Equidistribution-based training of Free Knot Splines and ReLU Neural Networks

We consider the problem of one-dimensional function approximation using shallow neural networks (NN) with a rectified linear unit (ReLU) activation function and compare their training with traditional methods such as univariate Free Knot Splines (FKS). ReLU NNs and FKS span the same function space, and thus have the same theoretical expressivity. In the case of ReLU NNs, we show that their ill-conditioning degrades rapidly as the width of the network increases. This often leads to significantly poorer approximation in contrast to the FKS representation, which remains well-conditioned as the number of knots increases. We leverage the theory of optimal piecewise linear interpolants to improve the training procedure for a ReLU NN. Using the equidistribution principle, we propose a two-level procedure for training the FKS by first solving the nonlinear problem of finding the optimal knot locations of the interpolating FKS. Determining the optimal knots then acts as a good starting point for training the weights of the FKS. The training of the FKS gives insights into how we can train a ReLU NN effectively to give an equally accurate approximation. More precisely, we combine the training of the ReLU NN with an equidistribution based loss to find the breakpoints of the ReLU functions, combined with preconditioning the ReLU NN approximation (to take an FKS form) to find the scalings of the ReLU functions, leads to a well-conditioned and reliable method of finding an accurate ReLU NN approximation to a target function. We test this method on a series or regular, singular, and rapidly varying target functions and obtain good results realising the expressivity of the network in this case.

Learning Governing Equations of Unobserved States in Dynamical Systems

Data driven modelling and scientific machine learning have been responsible for significant advances in determining suitable models to describe data. Within dynamical systems, neural ordinary differential equations (ODEs), where the system equations are set to be governed by a neural network, have become a popular tool for this challenge in recent years. However, less emphasis has been placed on systems that are only partially-observed. In this work, we employ a hybrid neural ODE structure, where the system equations are governed by a combination of a neural network and domain-specific knowledge, together with symbolic regression (SR), to learn governing equations of partially-observed dynamical systems. We test this approach on two case studies: A 3-dimensional model of the Lotka-Volterra system and a 5-dimensional model of the Lorenz system. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems, with robustness to measurement noise.

Investigating the use of publicly available natural videos to learn Dynamic MR image reconstruction

Purpose: To develop and assess a deep learning (DL) pipeline to learn dynamic MR image reconstruction from publicly available natural videos (Inter4K). Materials and Methods: Learning was performed for a range of DL architectures (VarNet, 3D UNet, FastDVDNet) and corresponding sampling patterns (Cartesian, radial, spiral) either from true multi-coil cardiac MR data (N=692) or from pseudo-MR data simulated from Inter4K natural videos (N=692). Real-time undersampled dynamic MR images were reconstructed using DL networks trained with cardiac data and natural videos, and compressed sensing (CS). Differences were assessed in simulations (N=104 datasets) in terms of MSE, PSNR, and SSIM and prospectively for cardiac (short axis, four chambers, N=20) and speech (N=10) data in terms of subjective image quality ranking, SNR and Edge sharpness. Friedman Chi Square tests with post-hoc Nemenyi analysis were performed to assess statistical significance. Results: For all simulation metrics, DL networks trained with cardiac data outperformed DL networks trained with natural videos, which outperformed CS (p<0.05). However, in prospective experiments DL reconstructions using both training datasets were ranked similarly (and higher than CS) and presented no statistical differences in SNR and Edge Sharpness for most conditions. Additionally, high SSIM was measured between the DL methods with cardiac data and natural videos (SSIM>0.85). Conclusion: The developed pipeline enabled learning dynamic MR reconstruction from natural videos preserving DL reconstruction advantages such as high quality fast and ultra-fast reconstructions while overcoming some limitations (data scarcity or sharing). The natural video dataset, code and pre-trained networks are made readily available on github. Key Words: real-time; dynamic MRI; deep learning; image reconstruction; machine learning;

Inverse Problems with Learned Forward Operators

Solving inverse problems requires knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants are desired that do not compromise reconstruction quality. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint.

Score-Based Generative Models for PET Image Reconstruction

Score-based generative models have demonstrated highly promising results for medical image reconstruction tasks in magnetic resonance imaging or computed tomography. However, their application to Positron Emission Tomography (PET) is still largely unexplored. PET image reconstruction involves a variety of challenges, including Poisson noise with high variance and a wide dynamic range. To address these challenges, we propose several PET-specific adaptations of score-based generative models. The proposed framework is developed for both 2D and 3D PET. In addition, we provide an extension to guided reconstruction using magnetic resonance images. We validate the approach through extensive 2D and 3D $\textit{in-silico}$ experiments with a model trained on patient-realistic data without lesions, and evaluate on data without lesions as well as out-of-distribution data with lesions. This demonstrates the proposed method's robustness and significant potential for improved PET reconstruction.

A Learned Born Series for Highly-Scattering Media

A new method for solving the wave equation is presented, called the learned Born series (LBS), which is derived from a convergent Born Series but its components are found through training. The LBS is shown to be significantly more accurate than the convergent Born series for the same number of iterations, in the presence of high contrast scatterers, while maintaining a comparable computational complexity. The LBS is able to generate a reasonable prediction of the global pressure field with a small number of iterations, and the errors decrease with the number of learned iterations.

Unsupervised denoising for sparse multi-spectral computed tomography

Multi-energy computed tomography (CT) with photon counting detectors (PCDs) enables spectral imaging as PCDs can assign the incoming photons to specific energy channels. However, PCDs with many spectral channels drastically increase the computational complexity of the CT reconstruction, and bespoke reconstruction algorithms need fine-tuning to varying noise statistics. \rev{Especially if many projections are taken, a large amount of data has to be collected and stored. Sparse view CT is one solution for data reduction. However, these issues are especially exacerbated when sparse imaging scenarios are encountered due to a significant reduction in photon counts.} In this work, we investigate the suitability of learning-based improvements to the challenging task of obtaining high-quality reconstructions from sparse measurements for a 64-channel PCD-CT. In particular, to overcome missing reference data for the training procedure, we propose an unsupervised denoising and artefact removal approach by exploiting different filter functions in the reconstruction and an explicit coupling of spectral channels with the nuclear norm. Performance is assessed on both simulated synthetic data and the openly available experimental Multi-Spectral Imaging via Computed Tomography (MUSIC) dataset. We compared the quality of our unsupervised method to iterative total nuclear variation regularized reconstructions and a supervised denoiser trained with reference data. We show that improved reconstruction quality can be achieved with flexibility on noise statistics and effective suppression of streaking artefacts when using unsupervised denoising with spectral coupling.

FReSCO: Flow Reconstruction and Segmentation for low latency Cardiac Output monitoring using deep artifact suppression and segmentation

Purpose: Real-time monitoring of cardiac output (CO) requires low latency reconstruction and segmentation of real-time phase contrast MR (PCMR), which has previously been difficult to perform. Here we propose a deep learning framework for 'Flow Reconstruction and Segmentation for low latency Cardiac Output monitoring' (FReSCO). Methods: Deep artifact suppression and segmentation U-Nets were independently trained. Breath hold spiral PCMR data (n=516) was synthetically undersampled using a variable density spiral sampling pattern and gridded to create aliased data for training of the artifact suppression U-net. A subset of the data (n=96) was segmented and used to train the segmentation U-net. Real-time spiral PCMR was prospectively acquired and then reconstructed and segmented using the trained models (FReSCO) at low latency at the scanner in 10 healthy subjects during rest, exercise and recovery periods. CO obtained via FReSCO was compared to a reference rest CO and rest and exercise Compressed Sensing (CS) CO. Results: FReSCO was demonstrated prospectively at the scanner. Beat-to-beat heartrate, stroke volume and CO could be visualized with a mean latency of 622ms. No significant differences were noted when compared to reference at rest (Bias = -0.21+-0.50 L/min, p=0.246) or CS at peak exercise (Bias=0.12+-0.48 L/min, p=0.458). Conclusion: FReSCO was successfully demonstrated for real-time monitoring of CO during exercise and could provide a convenient tool for assessment of the hemodynamic response to a range of stressors.

Neural Network Kalman filtering for 3D object tracking from linear array ultrasound data

Many interventional surgical procedures rely on medical imaging to visualise and track instruments. Such imaging methods not only need to be real-time capable, but also provide accurate and robust positional information. In ultrasound applications, typically only two-dimensional data from a linear array are available, and as such obtaining accurate positional estimation in three dimensions is non-trivial. In this work, we first train a neural network, using realistic synthetic training data, to estimate the out-of-plane offset of an object with the associated axial aberration in the reconstructed ultrasound image. The obtained estimate is then combined with a Kalman filtering approach that utilises positioning estimates obtained in previous time-frames to improve localisation robustness and reduce the impact of measurement noise. The accuracy of the proposed method is evaluated using simulations, and its practical applicability is demonstrated on experimental data obtained using a novel optical ultrasound imaging setup. Accurate and robust positional information is provided in real-time. Axial and lateral coordinates for out-of-plane objects are estimated with a mean error of 0.1mm for simulated data and a mean error of 0.2mm for experimental data. Three-dimensional localisation is most accurate for elevational distances larger than 1mm, with a maximum distance of 5mm considered for a 25mm aperture.

Quantifying Sources of Uncertainty in Deep Learning-Based Image Reconstruction

Image reconstruction methods based on deep neural networks have shown outstanding performance, equalling or exceeding the state-of-the-art results of conventional approaches, but often do not provide uncertainty information about the reconstruction. In this work we propose a scalable and efficient framework to simultaneously quantify aleatoric and epistemic uncertainties in learned iterative image reconstruction. We build on a Bayesian deep gradient descent method for quantifying epistemic uncertainty, and incorporate the heteroscedastic variance of the noise to account for the aleatoric uncertainty. We show that our method exhibits competitive performance against conventional benchmarks for computed tomography with both sparse view and limited angle data. The estimated uncertainty captures the variability in the reconstructions, caused by the restricted measurement model, and by missing information, due to the limited angle geometry.