Coherent X-ray diffraction imaging (CDI) is a powerful technique for characterizing the 3D internal structure of single particles, particularly Bragg CDI for 3D strain imaging. The key challenge in CDI is phase retrieval, the recovery of lost phase information from measured diffraction signals. Iterative methods, such as HIO, DM, and RAAR, are commonly used but struggle with noise and local minima, leading to ambiguous solutions and requiring extensive iterations and parameter tuning. Recent deep-learning-based methods offer advantages in speed but often rely on large training datasets, which are difficult to obtain experimentally. This research addresses these limitations by developing a 3D convolutional neural network (CNN)-based approach for CDI phase retrieval.

Existing iterative phase retrieval methods for CDI, while theoretically capable of unique inversion with sufficient oversampling, suffer from noise sensitivity and convergence issues in practice. These methods typically involve thousands of iterations and require expert tuning. Deep learning approaches have shown promise in accelerating 2D phase retrieval, but most utilize supervised learning, requiring substantial training data. Adaptive ML methods for 3D phase retrieval exist, but these also face data limitations. This work builds upon these existing methods, attempting to overcome the limitations of both iterative and supervised learning-based approaches.

The researchers developed a 3D CNN model with an encoder-decoder architecture. The input is the amplitude of a 3D coherent X-ray diffraction pattern, and the output is the real-space amplitude and phase images. The network uses 3D convolutional, max-pooling, and upsampling layers, with leaky rectified linear units (LRLU) for activation. Two approaches were used: supervised learning and unsupervised transfer learning. For supervised learning, a dataset of 30,000 simulated 3D diffraction patterns (from superellipsoid shapes with Gaussian-correlated phases) was used to train the model. The loss function combined relative root mean square error and modified Pearson correlation coefficient. For unsupervised transfer learning, the pre-trained model was further refined using a loss function minimizing the difference between the calculated diffraction intensity from the predicted particle and the measured intensity. The model was also tested with random initialization (without transfer learning) to evaluate the importance of pre-training. Experimental data from Bragg CDI experiments on SrTiO3, BaTiO3, Pd, and Au nanocrystals were used to validate the model's performance against traditional iterative methods.

The supervised learning approach yielded a highly accurate 3D CNN model capable of fast diffraction pattern inversion (~9 ms). The unsupervised transfer learning further improved reconstruction quality, achieving accuracy comparable to the best iterative algorithms. Crucially, the model performed well even without pre-training (using random initialization), demonstrating the ability of the network to directly retrieve phases. While pre-training sped up convergence, the final reconstruction quality was similar for both pre-trained and randomly initialized models. The model showed excellent agreement between experimental and calculated diffraction patterns for various nanocrystals. A comparison of reproducibility between the untrained CNN model and a conventional iterative method showed similar statistical errors, but the CNN model produced sharper and more well-defined features.

The results demonstrate a powerful ML-based approach for 3D CDI phase retrieval. The combination of supervised and unsupervised learning offers a robust solution, overcoming limitations of traditional methods. The ability to achieve high accuracy with or without pre-training is particularly significant, addressing the challenge of limited experimental training data. The flexibility of the self-defined loss function enables the model to be more robust to lower-quality data than iterative methods. The improved image quality compared to iterative algorithms highlights the potential of the method for resolving fine structural details.

This work presents a comprehensive ML-based approach for 3D CDI that provides accurate, rapid phase retrieval. The unsupervised learning approach allows for ab-initio phase retrieval even without extensive training data. Future research could explore more sophisticated loss functions and investigate the application of this method to other phase retrieval problems.

The simulation used idealized particle shapes (superellipsoids) and phase distributions (Gaussian correlated). While this allowed for generating a large training dataset, real-world particles may exhibit more complex morphologies and phase variations. The study focused on specific materials; further testing with a wider range of materials is needed to fully assess generalizability. The computational cost of unsupervised learning, although faster than many iterations of conventional methods, could be a limitation for extremely large datasets.