Robust contrast-transfer-function phase retrieval via flexible deep learning networks

On the use of deep learning for phase recovery

Phase recovery (PR) refers to calculating the phase of the light field from its intensity measurements. As exemplified from quantitative phase imaging and coherent diffraction imaging to adaptive optics, PR is essential for reconstructing the refractive index distribution or topography of an object and correcting the aberration of an imaging system. In recent years, deep learning (DL), often implemented through deep neural networks, has provided unprecedented support for computational imaging, leading to more efficient solutions for various PR problems. In this review, we first briefly introduce conventional methods for PR. Then, we review how DL provides support for PR from the following three stages, namely, pre-processing, in-processing, and post-processing. We also review how DL is used in phase image processing. Finally, we summarize the work in DL for PR and provide an outlook on how to better use DL to improve the reliability and efficiency of PR. Furthermore, we present a live-updating resource ( https://github.com/kqwang/phase-recovery ) for readers to learn more about PR.

Deep Gauss-Newton for phase retrieval

We propose the deep Gauss-Newton (DGN) algorithm. The DGN allows one to take into account the knowledge of the forward model in a deep neural network by unrolling a Gauss-Newton optimization method. No regularization or step size needs to be chosen; they are learned through convolutional neural networks. The proposed algorithm does not require an initial reconstruction and is able to retrieve simultaneously the phase and absorption from a single-distance diffraction pattern. The DGN method was applied to both simulated and experimental data and permitted large improvements of the reconstruction error and of the resolution compared with a state-of-the-art iterative method and another neural-network-based reconstruction algorithm.

Physics-informed neural network for phase imaging based on transport of intensity equation

Non-interferometric quantitative phase imaging based on Transport of Intensity Equation (TIE) has been widely used in bio-medical imaging. However, analytic TIE phase retrieval is prone to low-spatial frequency noise amplification, which is caused by the illposedness of inversion at the origin of the spectrum. There are also retrieval ambiguities resulting from the lack of sensitivity to the curl component of the Poynting vector occurring with strong absorption. Here, we establish a physics-informed neural network (PINN) to address these issues, by integrating the forward and inverse physics models into a cascaded deep neural network. We demonstrate that the proposed PINN is efficiently trained using a small set of sample data, enabling the conversion of noise-corrupted 2-shot TIE phase retrievals to high quality phase images under partially coherent LED illumination. The efficacy of the proposed approach is demonstrated by both simulation using a standard image database and experiment using human buccal epitehlial cells. In particular, high image quality (SSIM = 0.919) is achieved experimentally using a reduced size of labeled data (140 image pairs). We discuss the robustness of the proposed approach against insufficient training data, and demonstrate that the parallel architecture of PINN is efficient for transfer learning.

SiSPRNet: end-to-end learning for single-shot phase retrieval

With the success of deep learning methods in many image processing tasks, deep learning approaches have also been introduced to the phase retrieval problem recently. These approaches are different from the traditional iterative optimization methods in that they usually require only one intensity measurement and can reconstruct phase images in real-time. However, because of tremendous domain discrepancy, the quality of the reconstructed images given by these approaches still has much room to improve to meet the general application requirements. In this paper, we design a novel deep neural network structure named SiSPRNet for phase retrieval based on a single Fourier intensity measurement. To effectively utilize the spectral information of the measurements, we propose a new feature extraction unit using the Multi-Layer Perceptron (MLP) as the front end. It allows all pixels of the input intensity image to be considered together for exploring their global representation. The size of the MLP is carefully designed to facilitate the extraction of the representative features while reducing noises and outliers. A dropout layer is also equipped to mitigate the possible overfitting problem in training the MLP. To promote the global correlation in the reconstructed images, a self-attention mechanism is introduced to the Up-sampling and Reconstruction (UR) blocks of the proposed SiSPRNet. These UR blocks are inserted into a residual learning structure to prevent the weak information flow and vanishing gradient problems due to their complex layer structure. Extensive evaluations of the proposed model are performed using different testing datasets of phase-only images and images with linearly related magnitude and phase. Experiments were conducted on an optical experimentation platform (with defocusing to reduce the saturation problem) to understand the performance of different deep learning methods when working in a practical environment. The results demonstrate that the proposed approach consistently outperforms other deep learning methods in single-shot maskless phase retrieval. The source codes of the proposed method have been released in Github [see references].

Efficient sub-pixel convolutional neural network for terahertz image super-resolution

Terahertz waves are electromagnetic waves located at 0.1-10 THz, and terahertz imaging technology can be applied to security inspection, biomedicine, non-destructive testing of materials, and other fields. At present, terahertz images have unclear data and rough edges. Therefore, improving the resolution of terahertz images is one of the current hot research topics. This paper proposes an efficient terahertz image super-resolution model, which is used to extract low-resolution (LR) image features and learn the mapping of LR images to high-resolution (HR) images, and then introduce an attention mechanism to let the network pay attention to more information features. Finally, we use sub-pixel convolution to learn a set of scaling filters to upgrade the final LR feature map to an HR output, which not only reduces the model complexity, but also improves the quality of the terahertz image. The resolution reaches 31.67 db on the peak signal-to-noise ratio (PSNR) index and 0.86 on the structural similarity (SSIM) index. Experiments show that the efficient sub-pixel convolutional neural network used in this article achieves better accuracy and visual improvement compared with other terahertz image super-resolution algorithms.

Mixed scale dense convolutional networks for x-ray phase contrast imaging

X-ray in-line phase contrast imaging relies on the measurement of Fresnel diffraction intensity patterns due to the phase shift and the attenuation induced by the object. The recovery of phase and attenuation from one or several diffraction patterns is a nonlinear ill-posed inverse problem. In this work, we propose supervised learning approaches using mixed scale dense (MS-D) convolutional neural networks to simultaneously retrieve the phase and the attenuation from x-ray phase contrast images. This network architecture uses dilated convolutions to capture features at different image scales and densely connects all feature maps. The long range information in images becomes quickly available, and greater receptive field size can be obtained without losing resolution. This network architecture seems to account for the effect of the Fresnel operator very efficiently. We train the networks using simulated data of objects consisting of either homogeneous components, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, or heterogeneous components, consisting of various materials. We also train the networks in the image domain by applying a simple initial reconstruction using the adjoint of the Fréchet derivative. We compare the results obtained with the MS-D network to reconstructions using U-Net, another popular network architecture, as well as to reconstructions using the contrast transfer function method, a direct phase and attenuation retrieval method based on linearization of the direct problem. The networks are evaluated using simulated noisy data as well as images acquired at NanoMAX (MAX IV, Lund, Sweden). In all cases, large improvements of the reconstruction errors are obtained on simulated data compared to the linearized method. Moreover, on experimental data, the networks improve the reconstruction quantitatively, improving the low-frequency behavior and the resolution.

Phase retrieval with physics informed zero-shot network

Phase can be reliably estimated from a single diffracted intensity image if faithful prior information about the object is available. Examples include amplitude bounds, object support, sparsity in the spatial or transform domain, deep image prior, and the prior learned from labeled datasets by a deep neural network. Deep learning facilitates state-of-the-art reconstruction quality but requires a large labeled dataset (ground truth measurement pair acquired in the same experimental conditions) for training. To alleviate this data requirement problem, this Letter proposes a zero-shot learning method. The Letter demonstrates that the object prior learned by a deep neural network while being trained for a denoising task can also be utilized for phase retrieval if the diffraction physics is effectively enforced on the network output. The Letter additionally demonstrates that the incorporation of total variation in the proposed zero-shot framework facilitates reconstruction of similar quality in less time (e.g., ∼9 fold, for a test reported in this Letter).

Dynamic coherent diffractive imaging with a physics-driven untrained learning method

Reconstruction of a complex field from one single diffraction measurement remains a challenging task among the community of coherent diffraction imaging (CDI). Conventional iterative algorithms are time-consuming and struggle to converge to a feasible solution because of the inherent ambiguities. Recently, deep-learning-based methods have shown considerable success in computational imaging, but they require large amounts of training data that in many cases are difficult to obtain. Here, we introduce a physics-driven untrained learning method, termed Deep CDI, which addresses the above problem and can image a dynamic process with high confidence and fast reconstruction. Without any labeled data for pretraining, the Deep CDI can reconstruct a complex-valued object from a single diffraction pattern by combining a conventional artificial neural network with a real-world physical imaging model. To our knowledge, we are the first to demonstrate that the support region constraint, which is widely used in the iteration-algorithm-based method, can be utilized for loss calculation. The loss calculated from support constraint and free propagation constraint are summed up to optimize the network's weights. As a proof of principle, numerical simulations and optical experiments on a static sample are carried out to demonstrate the feasibility of our method. We then continuously collect 3600 diffraction patterns and demonstrate that our method can predict the dynamic process with an average reconstruction speed of 228 frames per second (FPS) using only a fraction of the diffraction data to train the weights.

DNN-FZA camera: a deep learning approach toward broadband FZA lensless imaging

In mask-based lensless imaging, iterative reconstruction methods based on the geometric optics model produce artifacts and are computationally expensive. We present a prototype of a lensless camera that uses a deep neural network (DNN) to realize rapid reconstruction for Fresnel zone aperture (FZA) imaging. A deep back-projection network (DBPN) is connected behind a U-Net providing an error feedback mechanism, which realizes the self-correction of features to recover the image detail. A diffraction model generates the training data under conditions of broadband incoherent imaging. In the reconstructed results, blur caused by diffraction is shown to have been ameliorated, while the computing time is 2 orders of magnitude faster than the traditional iterative image reconstruction algorithms. This strategy could drastically reduce the design and assembly costs of cameras, paving the way for integration of portable sensors and systems.

Robust contrast-transfer-function phase retrieval via flexible deep learning networks: publisher's note

This publisher's note contains corrections to Opt. Lett.44, 5141 (2019)OPLEDP0146-959210.1364/OL.44.005141.