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分数傅里叶变换近年来被越来越多地用于光学成像领域, 分数傅里叶变换情况下的相位恢复被广泛研究. 另外, 深度学习已经被广泛运用于各种光学计算成像中, 但由于在采集数据时光路本身的环境不会产生太大变化, 往往难以取得质量和数量足够的标记数据来训练, 并且训练时间也比较长. 近年来, 基于物理规律驱动的未训练神经网络用于计算成像的方法逐渐引起了研究人员的兴趣. 本文将这种未训练的神经网络深度学习应用于分数傅里叶变换成像, 通过将神经网络和光学模型相结合的方式完成分数傅里叶变换的相位恢复. 数值仿真和光学实验证明, 仅需2000次迭代, 该网络框架就能完成不同阶数的分数傅里叶变换重建, 包括强度物体和相位物体. 实验结果表明, 重建图像及原始图像的相似性, 即归一化互相关系数可达99.7%. 因此, 本工作框架为分数傅里叶变换的重建提供了一种新方法.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 4 加载在相位的测试图像 (a) Lenna; (b)猫
Fig. 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)
Fig. 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)
Fig. 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
Fig. 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
Fig. 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
Fig. 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
Fig. 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
Fig. 11. Variation of MSE between reconstructed phase and original phase with the number of iterations.
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