Matthias J. Ehrhardt

Matthias J. Ehrhardt
University of Bath | UB · Department of Mathematical Sciences

PhD in Medical Imaging, Diploma (Master) in Applied Mathematics

About

92
Publications
16,295
Reads
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1,605
Citations
Additional affiliations
February 2021 - present
University of Bath
Position
  • Professor (Full)
September 2018 - January 2021
University of Bath
Position
  • Fellow
January 2016 - September 2018
University of Cambridge
Position
  • Research Associate
Education
April 2012 - April 2015
University College London
Field of study
  • Medical Imaging
October 2006 - December 2011
Universität Bremen
Field of study
  • Industrial Mathematics

Publications

Publications (92)
Preprint
Full-text available
We address the optimization problem in a data-driven variational reconstruction framework, where the regularizer is parameterized by an input-convex neural network (ICNN). While gradient-based methods are commonly used to solve such problems, they struggle to effectively handle non-smoothness which often leads to slow convergence. Moreover, the nes...
Preprint
Full-text available
Motion correction aims to prevent motion artefacts which may be caused by respiration, heartbeat, or head movements for example. In a preliminary step, the measured data is divided in gates corresponding to motion states, and displacement maps from a reference state to each motion state are estimated. One common technique to perform motion correcti...
Article
Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by select...
Preprint
Full-text available
Stochastic optimisation algorithms are the de facto standard for machine learning with large amounts of data. Handling only a subset of available data in each optimisation step dramatically reduces the per-iteration computational costs, while still ensuring significant progress towards the solution. Driven by the need to solve large-scale optimisat...
Preprint
Full-text available
Image reconstruction for dynamic inverse problems with highly undersampled data poses a major challenge: not accounting for the dynamics of the process leads to a non-realistic motion with no time regularity. Variational approaches that penalize time derivatives or introduce motion model regularizers have been proposed to relate subsequent frames a...
Preprint
We explore the application of preconditioning in optimisation algorithms, specifically those appearing in Inverse Problems in imaging. Such problems often contain an ill-posed forward operator and are large-scale. Therefore, computationally efficient algorithms which converge quickly are desirable. To remedy these issues, learning-to-optimise lever...
Preprint
Full-text available
In this paper, we analyze a recently proposed algorithm for the problem of sampling from probability distributions $\mu^\ast$ in $\mathbb{R}^d$ with a Lebesgue density and potential of the form $f(Kx)+g(x)$, where $K$ is a linear operator and $f$, $g$ are convex and non-smooth. The algorithm is a generalization of the primal-dual hybrid gradient (P...
Article
Full-text available
In this work, we propose a new primal–dual algorithm with adaptive step sizes. The stochastic primal–dual hybrid gradient (SPDHG) algorithm with constant step sizes has become widely applied in large-scale convex optimization across many scientific fields due to its scalability. While the product of the primal and dual step sizes is subject to an u...
Article
Full-text available
Computed tomography (CT) imaging of the thorax is widely used for the detection and monitoring of pulmonary embolism (PE). However, CT images can contain artifacts due to the acquisition or the processes involved in image reconstruction. Radiologists often have to distinguish between such artifacts and actual PEs. We provide a proof of concept in t...
Article
Full-text available
Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for comp...
Article
Full-text available
Deep neural network approaches to inverse imaging problems have produced impressive results in the last few years. In this survey paper, we consider the use of generative models in a variational regularisation approach to inverse problems. The considered regularisers penalise images that are far from the range of a generative model that has learned...
Preprint
In various domains within imaging and data science, particularly when addressing tasks modeled utilizing the variational regularization approach, manually configuring regularization parameters presents a formidable challenge. The difficulty intensifies when employing regularizers involving a large number of hyperparameters. To overcome this challen...
Article
Full-text available
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...
Preprint
Full-text available
In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution. For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool....
Preprint
Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even...
Preprint
Full-text available
Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very differen...
Article
Full-text available
Demand for low carbon energy storage has highlighted the importance of imaging techniques for the characterization of electrode microstructures to determine key parameters associated with battery manufacture, operation, degradation, and failure both for next generation lithium and other novel battery systems. Here, recent progress and literature hi...
Chapter
Imaging with multiple modalities or multiple channels is becoming increasingly important for our modern society. A key tool for understanding and early diagnosis of cancer and dementia is PET-MR, a combined positron emission tomography and magnetic resonance imaging scanner which can simultaneously acquire functional and anatomical data. Similarly,...
Preprint
Full-text available
Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for comp...
Preprint
Computed tomography (CT) imaging of the thorax is widely used for the detection and monitoring of pulmonary embolism (PE). However, CT images can contain artifacts due to the acquisition or the processes involved in image reconstruction. Radiologists often have to distinguish between such artifacts and actual PEs. Our main contribution comes in the...
Preprint
Full-text available
In this work we propose a new primal-dual algorithm with adaptive step-sizes. The stochastic primal-dual hybrid gradient (SPDHG) algorithm with constant step-sizes has become widely applied in large-scale convex optimization across many scientific fields due to its scalability. While the product of the primal and dual step-sizes is subject to an up...
Article
From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry . Here symmetry refers to the invariance property of signal sets to transformations, such as translation, rotation, or scaling. Symmetry can also be incorporated into deep neural networks...
Preprint
Learned regularization for MRI reconstruction can 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 nois...
Preprint
From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry. Here symmetry refers to the invariance property of signal sets to transformations such as translation, rotation or scaling. Symmetry can also be incorporated into deep neural networks in t...
Preprint
In this work we propose a stochastic primal-dual preconditioned three-operator splitting algorithm for solving a class of convex three-composite optimization problems. Our proposed scheme is a direct three-operator splitting extension of the SPDHG algorithm [Chambolle et al. 2018]. We provide theoretical convergence analysis showing ergodic O(1/K)...
Preprint
The Stochastic Primal-Dual Hybrid Gradient or SPDHG is an algorithm proposed by Chambolle et al. to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals,...
Preprint
Full-text available
Variational regularization methods are commonly used to approximate solutions of inverse problems. In recent years, model-based variational regularization methods have often been replaced with data-driven ones such as the fields-of-expert model (Roth and Black, 2009). Training the parameters of such data-driven methods can be formulated as a bileve...
Article
Full-text available
Purpose Dynamic nuclear polarization is an emerging imaging method that allows noninvasive investigation of tissue metabolism. However, the relatively low metabolic spatial resolution that can be achieved limits some applications, and improving this resolution could have important implications for the technique. Methods We propose to enhance the 3...
Article
Full-text available
The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We p...
Preprint
Deep neural network approaches to inverse imaging problems have produced impressive results in the last few years. In this paper, we consider the use of generative models in a variational regularisation approach to inverse problems. The considered regularisers penalise images that are far from the range of a generative model that has learned to pro...
Article
We briefly review recent work where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We report here new preliminary experiments with implicit symplectic Runge-Kutta methods. In this paper, we discuss ongoing and future research in this area.
Article
Full-text available
This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image which keeps structural information shared by the energy...
Article
Full-text available
SIRF is a powerful PET/MR image reconstruction research tool for processing data and developing new algorithms. In this research, new developments to SIRF are presented, with focus on motion estimation and correction. SIRF’s recent inclusion of the adjoint of the resampling operator allows gradient propagation through resampling, enabling the MCIR...
Article
Full-text available
In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural netw...
Article
Full-text available
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by-now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a...
Article
Full-text available
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning: most deep learning methods require the solution of hard optimisation probl...
Article
Full-text available
Imaging is omnipresent in modern society with imaging devices based on a zoo of physical principles, probing a specimen across different wavelengths, energies and time. Recent years have seen a change in the imaging landscape with more and more imaging devices combining that which previously was used separately. Motivated by these hardware developm...
Chapter
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Reso...
Preprint
Full-text available
In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural netw...
Preprint
Full-text available
This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel, we propose to fuse this available data (represented by a single sinogram) to obtain a polyenergetic image which keeps structural information shared by the energy...
Preprint
Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is a practical tool to solve nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. The proof makes use of a classical supermartingale result, and also rewrit...
Preprint
Many machine learning solutions are framed as optimization problems which rely on good hyperparameters. Algorithms for tuning these hyperparameters usually assume access to exact solutions to the underlying learning problem, which is typically not practical. Here, we apply a recent dynamic accuracy derivative-free optimization method to hyperparame...
Preprint
Quantitative MRI (qMRI) is becoming increasingly important for research and clinical applications, however, state-of-the-art reconstruction methods for qMRI are computationally prohibitive. We propose a temporal multiscale approach to reduce computation times in qMRI. Instead of computing exact gradients of the qMRI likelihood, we propose a novel a...
Article
Full-text available
Variance reduction is a crucial tool for improving the slow convergence of stochastic gradient descent. Only a few variance-reduced methods, however, have yet been shown to directly benefit from Nesterov’s acceleration techniques to match the convergence rates of accelerated gradient methods. Such approaches rely on “negative momentum”, a technique...
Article
The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern...
Preprint
Imaging with multiple modalities or multiple channels is becoming increasingly important for our modern society. A key tool for understanding and early diagnosis of cancer and dementia is PET-MR, a combined positron emission tomography and magnetic resonance imaging scanner which can simultaneously acquire functional and anatomical data. Similarly...
Preprint
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a...
Preprint
Full-text available
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved in applying deep learning: most deep learning methods require the solution of hard optimisation probl...
Preprint
Full-text available
Multi-modality (or multi-channel) imaging is becoming increasingly important and more widely available, e.g. hyperspectral imaging in remote sensing, spectral CT in material sciences as well as multi-contrast MRI and PET-MR in medicine. Research in the last decades resulted in a plethora of mathematical methods to combine data from several modaliti...
Article
The combination of positron emission tomography (PET) with magnetic resonance (MR) imaging opens the way to more accurate diagnosis and improved patient management. At present, the data acquired by PET-MR scanners are essentially processed separately, but the opportunity to improve accuracy of the tomographic reconstruction via synergy of the two i...
Article
Full-text available
Uncompressed clinical data from modern positron emission tomography (PET) scanners are very large, exceeding 350 million data points (projection bins). The last decades have seen tremendous advancements in mathematical imaging tools many of which lead to non-smooth (i.e. non-differentiable) optimization problems which are much harder to solve than...
Preprint
Full-text available
Variance reduction is a crucial tool for improving the slow convergence of stochastic gradient descent. Only a few variance-reduced methods, however, have yet been shown to directly benefit from Nesterov's acceleration techniques to match the convergence rates of accelerated gradient methods. Such approaches rely on "negative momentum", a technique...
Preprint
Full-text available
The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern...
Conference Paper
Full-text available
Ill-posed image recovery requires regularisation to ensure stability. The presented open-source regularisation toolkit consists of state-of-the-art variational algorithms which can be embedded in a plug-and-play fashion into the general framework of proximal splitting methods. The packaged regularisers aim to satisfy various prior expectations of t...
Preprint
Full-text available
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretization...
Article
EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not...
Article
Full-text available
All imaging modalities such as computed tomography, emission tomography and magnetic resonance imaging require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. Recently, the idea of tackling both pr...
Article
Full-text available
The low spatial resolution of hyperspectral imaging can be significantly improved by fusing the hyperspectral image with a high resolution photograph. In most practical cases, however, the exact alignment between the fused images is not known a priori. In this work, we study how including a blind deconvolution approach in the mathematical model can...
Preprint
Full-text available
Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimisation problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by select...
Preprint
Full-text available
Uncompressed clinical data from modern positron emission tomography (PET) scanners are very large, exceeding 300 million data points. The last decades have seen tremendous advancements in mathematical imaging tools many of which lead to non-smooth optimization problems. Most of these tools have not been translated to clinical PET data, as the algor...
Preprint
Full-text available
The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We p...
Preprint
Full-text available
All imaging modalities such as computed tomography (CT), emission tomography and magnetic resonance imaging (MRI) require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. We explore a new approach t...
Article
Full-text available
This paper reports on the feasibility of using a quasi-Newton optimization algorithm, limited-memory Broyden- Fletcher-Goldfarb-Shanno with boundary constraints (L-BFGSB), for penalized image reconstruction problems in emission tomography (ET). For further acceleration, an additional preconditioning technique based on a diagonal approximation of th...
Article
Full-text available
We present a standalone, scalable and high-throughput software platform for PET image reconstruction and analysis. We focus on high fidelity modelling of the acquisition processes to provide high accuracy and precision quantitative imaging, especially for large axial field of view scanners. All the core routines are implemented using parallel compu...
Data
This software allows to reproduce all results from the accompanying paper. It is written in MATLAB and can be executed on a standard desktop with standard amount of RAM.
Article
Full-text available
We propose an extension of a special form of gradient descent --- in the literature known as linearised Bregman iteration --- to a larger class of non-convex functionals. We replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a proper, convex and lower semi-continuous functio...
Article
Full-text available
Hyperspectral imaging is a cutting-edge type of remote sensing used for mapping vegetation properties, rock minerals and other materials. A major drawback of hyperspectral imaging devices is their intrinsic low spatial resolution. In this paper, we propose a method for increasing the spatial resolution of a hyperspectral image by fusing it with an...