Ptychography using Blind Multi-Mode PMACE

Ptychography is an imaging technique that enables nanometer-scale reconstruction of complex transmittance images by scanning objects with overlapping illumination patterns. However, the illumination function is typically unknown, which presents challenges for reconstruction, especially when using partially coherent light sources. In this paper, we introduce Blind Multi-Mode Projected Multi-Agent Consensus Equilibrium (BM-PMACE) for blind ptychographic reconstruction. We extend the PMACE framework for distributed inverse problems to jointly estimate the complex transmittance image and multiple, unknown, partially coherent probe functions. Importantly, our method maintains local probe estimates to exploit complementary information at multiple probe locations. Our method also incorporates a dynamic strategy for integrating additional probe modes. Through experimental simulations and validations using both synthetic and measured data, we demonstrate that BM-PMACE outperforms existing approaches in reconstruction quality and convergence rate.

Plug-and-Play Priors as a Score-Based Method

Plug-and-play (PnP) methods are extensively used for solving imaging inverse problems by integrating physical measurement models with pre-trained deep denoisers as priors. Score-based diffusion models (SBMs) have recently emerged as a powerful framework for image generation by training deep denoisers to represent the score of the image prior. While both PnP and SBMs use deep denoisers, the score-based nature of PnP is unexplored in the literature due to its distinct origins rooted in proximal optimization. This letter introduces a novel view of PnP as a score-based method, a perspective that enables the re-use of powerful SBMs within classical PnP algorithms without retraining. We present a set of mathematical relationships for adapting popular SBMs as priors within PnP. We show that this approach enables a direct comparison between PnP and SBM-based reconstruction methods using the same neural network as the prior. Code is available at https://github.com/wustl-cig/score_pnp.

Closed-Form Approximation of the Total Variation Proximal Operator

Total variation (TV) is a widely used function for regularizing imaging inverse problems that is particularly appropriate for images whose underlying structure is piecewise constant. TV regularized optimization problems are typically solved using proximal methods, but the way in which they are applied is constrained by the absence of a closed-form expression for the proximal operator of the TV function. A closed-form approximation of the TV proximal operator has previously been proposed, but its accuracy was not theoretically explored in detail. We address this gap by making several new theoretical contributions, proving that the approximation leads to a proximal operator of some convex function, that it always decreases the TV function, and that its error can be fully characterized and controlled with its scaling parameter. We experimentally validate our theoretical results on image denoising and sparse-view computed tomography (CT) image reconstruction.

Random Walks with Tweedie: A Unified Framework for Diffusion Models

We present a simple template for designing generative diffusion model algorithms based on an interpretation of diffusion sampling as a sequence of random walks. Score-based diffusion models are widely used to generate high-quality images. Diffusion models have also been shown to yield state-of-the-art performance in many inverse problems. While these algorithms are often surprisingly simple, the theory behind them is not, and multiple complex theoretical justifications exist in the literature. Here, we provide a simple and largely self-contained theoretical justification for score-based-diffusion models that avoids using the theory of Markov chains or reverse diffusion, instead centering the theory of random walks and Tweedie's formula. This approach leads to unified algorithmic templates for network training and sampling. In particular, these templates cleanly separate training from sampling, e.g., the noise schedule used during training need not match the one used during sampling. We show that several existing diffusion models correspond to particular choices within this template and demonstrate that other, more straightforward algorithmic choices lead to effective diffusion models. The proposed framework has the added benefit of enabling conditional sampling without any likelihood approximation.

Physics and Deep Learning in Computational Wave Imaging

Computational wave imaging (CWI) extracts hidden structure and physical properties of a volume of material by analyzing wave signals that traverse that volume. Applications include seismic exploration of the Earth's subsurface, acoustic imaging and non-destructive testing in material science, and ultrasound computed tomography in medicine. Current approaches for solving CWI problems can be divided into two categories: those rooted in traditional physics, and those based on deep learning. Physics-based methods stand out for their ability to provide high-resolution and quantitatively accurate estimates of acoustic properties within the medium. However, they can be computationally intensive and are susceptible to ill-posedness and nonconvexity typical of CWI problems. Machine learning-based computational methods have recently emerged, offering a different perspective to address these challenges. Diverse scientific communities have independently pursued the integration of deep learning in CWI. This review delves into how contemporary scientific machine-learning (ML) techniques, and deep neural networks in particular, have been harnessed to tackle CWI problems. We present a structured framework that consolidates existing research spanning multiple domains, including computational imaging, wave physics, and data science. This study concludes with important lessons learned from existing ML-based methods and identifies technical hurdles and emerging trends through a systematic analysis of the extensive literature on this topic.

PtychoDV: Vision Transformer-Based Deep Unrolling Network for Ptychographic Image Reconstruction

Ptychography is an imaging technique that captures multiple overlapping snapshots of a sample, illuminated coherently by a moving localized probe. The image recovery from ptychographic data is generally achieved via an iterative algorithm that solves a nonlinear phase-field problem derived from measured diffraction patterns. However, these approaches have high computational cost. In this paper, we introduce PtychoDV, a novel deep model-based network designed for efficient, high-quality ptychographic image reconstruction. PtychoDV comprises a vision transformer that generates an initial image from the set of raw measurements, taking into consideration their mutual correlations. This is followed by a deep unrolling network that refines the initial image using learnable convolutional priors and the ptychography measurement model. Experimental results on simulated data demonstrate that PtychoDV is capable of outperforming existing deep learning methods for this problem, and significantly reduces computational cost compared to iterative methodologies, while maintaining competitive performance.

Learned Full Waveform Inversion Incorporating Task Information for Ultrasound Computed Tomography

Ultrasound computed tomography (USCT) is an emerging imaging modality that holds great promise for breast imaging. Full-waveform inversion (FWI)-based image reconstruction methods incorporate accurate wave physics to produce high spatial resolution quantitative images of speed of sound or other acoustic properties of the breast tissues from USCT measurement data. However, the high computational cost of FWI reconstruction represents a significant burden for its widespread application in a clinical setting. The research reported here investigates the use of a convolutional neural network (CNN) to learn a mapping from USCT waveform data to speed of sound estimates. The CNN was trained using a supervised approach with a task-informed loss function aiming at preserving features of the image that are relevant to the detection of lesions. A large set of anatomically and physiologically realistic numerical breast phantoms (NBPs) and corresponding simulated USCT measurements was employed during training. Once trained, the CNN can perform real-time FWI image reconstruction from USCT waveform data. The performance of the proposed method was assessed and compared against FWI using a hold-out sample of 41 NBPs and corresponding USCT data. Accuracy was measured using relative mean square error (RMSE), structural self-similarity index measure (SSIM), and lesion detection performance (DICE score). This numerical experiment demonstrates that a supervised learning model can achieve accuracy comparable to FWI in terms of RMSE and SSIM, and better performance in terms of task performance, while significantly reducing computational time.

Projected Multi-Agent Consensus Equilibrium (PMACE) for Distributed Reconstruction with Application to Ptychography

Multi-Agent Consensus Equilibrium (MACE) formulates an inverse imaging problem as a balance among multiple update agents such as data-fitting terms and denoisers. However, each such agent operates on a separate copy of the full image, leading to redundant memory use and slow convergence when each agent affects only a small subset of the full image. In this paper, we extend MACE to Projected Multi-Agent Consensus Equilibrium (PMACE), in which each agent updates only a projected component of the full image, thus greatly reducing memory use for some applications.We describe PMACE in terms of an equilibrium problem and an equivalent fixed point problem and show that in most cases the PMACE equilibrium is not the solution of an optimization problem. To demonstrate the value of PMACE, we apply it to the problem of ptychography, in which a sample is reconstructed from the diffraction patterns resulting from coherent X-ray illumination at multiple overlapping spots. In our PMACE formulation, each spot corresponds to a separate data-fitting agent, with the final solution found as an equilibrium among all the agents. Our results demonstrate that the PMACE reconstruction algorithm generates more accurate reconstructions at a lower computational cost than existing ptychography algorithms when the spots are sparsely sampled.

Deep Equilibrium Learning of Explicit Regularizers for Imaging Inverse Problems

There has been significant recent interest in the use of deep learning for regularizing imaging inverse problems. Most work in the area has focused on regularization imposed implicitly by convolutional neural networks (CNNs) pre-trained for image reconstruction. In this work, we follow an alternative line of work based on learning explicit regularization functionals that promote preferred solutions. We develop the Explicit Learned Deep Equilibrium Regularizer (ELDER) method for learning explicit regularizers that minimize a mean-squared error (MSE) metric. ELDER is based on a regularization functional parameterized by a CNN and a deep equilibrium learning (DEQ) method for training the functional to be MSE-optimal at the fixed points of the reconstruction algorithm. The explicit regularizer enables ELDER to directly inherit fundamental convergence results from optimization theory. On the other hand, DEQ training enables ELDER to improve over existing explicit regularizers without prohibitive memory complexity during training. We use ELDER to train several approaches to parameterizing explicit regularizers and test their performance on three distinct imaging inverse problems. Our results show that ELDER can greatly improve the quality of explicit regularizers compared to existing methods, and show that learning explicit regularizers does not compromise performance relative to methods based on implicit regularization.

TRINIDI: Time-of-Flight Resonance Imaging with Neutrons for Isotopic Density Inferenc

Accurate reconstruction of 2D and 3D isotope densities is a desired capability with great potential impact in applications such as evaluation and development of next-generation nuclear fuels. Neutron time-of-flight (TOF) resonance imaging offers a potential approach by exploiting the characteristic neutron adsorption spectra of each isotope. However, it is a major challenge to compute quantitatively accurate images due to a variety of confounding effects such as severe Poisson noise, background scatter, beam non-uniformity, absorption non-linearity, and extended source pulse duration. We present the TRINIDI algorithm which is based on a two-step process in which we first estimate the neutron flux and background counts, and then reconstruct the areal densities of each isotope and pixel. Both components are based on the inversion of a forward model that accounts for the highly non-linear absorption, energy-dependent emission profile, and Poisson noise, while also modeling the substantial spatio-temporal variation of the background and flux. To do this, we formulate the non-linear inverse problem as two optimization problems that are solved in sequence. We demonstrate on both synthetic and measured data that TRINIDI can reconstruct quantitatively accurate 2D views of isotopic areal density that can then be reconstructed into quantitatively accurate 3D volumes of isotopic volumetric density.