSE (K) | PSNR (dB) |
---|---|---|

IASR–CNN method | 9.4718 | 24.6375 |

Grid method | 15.3268 | 20.6268 |

AFF method | 14.8785 | 21.8099 |

**Table 2.**The evaluation metrics of the spectrum corresponding to the images reconstructed by three methods.

Method | RMSE |
---|---|

IASR–CNN method | 1.35 × 10^{−4} |

Grid method | 2.21 × 10^{−4} |

AFF method | 1.76 × 10^{−4} |

Method | Time (s) |
---|---|

IASR–CNN method | 0.016 |

Grid method | 0.19 |

AFF method | 0.85 |

Variable Noise Intensities | 50% | 60% | 70% | 80% | 90% |
---|---|---|---|---|---|

0 | 9.8924 | 9.7922 | 9.7757 | 9.6031 | 9.5327 |

0.1 | 10.7056 | 10.6691 | 10.5176 | 10.3349 | 10.2615 |

0.2 | 12.8341 | 12.7241 | 12.2785 | 12.0975 | 11.9740 |

0.3 | 15.3358 | 15.1270 | 14.9058 | 14.5469 | 14.1078 |

Variable Noise Intensities | 50% | 60% | 70% | 80% | 90% |
---|---|---|---|---|---|

0 | 23.9786 | 24.0357 | 24.2759 | 24.3040 | 24.3436 |

0.1 | 23.1031 | 23.2295 | 23.3485 | 23.4191 | 23.5625 |

0.2 | 22.0849 | 22.5657 | 22.6787 | 22.8003 | 22.9299 |

0.3 | 20.3953 | 20.8124 | 21.1419 | 21.3134 | 22.0037 |

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**MDPI and ACS Style**

Xiao, C.; Wang, X.; Dou, H.; Li, H.; Lv, R.; Wu, Y.; Song, G.; Wang, W.; Zhai, R. Non-Uniform Synthetic Aperture Radiometer Image Reconstruction Based on Deep Convolutional Neural Network. *Remote Sens.* **2022**, *14*, 2359.
https://doi.org/10.3390/rs14102359

**AMA Style**

Xiao C, Wang X, Dou H, Li H, Lv R, Wu Y, Song G, Wang W, Zhai R. Non-Uniform Synthetic Aperture Radiometer Image Reconstruction Based on Deep Convolutional Neural Network. *Remote Sensing*. 2022; 14(10):2359.
https://doi.org/10.3390/rs14102359

**Chicago/Turabian Style**

Xiao, Chengwang, Xi Wang, Haofeng Dou, Hao Li, Rongchuan Lv, Yuanchao Wu, Guangnan Song, Wenjin Wang, and Ren Zhai. 2022. "Non-Uniform Synthetic Aperture Radiometer Image Reconstruction Based on Deep Convolutional Neural Network" *Remote Sensing* 14, no. 10: 2359.
https://doi.org/10.3390/rs14102359