Matthias J. Ehrhardt

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

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

About

65
Publications
11,630
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
1,028
Citations
Citations since 2016
57 Research Items
1001 Citations
2016201720182019202020212022050100150200250
2016201720182019202020212022050100150200250
2016201720182019202020212022050100150200250
2016201720182019202020212022050100150200250
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 (65)
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 t...
Article
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
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
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
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...
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...
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...
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...
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
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
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...
Conference Paper
Full-text available
Image reconstruction in positron emission tomography (PET) is computationally challenging due to Poisson noise, constraints and potentially non-smooth priors-let alone the sheer size of the problem. An algorithm that can cope well with the first three of the aforementioned challenges is the primal-dual hybrid gradient algorithm (PDHG) studied by Ch...
Article
Full-text available
We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The analysis is carried out for general convex-concave saddle point problems and problems that are either partially smooth / strongly convex or fully smooth / st...
Article
Full-text available
We discuss a special form of gradient descent that in the literature has become known as the so-called linearised Bregman iteration. The idea is to replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a more general proper, convex and lower semi-continuous functional. Gradient...
Article
Full-text available
The combination of positron emission tomography (PET) and magnetic resonance imaging (MRI) offers unique possibilities. In this paper we aim to exploit the high spatial resolution of MRI to enhance the reconstruction of simultaneously acquired PET data. We propose a new prior to incorporate structural side information into a maximum a posteriori re...
Article
Full-text available
The combination of positron emission tomography (PET) and magnetic resonance imaging (MRI) offers unique possibilities. In this paper we aim to exploit the high spatial resolution of MRI to enhance the reconstruction of simultaneously acquired PET data. We propose a new prior to incorporate structural side information into a maximum a posteriori re...
Article
Full-text available
Magnetic resonance imaging (MRI) is a versatile imaging technique that allows different contrasts depending on the acquisition parameters. Many clinical imaging studies acquire MRI data for more than one of these contrasts—such as, for instance, T1 and T2 weighted images—which makes the overall scanning procedure very time consuming. As all of thes...
Article
Full-text available
Magnetic resonance imaging (MRI) is a versatile imaging technique that allows different contrasts depending on the acquisition parameters. Many clinical imaging studies acquire MRI data for more than one of these contrasts---such as for instance T1 and T2 weighted images---which makes the overall scanning procedure very time consuming. As all of th...
Conference Paper
A new algorithm (LBFGS-B-PC) which combines ideas of two existing convergent reconstruction algorithms, relaxed separable paraboloidal surrogate (SPS) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with boundary constraints (LBFGS-B), is proposed. Its performance is evaluated in terms of log-posterior value and regional recovery ratio. The res...
Thesis
Imaging is a powerful tool being used in many disciplines such as engineering, physics, biology and medicine to name a few. Recent years have seen a trend that imaging modalities have been combined to create multi-modality imaging tools where different modalities acquire complementary information. For example, in medical imaging, positron emission...
Article
Full-text available
Recent advances in technology have enabled the combination of positron emission tomography (PET) with magnetic resonance imaging (MRI). These PET-MRI scanners simultaneously acquire functional PET and anatomical or functional MRI data. As function and anatomy are not independent of one another the images to be reconstructed are likely to have share...
Article
Full-text available
Vector-valued images such as RGB color images or multimodal medical images show a strong interchannel correlation, which is not exploited by most image processing tools. We propose a new notion of treating vector-valued images which is based on the angle between the spatial gradients of their channels. Through minimizing a cost functional that pena...
Article
In recent years more and more long-term broadband data sets are collected in geosciences. Therefore there is an urgent need of algorithms which semi-automatically analyse and decompose these data into separate periods which are associated with different processes. Often Fourier and Wavelet Transform is used to decompose the data into short and long...

Network