The quality of inverse problem solutions obtained through deep learning is limited by the nature of the priors learned from examples presented during the training phase. Particularly in the case of quantitative phase retrieval, spatial frequencies that are underrepresented in the training database, most often at the high band, tend to be suppressed in the reconstruction. Ad hoc solutions have been proposed, such as pre-amplifying the high spatial frequencies in the examples; however, while that strategy improves the resolution, it also leads to high-frequency artefacts, as well as low-frequency distortions in the reconstructions. Here, we present a new approach that learns separately how to handle the two frequency bands, low and high, and learns how to synthesize these two bands into full-band reconstructions. We show that this “learning to synthesize” (LS) method yields phase reconstructions of high spatial resolution and without artefacts and that it is resilient to high-noise conditions, e.g., in the case of very low photon flux. In addition to the problem of quantitative phase retrieval, the LS method is applicable, in principle, to any inverse problem where the forward operator treats different frequency bands unevenly, i.e., is ill-posed.

Deep learning (DL) has been applied extensively in many computational imaging problems, often leading to superior performance over traditional iterative approaches. However, two important questions remain largely unanswered: first, how well can the trained neural network generalize to objects very different from the ones in training? This is particularly important in practice, since large-scale annotated examples similar to those of interest are often not available during training. Second, has the trained neural network learnt the underlying (inverse) physics model, or has it merely done something trivial, such as memorizing the examples or point-wise pattern matching? This pertains to the interpretability of machine-learning based algorithms. In this work, we use the Phase Extraction Neural Network (PhENN) [Optica 4, 1117-1125 (2017)], a deep neural network (DNN) for quantitative phase retrieval in a lensless phase imaging system as the standard platform and show that the two questions are related and share a common crux: the choice of the training examples. Moreover, we connect the strength of the regularization effect imposed by a training set to the training process with the Shannon entropy of images in the dataset. That is, the higher the entropy of the training images, the weaker the regularization effect can be imposed. We also discover that weaker regularization effect leads to better learning of the underlying propagation model, i.e. the weak object transfer function, applicable for weakly scattering objects under the weak object approximation. Finally, simulation and experimental results show that better cross-domain generalization performance can be achieved if DNN is trained on a higher-entropy database, e.g. the ImageNet, than if the same DNN is trained on a lower-entropy database, e.g. MNIST, as the former allows the underlying physics model be learned better than the latter.

Imaging with low-dose light is of importance in various fields, especially when minimizing radiation-induced damage onto samples is desirable. The raw image captured at the detector plane is then predominantly a Poisson random process with Gaussian noise added due to the quantum nature of photo-electric conversion. Under such noisy conditions, highly ill-posed problems such as phase retrieval from raw intensity measurements become prone to strong artifacts in the reconstructions; a situation that deep neural networks (DNNs) have already been shown to be useful at improving. Here, we demonstrate that random phase modulation on the optical field, also known as coherent modulation imaging (CMI), in conjunction with the phase extraction neural network (PhENN) and a Gerchberg-Saxton-Fienup (GSF) approximant, further improves resilience to noise of the phase-from-intensity imaging problem. We offer design guidelines for implementing the CMI hardware with the proposed computational reconstruction scheme and quantify reconstruction improvement as function of photon count.

Sensing and correction of low-order wavefront aberrations is critical for high-contrast astronomical imaging. State of the art coronagraph systems typically use image-based sensing methods that exploit the rejected on-axis light, such as Lyot-based low order wavefront sensors (LLOWFS); these methods rely on linear least-squares fitting to recover Zernike basis coefficients from intensity data. However, the dynamic range of linear recovery is limited. We propose the use of deep neural networks with residual learning techniques for non-linear wavefront sensing. The deep residual learning approach extends the usable range of the LLOWFS sensor by more than an order of magnitude compared to the conventional methods, and can improve closed-loop control of systems with large initial wavefront error. We demonstrate that the deep learning approach performs well even in low-photon regimes common to coronagraphic imaging of exoplanets.

Limited-angle tomography of an interior volume is a challenging, highly ill-posed problem with practical implications in medical and biological imaging, manufacturing, automation, and environmental and food security. Regularizing priors are necessary to reduce artifacts by improving the condition of such problems. Recently, it was shown that one effective way to learn the priors for strongly scattering yet highly structured 3D objects, e.g. layered and Manhattan, is by a static neural network [Goy et al. Proc. Natl. Acad. Sci. 116, 19848–19856 (2019)]. Here, we present a radically different approach where the collection of raw images from multiple angles is viewed analogously to a dynamical system driven by the object-dependent forward scattering operator. The sequence index in the angle of illumination plays the role of discrete time in the dynamical system analogy. Thus, the imaging problem turns into a problem of nonlinear system identification, which also suggests dynamical learning as a better fit to regularize the reconstructions. We devised a Recurrent Neural Network (RNN) architecture with a novel Separable-Convolution Gated Recurrent Unit (SC-GRU) as the fundamental building block. Through a comprehensive comparison of several quantitative metrics, we show that the dynamic method is suitable for a generic interior-volumetric reconstruction under a limited-angle scheme. We show that this approach accurately reconstructs volume interiors under two conditions: weak scattering, when the Radon transform approximation is applicable and the forward operator well defined; and strong scattering, which is nonlinear with respect to the 3D refractive index distribution and includes uncertainty in the forward operator.