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Fractional Fourier transform is an important branch of optical research, and it is widely used in optical encryption, optical filtering, image watermarking and other fields. The phase retrieval in the case of fractional Fourier transform is widely studied. Also, deep learning has been an intriguing method for optical computational imaging. However, in optical computational imaging, traditional deep learning methods possess some intrinsic disadvantages. In optical imaging experiments, it is often difficult to obtain sufficient quality and quantity of labeled data for training, thus leading to poor robustness of the trained neural network. Even with sufficient datasets, the training time can be particularly long. In recent years, there has been an increase in interest in physic-driven untrained neural networks for computational imaging. Herein we use such a method to study the fractional Fourier transform imaging, which combines neural networks with optical models to achieve phase retrieval of fractional Fourier transform. Unlike the traditional neural network training with the original image as the target, our network framework is used only a single intensity image for the phase retrieval of fractional Fourier transform images. The output image of the neural network will serve as an optical model through fractional Fourier transform, and then the output image of the optical model will be used as a loss function to drive the neural network training with the output image of the neural network. We study the fractional Fourier transform reconstruction for the cases where the fractional order is less than 1 and greater than 1. The simulations and experiments show that the network framework can implement the fractional Fourier transform reconstructions of the intensity objects and phase objects for different fraction orders, in which only 2000 iterations are needed. The experimental results show that the similarity between the reconstructed image and the original image, i.e. the number of normalized correlation coefficient, can reach 99.7%. Therefore, our work offers an efficient scheme for functional Fourier transform reconstruction with physics-enhanced deep neutral network.
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Keywords:
- fractional Fourier transform /
- deep learning /
- neural network /
- phase retrieval
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图 1 透镜的傅里叶变换光路图
Figure 1. Fourier transform optical path diagram of lens.
图 2 网络框架示意图
Figure 2. Schematic diagram of network framework.
图 3 神经网络原理图
Figure 3. Principle diagram of neural network.
图 4 加载在相位的测试图像 (a) Lenna; (b)猫
Figure 4. Test image loaded in phase: (a) Lenna; (b) cat.
图 5 不同阶数p的光强图像及其相位重建(p < 1)
Figure 5. Light intensity images of different orders p and their phase reconstruction (p < 1).
图 6 不同阶数p的光强图像及其相位重建(p > 1)
Figure 6. Light intensity images of different orders p and their phase reconstruction (p > 1).
图 7 分数傅里叶变换实验光路图
Figure 7. Fractional Fourier transform experimental optical path diagram.
图 8 (a)光路输出的强度图; (b)原始的强度图像; (c)重建的强度图像
Figure 8. (a) Intensity map of the optical path output; (b) the original intensity image; (c) reconstructed intensity image.
图 9 (a)光路输出的强度图; (b)加载在相位的原始图像; (c)重建的相位图像
Figure 9. (a) Intensity map of the optical path output; (b) load the original image in phase; (c) reconstructed phase image.
图 10 重建强度和原始强度间的MSE随迭代次数的变化
Figure 10. Variation of MSE between reconstructed intensity and original intensity with the number of iterations.
图 11 重建相位和原始相位间的MSE随迭代次数的变化
Figure 11. Variation of MSE between reconstructed phase and original phase with the number of iterations.
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