Pulse-echo ultrasound attenuation tomography

Objective.We present the first fully two-dimensional attenuation imaging technique developed for pulse-echo ultrasound systems. Unlike state-of-the-art techniques, which use line-by-line acquisitions, our method uses steered emissions to constrain attenuation values at each location with multiple crossing wave paths, essential to resolve the spatial variations of this tissue property.Approach.At every location, we compute normalized cross-correlations between the beamformed images that are obtained from emissions at different steering angles. We demonstrate that their log-amplitudes provide the changes between attenuation-induced amplitude losses undergone by the different incident waves. This allows us to formulate a linear tomographic problem, which we efficiently solve via a Tikhonov-regularized least-squares approach.Main results.The performance of our tomography technique is first validated in numerical examples and then experimentally demonstrated in custom-made tissue-mimicking phantoms with inclusions of varying size, echogenicity, and attenuation. We show that this technique is particularly good at resolving lateral variations in tissue attenuation and remains accurate in media with varying echogenicity.Significance.Based on a similar principle, this method can be easily combined with computed ultrasound tomography in echo mode for speed-of-sound imaging, paving the way towards a multi-modal ultrasound tomography framework characterizing multiple acoustic tissue properties simultaneously.

Monocular reconstruction of shapes of natural objects from orthographic and perspective images

Human subjects were tested in perception of shapes of 3D objects. The subjects reconstructed 3D shapes by viewing orthographic and perspective images. Perception of natural shapes was very close to veridical and was clearly better than perception of random symmetrical polyhedra. Viewing perspective images led to only slightly better performance than viewing orthographic images. In order to account for subjects' performance, we elaborated the previous computational models of 3D shape reconstruction. The previous models used as constraints mirror-symmetry and 3D compactness. The critical additional constraint was the use of a secondary mirror-symmetry that exists in most natural shapes. It is known that two planes of mirror symmetry are sufficient for a unique and veridical shape reconstruction. We also generalized the model so that it applies to both orthographic and perspective images. The results of our experiment suggest that the human visual system uses two planes of symmetry in addition to two forms of 3D compactness. Performance of the new model was highly correlated with subjects' performance with both orthographic and perspective images, which supports the claim that the most important 3D shape constraints that are used by the human visual system have been identified.

X-Ray Calc 3: improved software for simulation and inverse problem solving for X-ray reflectivity

This work introduces X-Ray Calc (XRC), an open-source software package designed to simulate X-ray reflectivity (XRR) and address the inverse problem of reconstructing film structures on the basis of measured XRR curves. XRC features a user-friendly graphical interface that facilitates interactive simulation and reconstruction. The software employs a recursive approach based on the Fresnel equations to calculate XRR and incorporates specialized tools for modeling periodic multilayer structures. This article presents the latest version of the X-Ray Calc software (XRC3), with notable improvements. These enhancements encompass an automatic fitting capability for XRR curves utilizing a modified flight particle swarm optimization algorithm. A novel cost function was also developed specifically for fitting XRR curves of periodic structures. Furthermore, the overall user experience has been enhanced by developing a new single-window interface.

Neural network analysis of neutron and X-ray reflectivity data incorporating prior knowledge

Due to the ambiguity related to the lack of phase information, determining the physical parameters of multilayer thin films from measured neutron and X-ray reflectivity curves is, on a fundamental level, an underdetermined inverse problem. This ambiguity poses limitations on standard neural networks, constraining the range and number of considered parameters in previous machine learning solutions. To overcome this challenge, a novel training procedure has been designed which incorporates dynamic prior boundaries for each physical parameter as additional inputs to the neural network. In this manner, the neural network can be trained simultaneously on all well-posed subintervals of a larger parameter space in which the inverse problem is underdetermined. During inference, users can flexibly input their own prior knowledge about the physical system to constrain the neural network prediction to distinct target subintervals in the parameter space. The effectiveness of the method is demonstrated in various scenarios, including multilayer structures with a box model parameterization and a physics-inspired special parameterization of the scattering length density profile for a multilayer structure. In contrast to previous methods, this approach scales favourably when increasing the complexity of the inverse problem, working properly even for a five-layer multilayer model and a periodic multilayer model with up to 17 open parameters.

Three-dimensional spatio-angular fluorescence microscopy with a polarized dual-view inverted selective-plane illumination microscope (pol-diSPIM)

Polarized fluorescence microscopy is a valuable tool for measuring molecular orientations, but techniques for recovering three-dimensional orientations and positions of fluorescent ensembles are limited. We report a polarized dual-view light-sheet system for determining the three-dimensional orientations and diffraction-limited positions of ensembles of fluorescent dipoles that label biological structures, and we share a set of visualization, histogram, and profiling tools for interpreting these positions and orientations. We model our samples, their excitation, and their detection using coarse-grained representations we call orientation distribution functions (ODFs). We apply ODFs to create physics-informed models of image formation with spatio-angular point-spread and transfer functions. We use theory and experiment to conclude that light-sheet tilting is a necessary part of our design for recovering all three-dimensional orientations. We use our system to extend known two-dimensional results to three dimensions in FM1-43-labelled giant unilamellar vesicles, fast-scarlet-labelled cellulose in xylem cells, and phalloidin-labelled actin in U2OS cells. Additionally, we observe phalloidin-labelled actin in mouse fibroblasts grown on grids of labelled nanowires and identify correlations between local actin alignment and global cell-scale orientation, indicating cellular coordination across length scales.

Constrained Plug-and-Play Priors for Image Restoration

The Plug-and-Play framework has demonstrated that a denoiser can implicitly serve as the image prior for model-based methods for solving various inverse problems such as image restoration tasks. This characteristic enables the integration of the flexibility of model-based methods with the effectiveness of learning-based denoisers. However, the regularization strength induced by denoisers in the traditional Plug-and-Play framework lacks a physical interpretation, necessitating demanding parameter tuning. This paper addresses this issue by introducing the Constrained Plug-and-Play (CPnP) method, which reformulates the traditional PnP as a constrained optimization problem. In this formulation, the regularization parameter directly corresponds to the amount of noise in the measurements. The solution to the constrained problem is obtained through the design of an efficient method based on the Alternating Direction Method of Multipliers (ADMM). Our experiments demonstrate that CPnP outperforms competing methods in terms of stability and robustness while also achieving competitive performance for image quality.

Robust Reconstruction of the Void Fraction from Noisy Magnetic Flux Density Using Invertible Neural Networks

Electrolysis stands as a pivotal method for environmentally sustainable hydrogen production. However, the formation of gas bubbles during the electrolysis process poses significant challenges by impeding the electrochemical reactions, diminishing cell efficiency, and dramatically increasing energy consumption. Furthermore, the inherent difficulty in detecting these bubbles arises from the non-transparency of the wall of electrolysis cells. Additionally, these gas bubbles induce alterations in the conductivity of the electrolyte, leading to corresponding fluctuations in the magnetic flux density outside of the electrolysis cell, which can be measured by externally placed magnetic sensors. By solving the inverse problem of the Biot-Savart Law, we can estimate the conductivity distribution as well as the void fraction within the cell. In this work, we study different approaches to solve the inverse problem including Invertible Neural Networks (INNs) and Tikhonov regularization. Our experiments demonstrate that INNs are much more robust to solving the inverse problem than Tikhonov regularization when the level of noise in the magnetic flux density measurements is not known or changes over space and time.

Estimation of Circular Statistics in the Presence of Measurement Bias

Circular statistics and Rayleigh tests are important tools for analyzing cyclic events. However, current methods are not robust to significant measurement bias, especially incomplete or otherwise non-uniform sampling. One example is studying 24-cyclicity but having data not recorded uniformly over the full 24-hour cycle. Our objective is to present a robust method to estimate circular statistics and their statistical significance in the presence of incomplete or otherwise non-uniform sampling. Our method is to solve the underlying Fredholm Integral Equation for the more general problem, estimating probability distributions in the context of imperfect measurements, with our circular statistics in the presence of incomplete/non-uniform sampling being one special case. The method is based on linear parameterizations of the underlying distributions. We simulated the estimation error of our approach for several toy examples as well as for a real-world example: analyzing the 24-hour cyclicity of an electrographic biomarker of epileptic tissue controlled for states of vigilance. We also evaluated the accuracy of the Rayleigh test statistic versus the direct simulation of statistical significance. Our method shows a very low estimation error. In the real-world example, the corrected moments had a root mean square error of [Formula: see text]. In contrast, the Rayleigh test statistic overestimated the statistical significance and was thus not reliable. The presented methods thus provide a robust solution to computing circular moments even with incomplete or otherwise non-uniform sampling. Since Rayleigh test statistics cannot be used in this circumstance, direct estimation of significance is the preferable option for estimating statistical significance.

Self-Guided Algorithm for Fast Image Reconstruction in Photo-Magnetic Imaging: Artificial Intelligence-Assisted Approach

Previously, we introduced photomagnetic imaging (PMI) that synergistically utilizes laser light to slightly elevate the tissue temperature and magnetic resonance thermometry (MRT) to measure the induced temperature. The MRT temperature maps are then converted into absorption maps using a dedicated PMI image reconstruction algorithm. In the MRT maps, the presence of abnormalities such as tumors would create a notable high contrast due to their higher hemoglobin levels. In this study, we present a new artificial intelligence-based image reconstruction algorithm that improves the accuracy and spatial resolution of the recovered absorption maps while reducing the recovery time. Technically, a supervised machine learning approach was used to detect and delineate the boundary of tumors directly from the MRT maps based on their temperature contrast to the background. This information was further utilized as a soft functional a priori in the standard PMI algorithm to enhance the absorption recovery. Our new method was evaluated on a tissue-like phantom with two inclusions representing tumors. The reconstructed absorption map showed that the well-trained neural network not only increased the PMI spatial resolution but also improved the accuracy of the recovered absorption to as low as a 2% percentage error, reduced the artifacts by 15%, and accelerated the image reconstruction process approximately 9-fold.

Quantitative photoacoustic tomography: modeling and inverse problems

Significance

Quantitative photoacoustic tomography (QPAT) exploits the photoacoustic effect with the aim of estimating images of clinically relevant quantities related to the tissue's optical absorption. The technique has two aspects: an acoustic part, where the initial acoustic pressure distribution is estimated from measured photoacoustic time-series, and an optical part, where the distributions of the optical parameters are estimated from the initial pressure.

Aim

Our study is focused on the optical part. In particular, computational modeling of light propagation (forward problem) and numerical solution methodologies of the image reconstruction (inverse problem) are discussed.

Approach

The commonly used mathematical models of how light and sound propagate in biological tissue are reviewed. A short overview of how the acoustic inverse problem is usually treated is given. The optical inverse problem and methods for its solution are reviewed. In addition, some limitations of real-life measurements and their effect on the inverse problems are discussed.

Results

An overview of QPAT with a focus on the optical part was given. Computational modeling and inverse problems of QPAT were addressed, and some key challenges were discussed. Furthermore, the developments for tackling these problems were reviewed. Although modeling of light transport is well-understood and there is a well-developed framework of inverse mathematics for approaching the inverse problem of QPAT, there are still challenges in taking these methodologies to practice.

Conclusions

Modeling and inverse problems of QPAT together were discussed. The scope was limited to the optical part, and the acoustic aspects were discussed only to the extent that they relate to the optical aspect.

Deep learning methods for blood flow reconstruction in a vessel with contrast enhanced x-ray computed tomography

The reconstruction of blood velocity in a vessel from contrast enhanced x-ray computed tomography projections is a complex inverse problem. It can be formulated as reconstruction problem with a partial differential equation constraint. A solution can be estimated with the a variational adjoint method and proper orthogonal decomposition (POD) basis. In this work, we investigate new inversion approaches based on PODs coupled with deep learning methods. The effectiveness of the reconstruction methods is shown with simulated realistic stationary blood flows in a vessel. The methods outperform the reduced adjoint method and show large speed-up at the online stage.

Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural networks

In recent years, advances in computing hardware and computational methods have prompted a wealth of activities for solving inverse problems in physics. These problems are often described by systems of partial differential equations (PDEs). The advent of machine learning has reinvigorated the interest in solving inverse problems using neural networks (NNs). In these efforts, the solution of the PDEs is expressed as NNs trained through the minimization of a loss function involving the PDE. Here, we show how to accelerate this approach by five orders of magnitude by deploying, instead of NNs, conventional PDE approximations. The framework of optimizing a discrete loss (ODIL) minimizes a cost function for discrete approximations of the PDEs using gradient-based and Newton's methods. The framework relies on grid-based discretizations of PDEs and inherits their accuracy, convergence, and conservation properties. The implementation of the method is facilitated by adopting machine-learning tools for automatic differentiation. We also propose a multigrid technique to accelerate the convergence of gradient-based optimizers. We present applications to PDE-constrained optimization, optical flow, system identification, and data assimilation. We compare ODIL with the popular method of physics-informed neural networks and show that it outperforms it by several orders of magnitude in computational speed while having better accuracy and convergence rates. We evaluate ODIL on inverse problems involving linear and nonlinear PDEs including the Navier-Stokes equations for flow reconstruction problems. ODIL bridges numerical methods and machine learning and presents a powerful tool for solving challenging, inverse problems across scientific domains.

Predicting Bone Adaptation in Astronauts during and after Spaceflight

A method was previously developed to identify participant-specific parameters in a model of trabecular bone adaptation from longitudinal computed tomography (CT) imaging. In this study, we use these numerical methods to estimate changes in astronaut bone health during the distinct phases of spaceflight and recovery on Earth. Astronauts (N = 16) received high-resolution peripheral quantitative CT (HR-pQCT) scans of their distal tibia prior to launch (L), upon their return from an approximately six-month stay on the international space station (R+0), and after six (R+6) and 12 (R+12) months of recovery. To model trabecular bone adaptation, we determined participant-specific parameters at each time interval and estimated their bone structure at R+0, R+6, and R+12. To assess the fit of our model to this population, we compared static and dynamic bone morphometry as well as the Dice coefficient and symmetric distance at each measurement. In general, modeled and observed static morphometry were highly correlated (R2> 0.94) and statistically different (p < 0.0001) but with errors close to HR-pQCT precision limits. Dynamic morphometry, which captures rates of bone adaptation, was poorly estimated by our model (p < 0.0001). The Dice coefficient and symmetric distance indicated a reasonable local fit between observed and predicted bone volumes. This work applies a general and versatile computational framework to test bone adaptation models. Future work can explore and test increasingly sophisticated models (e.g., those including load or physiological factors) on a participant-specific basis.

Domain-Aware Few-Shot Learning for Optical Coherence Tomography Noise Reduction

Speckle noise has long been an extensively studied problem in medical imaging. In recent years, there have been significant advances in leveraging deep learning methods for noise reduction. Nevertheless, adaptation of supervised learning models to unseen domains remains a challenging problem. Specifically, deep neural networks (DNNs) trained for computational imaging tasks are vulnerable to changes in the acquisition system's physical parameters, such as: sampling space, resolution, and contrast. Even within the same acquisition system, performance degrades across datasets of different biological tissues. In this work, we propose a few-shot supervised learning framework for optical coherence tomography (OCT) noise reduction, that offers high-speed training (of the order of seconds) and requires only a single image, or part of an image, and a corresponding speckle-suppressed ground truth, for training. Furthermore, we formulate the domain shift problem for OCT diverse imaging systems and prove that the output resolution of a despeckling trained model is determined by the source domain resolution. We also provide possible remedies. We propose different practical implementations of our approach, verify and compare their applicability, robustness, and computational efficiency. Our results demonstrate the potential to improve sample complexity, generalization, and time efficiency, for coherent and non-coherent noise reduction via supervised learning models, that can also be leveraged for other real-time computer vision applications.

Learning-Based Seismic Velocity Inversion with Synthetic and Field Data

Building accurate acoustic subsurface velocity models is essential for successful industrial exploration projects. Traditional inversion methods from field-recorded seismograms struggle in regions with complex geology. While deep learning (DL) presents a promising alternative, its robustness using field data in these complicated regions has not been sufficiently explored. In this study, we present a thorough analysis of DL's capability to harness labeled seismograms, whether field-recorded or synthetically generated, for accurate velocity model recovery in a challenging region of the Gulf of Mexico. Our evaluation centers on the impact of training data selection and data augmentation techniques on the DL model's ability to recover velocity profiles. Models trained on field data produced superior results to data obtained using quantitative metrics like Mean Squared Error (MSE), Structural Similarity Index Measure (SSIM), and R2 (R-squared). They also yielded more geologically plausible predictions and sharper geophysical migration images. Conversely, models trained on synthetic data, while less precise, highlighted the potential utility of synthetic training data, especially when labeled field data are scarce. Our work shows that the efficacy of synthetic data-driven models largely depends on bridging the domain gap between training and test data through the use of advanced wave equation solvers and geologic priors. Our results underscore DL's potential to advance velocity model-building workflows in industrial settings using previously labeled field-recorded seismograms. They also highlight the indispensable role of earth scientists' domain expertise in curating synthetic data when field data are lacking.

Reducing the risk of hallucinations with interpretable deep learning models for low-dose CT denoising: comparative performance analysis

Objective.Reducing CT radiation dose is an often proposed measure to enhance patient safety, which, however results in increased image noise, translating into degradation of clinical image quality. Several deep learning methods have been proposed for low-dose CT (LDCT) denoising. The high risks posed by possible hallucinations in clinical images necessitate methods which aid the interpretation of deep learning networks. In this study, we aim to use qualitative reader studies and quantitative radiomics studies to assess the perceived quality, signal preservation and statistical feature preservation of LDCT volumes denoised by deep learning. We aim to compare interpretable deep learning methods with classical deep neural networks in clinical denoising performance.Approach.We conducted an image quality analysis study to assess the image quality of the denoised volumes based on four criteria to assess the perceived image quality. We subsequently conduct a lesion detection/segmentation study to assess the impact of denoising on signal detectability. Finally, a radiomic analysis study was performed to observe the quantitative and statistical similarity of the denoised images to standard dose CT (SDCT) images.Main results.The use of specific deep learning based algorithms generate denoised volumes which are qualitatively inferior to SDCT volumes(p< 0.05). Contrary to previous literature, denoising the volumes did not reduce the accuracy of the segmentation (p> 0.05). The denoised volumes, in most cases, generated radiomics features which were statistically similar to those generated from SDCT volumes (p> 0.05).Significance.Our results show that the denoised volumes have a lower perceived quality than SDCT volumes. Noise and denoising do not significantly affect detectability of the abdominal lesions. Denoised volumes also contain statistically identical features to SDCT volumes.

Elliptic PDE learning is provably data-efficient

Partial differential equations (PDE) learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of three-dimensional uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.

Regularized reconstruction based on joint smoothly clipped absolute deviation regularization and graph manifold learning for fluorescence molecular tomography

Objective.Fluorescence molecular tomography (FMT) is an optical imaging modality that provides high sensitivity and low cost, which can offer the three-dimensional distribution of biomarkers by detecting the fluorescently labeled probe noninvasively. In the field of preclinical cancer diagnosis and treatment, FMT has gained significant traction. Nonetheless, the current FMT reconstruction results suffer from unsatisfactory morphology and location accuracy of the fluorescence distribution, primarily due to the light scattering effect and the ill-posed nature of the inverse problem.Approach.To address these challenges, a regularized reconstruction method based on joint smoothly clipped absolute deviation regularization and graph manifold learning (SCAD-GML) for FMT is presented in this paper. The SCAD-GML approach combines the sparsity of the fluorescent sources with the latent manifold structure of fluorescent source distribution to achieve more accurate and sparse reconstruction results. To obtain the reconstruction results efficiently, the non-convex gradient descent iterative method is employed to solve the established objective function. To assess the performance of the proposed SCAD-GML method, a comprehensive evaluation is conducted through numerical simulation experiments as well asin vivoexperiments.Main results.The results demonstrate that the SCAD-GML method outperforms other methods in terms of both location and shape recovery of fluorescence biomarkers distribution.Siginificance.These findings indicate that the SCAD-GML method has the potential to advance the application of FMT inin vivobiological research.

Stable diverse food webs become more common when interactions are more biologically constrained

Ecologists have long sought to understand how diversity and structure mediate the stability of whole ecosystems. For high-diversity food webs, the interactions between species are typically represented using matrices with randomly chosen interaction strengths. Unfortunately, this procedure tends to produce ecological systems with no underlying equilibrium solution, and so ecological inferences from this approach may be biased by nonbiological outcomes. Using recent computationally efficient methodological advances from metabolic networks, we employ for the first time an inverse approach to diversity-stability research. We compare classical random interaction matrices of realistic food web topology (hereafter the classical model) to feasible, biologically constrained, webs produced using the inverse approach. We show that an energetically constrained feasible model yields a far higher proportion of stable high-diversity webs than the classical random matrix approach. When we examine the energetically constrained interaction strength distributions of these matrix models, we find that although these diverse webs have consistent negative self-regulation, they do not require strong self-regulation to persist. These energetically constrained diverse webs instead show an increasing preponderance of weak interactions that are known to increase local stability. Further examination shows that some of these weak interactions naturally appear to arise in the model food webs from a constraint-generated realistic generalist-specialist trade-off, whereby generalist predators have weaker interactions than more specialized species. Additionally, the inverse technique we present here has enormous promise for understanding the role of the biological structure behind stable high-diversity webs and for linking empirical data to the theory.

Material Data Identification in an Induction Hardening Test Rig with Physics-Informed Neural Networks

Physics-Informed neural networks (PINNs) have demonstrated remarkable performance in solving partial differential equations (PDEs) by incorporating the governing PDEs into the network's loss function during optimization. PINNs have been successfully applied to diverse inverse and forward problems. This study investigates the feasibility of using PINNs for material data identification in an induction hardening test rig. By utilizing temperature sensor data and imposing the heat equation with initial and boundary conditions, thermo-physical material properties, such as specific heat, thermal conductivity, and the heat convection coefficient, were estimated. To validate the effectiveness of the PINNs in material data estimation, benchmark data generated by a finite element model (FEM) of an air-cooled cylindrical sample were used. The accurate identification of the material data using only a limited number of virtual temperature sensor data points was demonstrated. The influence of the sensor positions and measurement noise on the uncertainty of the estimated parameters was examined. The study confirms the robustness and accuracy of this approach in the presence of measurement noise, albeit with lower efficiency, thereby requiring more time to converge. Lastly, the applicability of the presented approach to real measurement data obtained from an air-cooled cylindrical sample heated in an induction heating test rig was discussed. This research contributes to the accurate offline estimation of material data and has implications for optimizing induction heat treatments.

VAEs with structured image covariance applied to compressed sensing MRI

OBJECTIVE

This paper investigates how generative models, trained on ground-truth images, can be used \changes{as} priors for inverse problems, penalizing reconstructions far from images the generator can produce. The aim is that learned regularization will provide complex data-driven priors to inverse problems while still retaining the control and insight of a variational regularization method. Moreover, unsupervised learning, without paired training data, allows the learned regularizer to remain flexible to changes in the forward problem such as noise level, sampling pattern or coil sensitivities in MRI. &#xD;&#xD;Approach: We utilize variational autoencoders (VAEs) that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images. &#xD;&#xD;Main results: We evaluate these novel generative regularizers on retrospectively sub-sampled real-valued MRI measurements from the fastMRI dataset. We compare our proposed learned regularization against other \changes{unlearned regularization approaches and unsupervised and supervised deep learning methods}. &#xD;&#xD;Significance: Our results show that the proposed method is competitive with other state-of-the-art methods and behaves consistently with changing sampling patterns and noise levels.

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

Analysis of relative error in perturbation Monte Carlo simulations of radiative transport

Significance

Perturbation and differential Monte Carlo (pMC/dMC) methods, used in conjunction with nonlinear optimization methods, have been successfully applied to solve inverse problems in diffuse optics. Application of pMC to systems over a large range of optical properties requires optimal "placement" of baseline conventional Monte Carlo (cMC) simulations to minimize the pMC variance. The inability to predict the growth in pMC solution uncertainty with perturbation size limits the application of pMC, especially for multispectral datasets where the variation of optical properties can be substantial.

Aim

We aim to predict the variation of pMC variance with perturbation size without explicit computation of perturbed photon weights. Our proposed method can be used to determine the range of optical properties over which pMC predictions provide sufficient accuracy. This method can be used to specify the optical properties for the reference cMC simulations that pMC utilizes to provide accurate predictions over a desired optical property range.

Approach

We utilize a conventional error propagation methodology to calculate changes in pMC relative error for Monte Carlo simulations. We demonstrate this methodology for spatially resolved diffuse reflectance measurements with ±20% scattering perturbations. We examine the performance of our method for reference simulations spanning a broad range of optical properties relevant for diffuse optical imaging of biological tissues. Our predictions are computed using the variance, covariance, and skewness of the photon weight, path length, and collision distributions generated by the reference simulation.

Results

We find that our methodology performs best when used in conjunction with reference cMC simulations that utilize Russian Roulette (RR) method. Specifically, we demonstrate that for a proximal detector placed immediately adjacent to the source, we can estimate the pMC relative error within 5% of the true value for scattering perturbations in the range of [ - 15 % , + 20 % ] . For a distal detector placed at ∼ 3 transport mean free paths relative to the source, our method provides relative error estimates within 20% for scattering perturbations in the range of [ - 8 % , + 15 % ] . Moreover, reference simulations performed at lower ( μ s ' / μ a ) values showed better performance for both proximal and distal detectors.

Conclusions

These findings indicate that reference simulations utilizing continuous absorption weighting (CAW) with the Russian Roulette method and executed using optical properties with a low ( μ s ' / μ a ) ratio spanning the desired range of μ s values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.

On Krylov methods for large-scale CBCT reconstruction

Krylov subspace methods are a powerful family of iterative solvers for linear systems of equations, which are commonly used for inverse problems due to their intrinsic regularization properties. Moreover, these methods are naturally suited to solve large-scale problems, as they only require matrix-vector products with the system matrix (and its adjoint) to compute approximate solutions, and they display a very fast convergence. &#xD;Even if this class of methods has been widely researched and studied in the numerical linear algebra community, its use in applied medical physics and applied engineering is still very limited. e.g. in realistic large-scale Computed Tomography (CT) problems, and more specifically in Cone Beam CT (CBCT). This work attempts to breach this gap by providing a general framework for the most relevant Krylov subspace methods applied to 3D CT problems, including the most well-known Krylov solvers for non-square systems (CGLS, LSQR, LSMR), possibly in combination with Tikhonov regularization, and methods that incorporate total variation (TV) regularization. This is provided within an open source framework: the Tomographic Iterative GPU-based Reconstruction (TIGRE) toolbox, with the idea of promoting accessibility and reproducibility of the results for the algorithms presented. Finally, numerical results in synthetic and real-world 3D CT applications (medical CBCT and μ-CT datasets) are provided to showcase and compare the different Krylov subspace methods presented in the paper, as well as their suitability for different kinds of problems.&#xD.

Momentum-Net: Fast and Convergent Iterative Neural Network for Inverse Problems

Iterative neural networks (INN) are rapidly gaining attention for solving inverse problems in imaging, image processing, and computer vision. INNs combine regression NNs and an iterative model-based image reconstruction (MBIR) algorithm, often leading to both good generalization capability and outperforming reconstruction quality over existing MBIR optimization models. This paper proposes the first fast and convergent INN architecture, Momentum-Net, by generalizing a block-wise MBIR algorithm that uses momentum and majorizers with regression NNs. For fast MBIR, Momentum-Net uses momentum terms in extrapolation modules, and noniterative MBIR modules at each iteration by using majorizers, where each iteration of Momentum-Net consists of three core modules: image refining, extrapolation, and MBIR. Momentum-Net guarantees convergence to a fixed-point for general differentiable (non)convex MBIR functions (or data-fit terms) and convex feasible sets, under two asymptomatic conditions. To consider data-fit variations across training and testing samples, we also propose a regularization parameter selection scheme based on the "spectral spread" of majorization matrices. Numerical experiments for light-field photography using a focal stack and sparse-view computational tomography demonstrate that, given identical regression NN architectures, Momentum-Net significantly improves MBIR speed and accuracy over several existing INNs; it significantly improves reconstruction quality compared to a state-of-the-art MBIR method in each application.

ENSURE: A General Approach for Unsupervised Training of Deep Image Reconstruction Algorithms

Image reconstruction using deep learning algorithms offers improved reconstruction quality and lower reconstruction time than classical compressed sensing and model-based algorithms. Unfortunately, clean and fully sampled ground-truth data to train the deep networks is often unavailable in several applications, restricting the applicability of the above methods. We introduce a novel metric termed the ENsemble Stein's Unbiased Risk Estimate (ENSURE) framework, which can be used to train deep image reconstruction algorithms without fully sampled and noise-free images. The proposed framework is the generalization of the classical SURE and GSURE formulation to the setting where the images are sampled by different measurement operators, chosen randomly from a set. We evaluate the expectation of the GSURE loss functions over the sampling patterns to obtain the ENSURE loss function. We show that this loss is an unbiased estimate for the true mean-square error, which offers a better alternative to GSURE, which only offers an unbiased estimate for the projected error. Our experiments show that the networks trained with this loss function can offer reconstructions comparable to the supervised setting. While we demonstrate this framework in the context of MR image recovery, the ENSURE framework is generally applicable to arbitrary inverse problems.

Machine Learning Diffuse Optical Tomography Using Extreme Gradient Boosting and Genetic Programming

Diffuse optical tomography (DOT) is a non-invasive method for detecting breast cancer; however, it struggles to produce high-quality images due to the complexity of scattered light and the limitations of traditional image reconstruction algorithms. These algorithms can be affected by boundary conditions and have a low imaging accuracy, a shallow imaging depth, a long computation time, and a high signal-to-noise ratio. However, machine learning can potentially improve the performance of DOT by being better equipped to solve inverse problems, perform regression, classify medical images, and reconstruct biomedical images. In this study, we utilized a machine learning model called "XGBoost" to detect tumors in inhomogeneous breasts and applied a post-processing technique based on genetic programming to improve accuracy. The proposed algorithm was tested using simulated DOT measurements from complex inhomogeneous breasts and evaluated using the cosine similarity metrics and root mean square error loss. The results showed that the use of XGBoost and genetic programming in DOT could lead to more accurate and non-invasive detection of tumors in inhomogeneous breasts compared to traditional methods, with the reconstructed breasts having an average cosine similarity of more than 0.97 ± 0.07 and average root mean square error of around 0.1270 ± 0.0031 compared to the ground truth.

Bayesian MRI reconstruction with joint uncertainty estimation using diffusion models

PURPOSE

We introduce a framework that enables efficient sampling from learned probability distributions for MRI reconstruction.

METHOD

Samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method, different from conventional deep learning-based MRI reconstruction techniques. In addition to the maximum a posteriori estimate for the image, which can be obtained by maximizing the log-likelihood indirectly or directly, the minimum mean square error estimate and uncertainty maps can also be computed from those drawn samples. The data-driven Markov chains are constructed with the score-based generative model learned from a given image database and are independent of the forward operator that is used to model the k-space measurement.

RESULTS

We numerically investigate the framework from these perspectives: (1) the interpretation of the uncertainty of the image reconstructed from undersampled k-space; (2) the effect of the number of noise scales used to train the generative models; (3) using a burn-in phase in MCMC sampling to reduce computation; (4) the comparison to conventional ℓ 1 $$ {\ell}_1 $$ -wavelet regularized reconstruction; (5) the transferability of learned information; and (6) the comparison to fastMRI challenge.

CONCLUSION

A framework is described that connects the diffusion process and advanced generative models with Markov chains. We demonstrate its flexibility in terms of contrasts and sampling patterns using advanced generative priors and the benefits of also quantifying the uncertainty for every pixel.

Implicit Solutions of the Electrical Impedance Tomography Inverse Problem in the Continuous Domain with Deep Neural Networks

Electrical impedance tomography (EIT) is a non-invasive imaging modality used for estimating the conductivity of an object Ω from boundary electrode measurements. In recent years, researchers achieved substantial progress in analytical and numerical methods for the EIT inverse problem. Despite the success, numerical instability is still a major hurdle due to many factors, including the discretization error of the problem. Furthermore, most algorithms with good performance are relatively time consuming and do not allow real-time applications. In our approach, the goal is to separate the unknown conductivity into two regions, namely the region of homogeneous background conductivity and the region of non-homogeneous conductivity. Therefore, we pose and solve the problem of shape reconstruction using machine learning. We propose a novel and simple jet intriguing neural network architecture capable of solving the EIT inverse problem. It addresses previous difficulties, including instability, and is easily adaptable to other ill-posed coefficient inverse problems. That is, the proposed model estimates the probability for a point of whether the conductivity belongs to the background region or to the non-homogeneous region on the continuous space Rd&cap;Ω with d∈{2,3}. The proposed model does not make assumptions about the forward model and allows for solving the inverse problem in real time. The proposed machine learning approach for shape reconstruction is also used to improve gradient-based methods for estimating the unknown conductivity. In this paper, we propose a piece-wise constant reconstruction method that is novel in the inverse problem setting but inspired by recent approaches from the 3D vision community. We also extend this method into a novel constrained reconstruction method. We present extensive numerical experiments to show the performance of the architecture and compare the proposed method with previous analytic algorithms, mainly the monotonicity-based shape reconstruction algorithm and iteratively regularized Gauss-Newton method.

In silicostudy of the effects of cerebral circulation on source localization using a dynamical anatomical atlas of the human head

Objective.This study focuses on the effects of dynamical vascular modeling on source localization errors in electroencephalography (EEG). Our aim of thisin silicostudy is to (a) find out the effects of cerebral circulation on the accuracy of EEG source localization estimates, and (b) evaluate its relevance with respect to measurement noise and interpatient variation.Approach.We employ a four-dimensional (3D + T) statistical atlas of the electrical properties of the human head with a cerebral circulation model to generate virtual patients with different cerebral circulatory conditions for EEG source localization analysis. As source reconstruction techniques, we use the linearly constraint minimum variance (LCMV) beamformer, standardized low-resolution brain electromagnetic tomography (sLORETA), and the dipole scan (DS).Main results.Results indicate that arterial blood flow affects source localization at different depths and with varying significance. The average flow rate plays an important role in source localization performance, while the pulsatility effects are very small. In cases where a personalized model of the head is available, blood circulation mismodeling causes localization errors, especially in the deep structures of the brain where the main cerebral arteries are located. When interpatient variations are considered, the results show differences up to 15 mm for sLORETA and LCMV beamformer and 10 mm for DS in the brainstem and entorhinal cortices regions. In regions far from the main arteries vessels, the discrepancies are smaller than 3 mm. When measurement noise is added and interpatient differences are considered in a deep dipolar source, the results indicate that the effects of conductivity mismatch are detectable even for moderate measurement noise. The signal-to-noise ratio limit for sLORETA and LCMV beamformer is 15 dB, while the limit is under 30 dB for DS.Significance.Localization of the brain activity via EEG constitutes an ill-posed inverse problem, where any modeling uncertainty, e.g. a slight amount of noise in the data or material parameter discrepancies, can lead to a significant deviation of the estimated activity, especially in the deep structures of the brain. Proper modeling of the conductivity distribution is necessary in order to obtain an appropriate source localization. In this study, we show that the conductivity of the deep brain structures is particularly impacted by blood flow-induced changes in conductivity because large arteries and veins access the brain through that region.

VTDCE-Net: A time invariant deep neural network for direct estimation of pharmacokinetic parameters from undersampled DCE MRI data

PURPOSE

To propose a robust time and space invariant deep learning (DL) method to directly estimate the pharmacokinetic/tracer kinetic (PK/TK) parameters from undersampled dynamic contrast-enhanced (DCE) magnetic resonance imaging (MRI) data.

METHODS

DCE-MRI consists of 4D (3D-spatial + temporal) data and has been utilized to estimate 3D (spatial) tracer kinetic maps. Existing DL architecture for this task needs retraining for variation in temporal and/or spatial dimensions. This work proposes a DL algorithm that is invariant to training and testing in both temporal and spatial dimensions. The proposed network was based on a 2.5-dimensional Unet architecture, where the encoder consists of a 3D convolutional layer and the decoder consists of a 2D convolutional layer. The proposed VTDCE-Net was evaluated for solving the ill-posed inverse problem of directly estimating TK parameters from undersampled k - t $k-t$ space data of breast cancer patients, and the results were systematically compared with a total variation (TV) regularization based direct parameter estimation scheme. In the breast dataset, the training was performed on patients with 32 time samples, and testing was carried out on patients with 26 and 32 time samples. Translation of the proposed VTDCE-Net for brain dataset to show the generalizability was also carried out. Undersampling rates (R) of 8× , 12× , and 20× were utilized with PSNR and SSIM as the figures of merit in this evaluation. TK parameter maps estimated from fully sampled data were utilized as ground truth.

RESULTS

Experiments carried out in this work demonstrate that the proposed VTDCE-Net outperforms the TV scheme on both breast and brain datasets across all undersampling rates. For K trans $\mathbf {K_{trans}}$ and V p $\mathbf {V_{p}}$ maps, the improvement over TV is as high as 2 and 5 dB, respectively, using the proposed VTDCE-Net.

CONCLUSION

Temporal points invariant DL network that was proposed in this work to estimate the TK-parameters using DCE-MRI data has provided state-of-the-art performance compared to standard image reconstruction methods and is shown to work across all undersampling rates.

Improving Generalization by Learning Geometry-Dependent and Physics-Based Reconstruction of Image Sequences

Deep neural networks have shown promise in image reconstruction tasks, although often on the premise of large amounts of training data. In this paper, we present a new approach to exploit the geometry and physics underlying electrocardiographic imaging (ECGI) to learn efficiently with a relatively small dataset. We first introduce a non-Euclidean encoding-decoding network that allows us to describe the unknown and measurement variables over their respective geometrical domains. We then explicitly model the geometry-dependent physics in between the two domains via a bipartite graph over their graphical embeddings. We applied the resulting network to reconstruct electrical activity on the heart surface from body-surface potentials. In a series of generalization tasks with increasing difficulty, we demonstrated the improved ability of the network to generalize across geometrical changes underlying the data using less than 10% of training data and fewer variations of training geometry in comparison to its Euclidean alternatives. In both simulation and real-data experiments, we further demonstrated its ability to be quickly fine-tuned to new geometry using a modest amount of data.

Deep, deep learning with BART

PURPOSE

To develop a deep-learning-based image reconstruction framework for reproducible research in MRI.

METHODS

The BART toolbox offers a rich set of implementations of calibration and reconstruction algorithms for parallel imaging and compressed sensing. In this work, BART was extended by a nonlinear operator framework that provides automatic differentiation to allow computation of gradients. Existing MRI-specific operators of BART, such as the nonuniform fast Fourier transform, are directly integrated into this framework and are complemented by common building blocks used in neural networks. To evaluate the use of the framework for advanced deep-learning-based reconstruction, two state-of-the-art unrolled reconstruction networks, namely the Variational Network and MoDL, were implemented.

RESULTS

State-of-the-art deep image-reconstruction networks can be constructed and trained using BART's gradient-based optimization algorithms. The BART implementation achieves a similar performance in terms of training time and reconstruction quality compared to the original implementations based on TensorFlow.

CONCLUSION

By integrating nonlinear operators and neural networks into BART, we provide a general framework for deep-learning-based reconstruction in MRI.

Application of the Model of Spots for Inverse Problems

This article proposes the application of a new mathematical model of spots for solving inverse problems using a learning method, which is similar to using deep learning. In general, the spots represent vague figures in abstract "information spaces" or crisp figures with a lack of information about their shapes. However, crisp figures are regarded as a special and limiting case of spots. A basic mathematical apparatus, based on L4 numbers, has been developed for the representation and processing of qualitative information of elementary spatial relations between spots. Moreover, we defined L4 vectors, L4 matrices, and mathematical operations on them. The developed apparatus can be used in Artificial Intelligence, in particular, for knowledge representation and for modeling qualitative reasoning and learning. Another application area is the solution of inverse problems by learning. For example, this can be applied to image reconstruction using ultrasound, X-ray, magnetic resonance, or radar scan data. The introduced apparatus was verified by solving problems of reconstruction of images, utilizing only qualitative data of its elementary relations with some scanning figures. This article also demonstrates the application of a spot-based inverse Radon algorithm for binary image reconstruction. In both cases, the spot-based algorithms have demonstrated an effective denoising property.

Magnetic Induction Tomography: Separation of the Ill-Posed and Non-Linear Inverse Problem into a Series of Isolated and Less Demanding Subproblems

Magnetic induction tomography (MIT) is based on remotely excited eddy currents inside a measurement object. The conductivity distribution shapes the eddies, and their secondary fields are detected and used to reconstruct the conductivities. While the forward problem from given conductivities to detected signals can be unambiguously simulated, the inverse problem from received signals back to searched conductivities is a non-linear ill-posed problem that compromises MIT and results in rather blurry imaging. An MIT inversion is commonly applied over the entire process (i.e., localized conductivities are directly determined from specific signal features), but this involves considerable computation. The present more theoretical work treats the inverse problem as a non-retroactive series of four individual subproblems, each one less difficult by itself. The decoupled tasks yield better insights and control and promote more efficient computation. The overall problem is divided into an ill-posed but linear problem for reconstructing eddy currents from given signals and a nonlinear but benign problem for reconstructing conductivities from given eddies. The separated approach is unsuitable for common and circular MIT designs, as it merely fits the data structure of a recently presented and planar 3D MIT realization for large biomedical phantoms. For this MIT scanner, in discretization, the number of unknown and independent eddy current elements reflects the number of ultimately searched conductivities. For clarity and better representation, representative 2D bodies are used here and measured at the depth of the 3D scanner. The overall difficulty is not substantially smaller or different than for 3D bodies. In summary, the linear problem from signals to eddies dominates the overall MIT performance.

Developments of inverse analysis by Kalman filters and Bayesian methods applied to geotechnical engineering

The present paper reviews recent activities on inverse analysis strategies in geotechnical engineering using Kalman filters, nonlinear Kalman filters, and Markov chain Monte Carlo (MCMC)/Hamiltonian Monte Carlo (HMC) methods. Nonlinear Kalman filters with finite element method (FEM) broaden the choices of unknowns to be determined for not only parameters but also initial and/or boundary conditions, and the use of the posterior probability of the state variables can be widely applied to, for example, the decision making for design changes. The relevance of the unknowns and the observed values and the selection of the best sensor locations are some of the considerations made while using the Kalman filter FEM. This paper demonstrates several real-world geotechnical applications of the nonlinear Kalman filter and the MCMC with FEM. Future studies should focus on the following areas: attaining excellent performance for long-term forecasts using short-term observation and developing a viable method for selecting equations that describe physical phenomena and constitutive models.

Encoder-Decoder Architecture for 3D Seismic Inversion

Inverting seismic data to build 3D geological structures is a challenging task due to the overwhelming amount of acquired seismic data, and the very-high computational load due to iterative numerical solutions of the wave equation, as required by industry-standard tools such as Full Waveform Inversion (FWI). For example, in an area with surface dimensions of 4.5 km × 4.5 km, hundreds of seismic shot-gather cubes are required for 3D model reconstruction, leading to Terabytes of recorded data. This paper presents a deep learning solution for the reconstruction of realistic 3D models in the presence of field noise recorded in seismic surveys. We implement and analyze a convolutional encoder-decoder architecture that efficiently processes the entire collection of hundreds of seismic shot-gather cubes. The proposed solution demonstrates that realistic 3D models can be reconstructed with a structural similarity index measure (SSIM) of 0.9143 (out of 1.0) in the presence of field noise at 10 dB signal-to-noise ratio.

ABPO-TVSCAD: alternating Bregman proximity operators approach based on TVSCAD regularization for bioluminescence tomography

Objective.Bioluminescence tomography (BLT) is a promising non-invasive optical medical imaging technique, which can visualize and quantitatively analyze the distribution of tumor cells in living tissues. However, due to the influence of photon scattering effect and ill-conditioned inverse problem, the reconstruction result is unsatisfactory. The purpose of this study is to improve the reconstruction performance of BLT.Approach.An alternating Bregman proximity operators (ABPO) method based on TVSCAD regularization is proposed for BLT reconstruction. TVSCAD combines the anisotropic total variation (TV) regularization constraints and the non-convex smoothly clipped absolute deviation (SCAD) penalty constraints, to make a trade-off between the sparsity and edge preservation of the source. ABPO approach is used to solve the TVSCAD model (ABPO-TVSCAD for short). In addition, to accelerate the convergence speed of the ABPO, we adapt the strategy of shrinking the permission source region, which further improves the performance of ABPO-TVSCAD.Main results.The results of numerical simulations andin vivoxenograft mouse experiment show that our proposed method achieved superior accuracy in spatial localization and morphological reconstruction of bioluminescent source.Significance.ABPO-TVSCAD is an effective and robust reconstruction method for BLT, and we hope that this method can promote the development of optical molecular tomography.

A Deep Residual Neural Network for Image Reconstruction in Biomedical 3D Magnetic Induction Tomography

In recent years, it has become increasingly popular to solve inverse problems of various tomography methods with deep learning techniques. Here, a deep residual neural network (ResNet) is introduced to reconstruct the conductivity distribution of a biomedical, voluminous body in magnetic induction tomography (MIT). MIT is a relatively new, contactless and noninvasive tomography method. However, the ill-conditioned inverse problem of MIT is challenging to solve, especially for voluminous bodies with conductivities in the range of biological tissue. The proposed ResNet can reconstruct up to two cuboid perturbation objects with conductivities of 0.0 and 1.0 S/m in the whole voluminous body, even in the difficult-to-detect centre. The dataset used for training and testing contained simulated signals of cuboid perturbation objects with randomised lengths and positions. Furthermore, special care went into avoiding the inverse crime while creating the dataset. The calculated metrics showed good results over the test dataset, with an average correlation coefficient of 0.87 and mean squared error of 0.001. Robustness was tested on three special test cases containing unknown shapes, conductivities and a real measurement that showed error results well within the margin of the metrics of the test dataset. This indicates that a good approximation of the inverse function in MIT for up to two perturbation objects was achieved and the inverse crime was avoided.

Optimization of Initial Drug Distribution in Spherical Capsules for Personalized Release

OBJECTIVE

Customization of the rate of drug delivered based on individual patient requirements is of paramount importance in the design of drug delivery devices. Advances in manufacturing may enable multilayer drug delivery devices with different initial drug distributions in each layer. However, a robust mathematical understanding of how to optimize such capabilities is critically needed. The objective of this work is to determine the initial drug distribution needed in a spherical drug delivery device such as a capsule in order to obtain a desired drug release profile.

METHODS

This optimization problem is posed as an inverse mass transfer problem, and optimization is carried out using the solution of the forward problem. Both non-erodible and erodible multilayer spheres are analyzed. Cases with polynomial forms of initial drug distribution are also analyzed. Optimization is also carried out for a case where an initial burst in drug release rate is desired, followed by a constant drug release rate.

RESULTS

More than 60% reduction in root-mean-square deviation of the actual drug release rate from the ideal constant drug release rate is reported. Typically, the optimized initial drug distribution in these cases prevents or minimizes large drug release rate at early times, leading to a much more uniform drug release overall.

CONCLUSIONS

Results demonstrate potential for obtaining a desired drug delivery profile over time by carefully engineering the drug distribution in the drug delivery device. These results may help engineer devices that offer customized drug delivery by combining advanced manufacturing with mathematical optimization.

Deformation-Compensated Learning for Image Reconstruction Without Ground Truth

Deep neural networks for medical image reconstruction are traditionally trained using high-quality ground-truth images as training targets. Recent work on Noise2Noise (N2N) has shown the potential of using multiple noisy measurements of the same object as an alternative to having a ground-truth. However, existing N2N-based methods are not suitable for learning from the measurements of an object undergoing nonrigid deformation. This paper addresses this issue by proposing the deformation-compensated learning (DeCoLearn) method for training deep reconstruction networks by compensating for object deformations. A key component of DeCoLearn is a deep registration module, which is jointly trained with the deep reconstruction network without any ground-truth supervision. We validate DeCoLearn on both simulated and experimentally collected magnetic resonance imaging (MRI) data and show that it significantly improves imaging quality.

Revisiting [Formula: see text]-wavelet compressed-sensing MRI in the era of deep learning

Following their success in numerous imaging and computer vision applications, deep-learning (DL) techniques have emerged as one of the most prominent strategies for accelerated MRI reconstruction. These methods have been shown to outperform conventional regularized methods based on compressed sensing (CS). However, in most comparisons, CS is implemented with two or three hand-tuned parameters, while DL methods enjoy a plethora of advanced data science tools. In this work, we revisit [Formula: see text]-wavelet CS reconstruction using these modern tools. Using ideas such as algorithm unrolling and advanced optimization methods over large databases that DL algorithms utilize, along with conventional insights from wavelet representations and CS theory, we show that [Formula: see text]-wavelet CS can be fine-tuned to a level close to DL reconstruction for accelerated MRI. The optimized [Formula: see text]-wavelet CS method uses only 128 parameters compared to >500,000 for DL, employs a convex reconstruction at inference time, and performs within <1% of a DL approach that has been used in multiple studies in terms of quantitative quality metrics.

Inverse problems in blood flow modeling: A review

Mathematical and computational modeling of the cardiovascular system is increasingly providing non-invasive alternatives to traditional invasive clinical procedures. Moreover, it has the potential for generating additional diagnostic markers. In blood flow computations, the personalization of spatially distributed (i.e., 3D) models is a key step which relies on the formulation and numerical solution of inverse problems using clinical data, typically medical images for measuring both anatomy and function of the vasculature. In the last years, the development and application of inverse methods has rapidly expanded most likely due to the increased availability of data in clinical centers and the growing interest of modelers and clinicians in collaborating. Therefore, this work aims to provide a wide and comparative overview of literature within the last decade. We review the current state of the art of inverse problems in blood flows, focusing on studies considering fully dimensional fluid and fluid-solid models. The relevant physical models and hemodynamic measurement techniques are introduced, followed by a survey of mathematical data assimilation approaches used to solve different kinds of inverse problems, namely state and parameter estimation. An exhaustive discussion of the literature of the last decade is presented, structured by types of problems, models and available data.

Multi-Coil MRI Reconstruction Challenge-Assessing Brain MRI Reconstruction Models and Their Generalizability to Varying Coil Configurations

Deep-learning-based brain magnetic resonance imaging (MRI) reconstruction methods have the potential to accelerate the MRI acquisition process. Nevertheless, the scientific community lacks appropriate benchmarks to assess the MRI reconstruction quality of high-resolution brain images, and evaluate how these proposed algorithms will behave in the presence of small, but expected data distribution shifts. The multi-coil MRI (MC-MRI) reconstruction challenge provides a benchmark that aims at addressing these issues, using a large dataset of high-resolution, three-dimensional, T1-weighted MRI scans. The challenge has two primary goals: (1) to compare different MRI reconstruction models on this dataset and (2) to assess the generalizability of these models to data acquired with a different number of receiver coils. In this paper, we describe the challenge experimental design and summarize the results of a set of baseline and state-of-the-art brain MRI reconstruction models. We provide relevant comparative information on the current MRI reconstruction state-of-the-art and highlight the challenges of obtaining generalizable models that are required prior to broader clinical adoption. The MC-MRI benchmark data, evaluation code, and current challenge leaderboard are publicly available. They provide an objective performance assessment for future developments in the field of brain MRI reconstruction.

Improved resolution of D-bar images of ventilation using a Schur complement property and an anatomical atlas

BACKGROUND

Electrical impedance tomography (EIT) is a nonionizing imaging technique for real-time imaging of ventilation of patients with respiratory distress. Cross-sectional dynamic images are formed by reconstructing the conductivity distribution from measured voltage data arising from applied alternating currents on electrodes placed circumferentially around the chest. Since the conductivity of lung tissue depends on air content, blood flow, and the presence of pathology, the dynamic images provide regional information about ventilation, pulsatile perfusion, and abnormalities. However, due to the ill-posedness of the inverse conductivity problem, EIT images have a coarse spatial resolution. One method of improving the resolution is to include prior information in the reconstruction.

PURPOSE

In this work, we propose a technique in which a statistical prior built from an anatomical atlas is used to postprocess EIT reconstructions of human chest data. The effectiveness of the method is demonstrated on data from two patients with cystic fibrosis.

METHODS

A direct reconstruction algorithm known as the D-bar method was used to compute a two-dimensional reconstruction of the conductivity distribution in the plane of the electrodes. Reconstructions using one step in an iterative (regularized) Newton's method were also computed for comparison. An anatomical atlas consisting of 1589 synthetic EIT images computed from X-ray computed tomography (CT) scans of 74 adult male subjects was computed for use as a statistical prior. The resolution of the D-bar images was then improved by maximizing the conditional probability density function of an image that is consistent with the a priori information and the statistical model. A new method to evaluate the accuracy of the EIT images using CT scans of the imaged patient as ground truth is presented. The novel approach is tested on data from two patients with cystic fibrosis.

RESULTS AND CONCLUSIONS

The D-bar images resulted in better structural similarity index measures (SSIM) and multiscale (MS) SSIM measures for both subjects using the mask or amplitude evaluation approach than the one-step (regularized) Newton's method. Further improvement was achieved using the Schur complement (SC) approach, with MS-SSIM values of 0.718 and 0.682 using SC evaluated with the mask and amplitude approach, respectively, for Patient 1, and MS-SSIM values of 0.726 and 0.692 using SC evaluated with the mask and amplitude approach, respectively, for Patient 2. The results from applying an anatomical atlas and statistical prior to EIT data from two patients with cystic fibrosis suggest that the spatial resolution of the EIT image can be improved to reveal pathology that may be difficult to discern in the original EIT image. The novel metric of evaluation is consistent with the appearance of improved spatial resolution and provides a new way to evaluate the accuracy of EIT reconstructions when a CT scan is available.

Multi-domain integrative Swin transformer network for sparse-view tomographic reconstruction

Decreasing projection views to a lower X-ray radiation dose usually leads to severe streak artifacts. To improve image quality from sparse-view data, a multi-domain integrative Swin transformer network (MIST-net) was developed and is reported in this article. First, MIST-net incorporated lavish domain features from data, residual data, image, and residual image using flexible network architectures, where a residual data and residual image sub-network was considered as a data consistency module to eliminate interpolation and reconstruction errors. Second, a trainable edge enhancement filter was incorporated to detect and protect image edges. Third, a high-quality reconstruction Swin transformer (i.e., Recformer) was designed to capture image global features. The experimental results on numerical and real cardiac clinical datasets with 48 views demonstrated that our proposed MIST-net provided better image quality with more small features and sharp edges than other competitors.

Calibration by differentiation - Self-supervised calibration for X-ray microscopy using a differentiable cone-beam reconstruction operator

High-resolution X-ray microscopy (XRM) is gaining interest for biological investigations of extremely small-scale structures. XRM imaging of bones in living mice could provide new insights into the emergence and treatment of osteoporosis by observing osteocyte lacunae, which are holes in the bone of few micrometers in size. Imaging living animals at that resolution, however, is extremely challenging and requires very sophisticated data processing converting the raw XRM detector output into reconstructed images. This paper presents an open-source, differentiable reconstruction pipeline for XRM data which analytically computes the final image from the raw measurements. In contrast to most proprietary reconstruction software, it offers the user full control over each processing step and, additionally, makes the entire pipeline deep learning compatible by ensuring differentiability. This allows fitting trainable modules both before and after the actual reconstruction step in a purely data-driven way using the gradient-based optimizers of common deep learning frameworks. The value of such differentiability is demonstrated by calibrating the parameters of a simple cupping correction module operating on the raw projection images using only a self-supervisory quality metric based on the reconstructed volume and no further calibration measurements. The retrospective calibration directly improves image quality as it avoids cupping artifacts and decreases the difference in gray values between outer and inner bone by 68% to 94%. Furthermore, it makes the reconstruction process entirely independent of the XRM manufacturer and paves the way to explore modern deep learning reconstruction methods for arbitrary XRM and, potentially, other flat-panel CT systems. This exemplifies how differentiable reconstruction can be leveraged in the context of XRM and, hence, is an important step toward the goal of reducing the resolution limit of in-vivo bone imaging to the single micrometer domain. This article is protected by copyright. All rights reserved.

Adaptive stochastic Gauss-Newton method with optical Monte Carlo for quantitative photoacoustic tomography

SIGNIFICANCE

The image reconstruction problem in quantitative photoacoustic tomography (QPAT) is an ill-posed inverse problem. Monte Carlo method for light transport can be utilized in solving this image reconstruction problem.

AIM

The aim was to develop an adaptive image reconstruction method where the number of photon packets in Monte Carlo simulation is varied to achieve a sufficient accuracy with reduced computational burden.

APPROACH

The image reconstruction problem was formulated as a minimization problem. An adaptive stochastic Gauss-Newton (A-SGN) method combined with Monte Carlo method for light transport was developed. In the algorithm, the number of photon packets used on Gauss-Newton (GN) iteration was varied utilizing a so-called norm test.

RESULTS

The approach was evaluated with numerical simulations. With the proposed approach, the number of photon packets needed for solving the inverse problem was significantly smaller than in a conventional approach where the number of photon packets was fixed for each GN iteration.

CONCLUSIONS

The A-SGN method with a norm test can be utilized in QPAT to provide accurate and computationally efficient solutions.

Deep Prior Approach for Room Impulse Response Reconstruction

In this paper, we propose a data-driven approach for the reconstruction of unknown room impulse responses (RIRs) based on the deep prior paradigm. We formulate RIR reconstruction as an inverse problem. More specifically, a convolutional neural network (CNN) is employed prior, in order to obtain a regularized solution to the RIR reconstruction problem for uniform linear arrays. This approach allows us to avoid assumptions on sound wave propagation, acoustic environment, or measuring setting made in state-of-the-art RIR reconstruction algorithms. Moreover, differently from classical deep learning solutions in the literature, the deep prior approach employs a per-element training. Therefore, the proposed method does not require training data sets, and it can be applied to RIRs independently from available data or environments. Results on simulated data demonstrate that the proposed technique is able to provide accurate results in a wide range of scenarios, including variable direction of arrival of the source, room T60, and SNR at the sensors. The devised technique is also applied to real measurements, resulting in accurate RIR reconstruction and robustness to noise compared to state-of-the-art solutions.

The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem

Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite the existence of stable ones. This foundational issue relates to Smale’s 18th mathematical problem for the 21st century on the limits of AI. By expanding methodologies initiated by Gödel and Turing, we demonstrate limitations on the existence of (even randomized) algorithms for computing NNs. Despite numerous existence results of NNs with great approximation properties, only in specific cases do there also exist algorithms that can compute them. We initiate a classification theory on which NNs can be trained and introduce NNs that—under suitable conditions—are robust to perturbations and exponentially accurate in the number of hidden layers.

Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities; however, there does not exist any algorithm, even randomized, that can train (or compute) such a NN. For any positive integers K>2 and L, there are cases where simultaneously 1) no randomized training algorithm can compute a NN correct to K digits with probability greater than 1/2; 2) there exists a deterministic training algorithm that computes a NN with K –1 correct digits, but any such (even randomized) algorithm needs arbitrarily many training data; and 3) there exists a deterministic training algorithm that computes a NN with K –2 correct digits using no more than L training samples. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. We introduce fast iterative restarted networks (FIRENETs), which we both prove and numerically verify are stable. Moreover, we prove that only O(|log (ϵ)|) layers are needed for an ϵ-accurate solution to the inverse problem.