In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize the aspects that play an important role in practical problems. We confine ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation. The techniques discussed in the intro-ductory chapters, for instance interpolation, numerical quadrature and the solution to nonlinear equations, may also be used outside the context of differential equations. They have been in-cluded to make the book self-contained as far as the numerical aspects are concerned. Chapters, sections and exercises marked with a * are not part of the Delft Institutional Package. The numerical examples in this book were implemented in Matlab, but also Python or any other programming language could be used. A list of references to background knowledge and related literature can be found at the end of this book. Extra information about this course can be found at http://NMODE.ewi.tudelft.nl, among which old exams, answers to the exercises, and a link to an online education platform. We thank Matthias Moller for his thorough reading of the draft of this book and his helpful suggestions.@en

This paper studies time-domain parallelisation using Parareal to speed up the computations of anisotropic diffusion filtering. We consider both explicit and implicit Euler based method for the propagation in time for Parareal. The Preconditioned Conjugate Gradient (PCG) method is used to solve the systems that arise in the implicit based method. The estimation of the iteration numbers of PCG allows us to predict the running time of Parareal calculation, which further guides us in the experimental stage. Parallelisation of the method is implemented using Coarray Fortran. We illustrate the experimental results on 3D low-field MRI images using up to 960 cores. The computational improvement in time is achieved.

@en

Purpose: To describe the construction and testing of a portable point-of-care low-field MRI system on site in Africa. Methods: All of the components to assemble a 50 mT Halbach magnet-based system, together with the necessary tools, were air-freighted from the Netherlands to Uganda. The construction steps included individual magnet sorting, filling of each ring of the magnet assembly, fine-tuning the inter-ring separations of the 23-ring magnet assembly, gradient coil construction, integration of gradient coils and magnet assembly, construction of the portable aluminum trolley and finally testing of the entire system with an open source MR spectrometer. Results: With four instructors and six untrained personnel, the complete project from delivery to first image took approximately 11 days. Conclusions: An important step in translating scientific developments in the western world from high-income industrialized countries to low- and middle-income countries (LMICs) is to produce technology that can be assembled and ultimately constructed locally. Local assembly and construction are associated with skill development, low costs and jobs. Point-of-care systems have a large potential to increase the accessibility and sustainability of MRI in LMICs, and this work demonstrates that technology and knowledge transfer can be performed relatively seamlessly.

@en

Purpose: To develop a method for MR Fingerprinting (MRF) sequence optimization that takes both the applied undersampling pattern and a realistic reference map into account. Methods: A predictive model for the undersampling error leveraging on perturbation theory was exploited to optimize the MRF flip angle sequence for improved robustness against undersampling artifacts. In this framework parameter maps from a previously acquired MRF scan were used as reference. Sequences were optimized for different sequence lengths, smoothness constraints and undersampling factors. Numerical simulations and in vivo measurements in eight healthy subjects were performed to assess the effect of the performed optimization. The optimized MRF sequences were compared to a conventionally shaped flip angle pattern and an optimized pattern based on the Cramér–Rao lower bound (CRB). Results: Numerical simulations and in vivo results demonstrate that the undersampling errors can be suppressed by flip angle optimization. Analysis of the in vivo results show that a sequence optimized for improved robustness against undersampling with a flip angle train of length 400 yielded significantly lower median absolute errors in (Formula presented.) : (Formula presented.) and (Formula presented.) : (Formula presented.) compared to the conventional ((Formula presented.) : (Formula presented.), (Formula presented.) : (Formula presented.)) and CRB-based ((Formula presented.) : (Formula presented.), (Formula presented.) : (Formula presented.)) sequences. Conclusion: The proposed method is able to optimize the MRF flip angle pattern such that significant mitigation of the artifacts from strong k-space undersampling in MRF is achieved.

@en

We study efficient implicit methods to denoise low-field MR images using a nonlinear diffusion operator as a regularizer. This problem can be formulated as solving a nonlinear reaction–diffusion equation. After discretization, a lagged-diffusion approach is used which requires a linear system solve in every nonlinear iteration. The choice of diffusion model determines the denoising properties, but it also influences the conditioning of the linear systems. As a solution method, we use Conjugate Gradient (CG) in combination with a suitable preconditioner and deflation technique. We consider four different preconditioners in combination with subdomain deflation. We evaluate the methods for four commonly used denoising operators: standard Laplace operator, two Perona–Malik type operators, and the Total Variation (TV) operator. We show that a Discrete Cosine Transform (DCT) preconditioner works best for problems with a slowly varying diffusion coefficient, while Jacobi preconditioning with subdomain deflation works best for a strongly varying diffusion, as happens for the TV operator. This research is part of a larger effort that aims to provide low-cost MR imaging capabilities for low-resource settings. We have evaluated the algorithms on low-field MRI images using inexpensive commodity hardware. With a suitable preconditioner for the chosen diffusion model, we are able to limit the time to denoise three-dimensional images of more than 2 million pixels to less than 15 s, which is fast enough to be used in practice.

@en

Low-field MRI scanners are significantly less expensive than their high-field counterparts, which gives them the potential to make MRI technology more accessible all around the world. In general, images acquired using low-field MRI scanners tend to be of a relatively low resolution, as signal-to-noise ratios are lower. The aim of this work is to improve the resolution of these images. To this end, we present a deep learning-based approach to transform low-resolution low-field MR images into high-resolution ones. A convolutional neural network was trained to carry out single image super-resolution reconstruction using pairs of noisy low-resolution images and their noise-free high-resolution counterparts, which were obtained from the publicly available NYU fastMRI database. This network was subsequently applied to noisy images acquired using a low-field MRI scanner. The trained convolutional network yielded sharp super-resolution images in which most of the high-frequency components were recovered. In conclusion, we showed that a deep learning-based approach has great potential when it comes to increasing the resolution of low-field MR images.

@en

Medical image reconstruction aims to acquire high-quality medical images for clinical usage at minimal cost and risk to the patients. Deep learning and its applications in medical imaging, especially in image reconstruction have received considerable attention in the literature in recent years. This study reviews records obtained electronically through the leading scientific databases (Magnetic Resonance Imaging journal, Google Scholar, Scopus, Science Direct, Elsevier, and from other journal publications) searched using three sets of keywords: (1) Deep learning, image reconstruction, medical imaging; (2) Medical imaging, Deep learning, Image reconstruction; (3) Open science, Open imaging data, Open software. The articles reviewed revealed that deep learning-based reconstruction methods improve the quality of reconstructed images qualitatively and quantitatively. However, deep learning techniques are generally computationally expensive, require large amounts of training datasets, lack decent theory to explain why the algorithms work, and have issues of generalization and robustness. The challenge of lack of enough training datasets is currently being addressed by using transfer learning techniques.

@en

Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.

@en

We investigate the adaptation of the spectral clustering algorithm to the privacy preserving domain. Spectral clustering is a data mining technique that divides points according to a measure of connectivity in a data graph. When the matrix data are privacy sensitive, cryptographic techniques can be applied to protect the data. A pivotal part of spectral clustering is the partial eigendecomposition of the graph Laplacian. The Lanczos algorithm is used to approximate the eigenvectors of the Laplacian. Many cryptographic techniques are designed to work with positive integers, whereas the numerical algorithms are generally applied in the real domain. To overcome this problem, the Lanczos algorithm is adapted to be performed with fixed-point arithmetic. Square roots are eliminated and floating-point computations are transformed to fixed-point computations. The effects of these adaptations on the accuracy and stability of the algorithm are investigated using standard datasets. The performance of the original and the adapted algorithm is similar when few eigenvectors are needed. For a large number of eigenvectors loss of orthogonality affects the results.

@en

In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle-point system. The Schur complement for the generalized saddle-point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle-point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state-of-the-art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

@en

In this paper we present a magnetic resonance imaging (MRI) technique that is based on multiplicative regularization. Instead of adding a regularizing objective function to a data fidelity term, we multiply by such a regularizing function. By following this approach, no regularization parameter needs to be determined for each new data set that is acquired. Reconstructions are obtained by iteratively updating the images using short-term conjugate gradient-type update formulas and Polak-Ribière update directions. We show that the algorithm can be used as an image reconstruction algorithm and as a denoising algorithm. We illustrate the performance of the algorithm on two-dimensional simulated low-field MR data that is corrupted by noise and on three-dimensional measured data obtained from a low-field MR scanner. Our reconstruction results show that the algorithm effectively suppresses noise and produces accurate reconstructions even for low-field MR signals with a low signal-to-noise ratio.

@en

Currently, many children with hydrocephalus in East Africa and other resource-constrained countries do not have access to Magnetic Resonance Imaging (MRI) scanners, the preferred imaging tool during the disease administration and treatment. Conventional MRI scanners are costly to buy and manage, which limits their utilization in low-income countries. Low-field MRI scanners can offer an affordable, sustainable, and safe imaging alternative to high-field MRI. However, they are associated with a low signal-to-noise ratio (SNR), and therefore the images obtained are noisy. In this study, we propose an algorithm that may help to alleviate the drawbacks of low-field MRI by improving the quality of images obtained. The proposed algorithm combines our previous proposed algorithm known as AS-DLMRI for image reconstruction and a nonlinear diffusion filter for image denoising. The formulation is capable of removing additive zero-mean white and homogeneous Gaussian noise, as well as other noise types that could be present in the original signal. Experiments on visual quality revealed that the proposed algorithm is effective in denoising images during reconstruction. The proposed algorithm effectively denoised a noisy phantom, and a noisy MRI image, and had better performance when compared to DLMRI and AS-DLMRI in terms of Peak Signal to Noise ratio (PSNR) and High-Frequency Error Norm (HFEN). Integrating AS-DLMRI and the nonlinear diffusion filter proved to be effective in improving the quality of the images during the experiments performed. The hybrid algorithm may be of great use in imaging modalities like low-field MRI that are associated with low SNR.@en

In an earlier paper, we generalized the CGME (Conjugate Gradient Minimal Error) algorithm to the ℓ₂-regularized weighted least-squares problem. Here, we use this Generalized CGME method to reconstruct images from actual signals measured using a low-field MRI scanner. We analyze the convergence of both GCGME and the classical Generalized Conjugate Gradient Least Squares (GCGLS) method for the simple case when a Laplace operator is used as a regularizer and indicate when GCGME is to be preferred in terms of convergence speed. We also consider a more complicated ℓ₁-penalty in a compressed sensing framework.

@en

Compared to other parts of the world, the incidence of hydrocephalus in children is very high in sub-Saharan Africa. Magnetic resonance imaging (MRI) would be the preferred diagnostic method for infant hydrocephaleus. However, in practice, MRI is seldom used in sub-Saharan Africa due to its high prize, low mobility, and high power consumption. A low-cost MRI technology is under development by reducing the strength of the magnetic field and the use of alternative technologies to create the magnetic field. This paper describes the embodiment design process to match this new MRI technology under development with the specific characteristics of the healthcare system in Uganda.A context exploration was performed to identify factors that may affect the design and implementation of the low-field MRI in Ugandan hospitals and Ugandan healthcare environment. The key-insights from the technology- A nd context-exploration were translated into requirements which were the starting point for the design process. The concept development did have a focus on Cost-effective design, Design for durability reliability, and Design for repairability. The final design was validated by stakeholders from the Ugandan Healthcare context.

@en

Compared to other parts of the world, the incidence of hydrocephalus in children is very high in sub-Saharan Africa. Magnetic resonance imaging (MRI) would be the preferred diagnostic method for infant hydrocephaleus. However, in practice, MRI is seldom used in sub-Saharan Africa due to its high prize, low mobility, and high power consumption. A low-cost MRI technology is under development by reducing the strength of the magnetic field and the use of alternative technologies to create the magnetic field. This paper describes the embodiment design process to match this new MRI technology under development with the specific characteristics of the healthcare system in Uganda.A context exploration was performed to identify factors that may affect the design and implementation of the low-field MRI in Ugandan hospitals and Ugandan healthcare environment. The key-insights from the technology- A nd context-exploration were translated into requirements which were the starting point for the design process. The concept development did have a focus on Cost-effective design, Design for durability reliability, and Design for repairability. The final design was validated by stakeholders from the Ugandan Healthcare context.

@en

Compared to other parts of the world, the incidence of hydrocephalus in children is very high in sub- Saharan Africa. Magnetic resonance imaging (MRI) would be the preferred diagnostic method for infant hydrocephaleus. However, in practice, MRI is seldom used in sub-Saharan Africa due to its high prize, low mobility, and high power consumption. A low-cost MRI technology is under development by reducing the strength of the magnetic field and the use of alternative technologies to create the magnetic field. This paper describes the embodiment design process to match this new MRI technology under development with the specific characteristics of thehealthcare system in Uganda. A context exploration was performed to identify factors that may affect the design and implementation of the low-field MRI in Ugandan hospitals and Ugandan healthcare environment. The key-insights from the technology- and context-exploration were translated into requirements which were the starting point for the design process. The concept development did have a focus on Cost-effective design, Design for durability & reliability, and Design for repairability. The final design was validated by stakeholders from the Ugandan Healthcare context

@en

Objective: Image denoising has been considered as a separate procedure from image reconstruction which could otherwise be combined with acquisition and reconstruction. This paper discusses a joint image reconstruction and denoising algorithm in low-field MRI using a dictionary learning approach. Method: Our proposed algorithm uses a two-level Bregman iterative method for image reconstruction and image denoising procedure using OMP for sparse coding and SimCO for Dictionary Update and Learning. Results: Experiments were done on a noisy phantom that was obtained from a low field MRI scanner. Results demonstrate that our proposed algorithm performs superior image reconstructions that are almost noise-free. Our proposed method also performed better than the TBMDU algorithm, which performed better than DLMRI, a technique that substantially outperformed other CSMRI based reconstruction methods. However, the TBMDU algorithm is faster than our proposed algorithm due to additional iterations required during the denoising step. Conclusion: An algorithm that jointly performs reconstruction and denoising is essential in medical imaging modalities where image denoising has been a separate process from the reconstruction. Combining the two could save time and could avoid image details to be lost due to having two separate operations. This formulation is essential in imaging modalities like low-field MRI where the image signal is noisy and therefore performing a joint reconstruction and denoising could help improve the quality of the images obtained.

@en

More than 6,000 infants develop hydrocephalus in East Africa every year. Magnetic Resonance Imaging is the preferred technique to diagnose hydrocephalus. In countries such as Uganda, MRI is unaffordable at even major referral hospitals. In order to provide a sustainable diagnostic tool we are developing an inexpensive and easy-to-use MRI system that yields images of sufficient quality to diagnose hydrocephalus. This paper describes our first prototype of such a scanner. We explain the lessons that we have learned from this prototype and how we used these to come up with an improved design. We also describe a dataset that has been obtained with this scanner that will be made publically available.@en

Background: Magnetic resonance imaging (MRI) is a safe non-invasive and nonionizing medical imaging modality that is used to visualize the structure of human anatomy. Conventional (high-field) MRI scanners are very expensive to purchase, operate and maintain, which limit their use in many developing countries. This study is part of a project that aims at addressing these challenges and is carried out by teams from Mbarara University of Science and Technology (MUST) in Uganda, Leiden University Medical Center (LUMC) in the Netherlands, Delft University of Technology (TU Delft) in the Netherlands and Pennsylvania State University (PSU) in the USA. These are working on developing affordable, portable and low-field MRI scanners to diagnose children in developing countries with hydrocephalus. The challenges faced by the teams are that the low-field MRI scanners currently under development are characterized by low Signal-to-Noise Ratio (SNR), and long scan times. Methods: We propose an algorithm called adaptive-size dictionary learning algorithm (AS-DLMRI) that integrates information-theoretic criteria (ITC) and Dictionary learning approaches. The result of the integration is an adaptive-size dictionary that is optimal for any input signal. AS-DLMRI may help to reduce the scan time and improve the SNR of the generated images, thereby improving the image quality. Results: We compared our proposed algorithm AS-DLMRI with adaptive patch-based algorithm known as DLMRI and non-adaptive CSMRI technique known as LDP. DLMRI and LDP have been used as the baseline algorithms in other related studies. The results of AS-DLMRI are consistently slightly better in terms of PSNR, SNR and HFEN than for DLMRI, and are significantly better than for LDP. Moreover, AS-DLMRI is faster than DLMRI. Conclusion: Using a dictionary size that is appropriate to the input data could reduce the computational complexity, and also the construction quality since only dictionary atoms that are relevant to the task are included in the dictionary and are used during the reconstruction. However, AS-DLMRI did not completely remove noise during the experiments with the noisy phantom. Our next step in our research is to integrate our proposed algorithm with an image denoising function.

@en