arXiv subjects

Martin Benning

Publications and source records attributed to Martin Benning.

At least 37 records · Page 2Linked to original sources

arXiv2021

Timbre Transfer with Variational Auto Encoding and Cycle-Consistent Adversarial Networks

This research project investigates the application of deep learning to timbre transfer, where the timbre of a source audio can be converted to the timbre of a target audio with minimal loss in quality. The adopted approach combines Variational Autoencoders with Generative Adversarial Networks to construct meaningful representations of the source audio and produce realistic generations of the target audio and is applied to the Flickr 8k Audio dataset for transferring the vocal timbre between speakers and the URMP dataset for transferring the musical timbre between instruments. Furthermore, variations of the adopted approach are trained, and generalised performance is compared using the metrics SSIM (Structural Similarity Index) and FAD (Frechét Audio Distance). It was found that a many-to-many approach supersedes a one-to-one approach in terms of reconstructive capabilities, and that the adoption of a basic over a bottleneck residual block design is more suitable for enriching content information about a latent space. It was also found that the decision on whether cyclic loss takes on a variational autoencoder or vanilla autoencoder approach does not have a significant impact on reconstructive and adversarial translation aspects of the model.

cs.SD↗

arXiv2021

Choose your path wisely: gradient descent in a Bregman distance framework

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 functions. 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 function. The algorithm's global convergence is proven for functions that satisfy the Kurdyka-Łojasiewicz property. Examples illustrate that features of different scale are being introduced throughout the iteration, transitioning from coarse to fine. This coarse-to-fine approach with respect to scale allows to recover solutions of non-convex optimisation problems that are superior to those obtained with conventional gradient descent, or even projected and proximal gradient descent. The effectiveness of the linearised Bregman iteration in combination with early stopping is illustrated for the applications of parallel magnetic resonance imaging, blind deconvolution as well as image classification with neural networks.

math.OC↗

arXiv2020

Generalised Perceptron Learning

We present a generalisation of Rosenblatt's traditional perceptron learning algorithm to the class of proximal activation functions and demonstrate how this generalisation can be interpreted as an incremental gradient method applied to a novel energy function. This novel energy function is based on a generalised Bregman distance, for which the gradient with respect to the weights and biases does not require the differentiation of the activation function. The interpretation as an energy minimisation algorithm paves the way for many new algorithms, of which we explore a novel variant of the iterative soft-thresholding algorithm for the learning of sparse perceptrons.

cs.LG↗

arXiv2020

Scanning electron diffraction tomography of strain

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with scanning precession electron diffraction data.

math.NA↗

arXiv2020

Learning the Sampling Pattern for MRI

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 that can be used to optimally balance acquisition time versus quality of the reconstructed image. We use a supervised learning approach, making the assumption that our training data is representative enough of new data acquisitions. We demonstrate that this is indeed the case, even if the training data consists of just 7 training pairs of measurements and ground-truth images; with a training set of brain images of size 192 by 192, for instance, one of the learned patterns samples only 35% of k-space, however results in reconstructions with mean SSIM 0.914 on a test set of similar images. The proposed framework is general enough to learn arbitrary sampling patterns, including common patterns such as Cartesian, spiral and radial sampling.

eess.IV↗

arXiv2020

Bregman Itoh--Abe methods for sparse optimisation

In this paper we propose optimisation methods for variational regularisation problems based on discretising the inverse scale space flow with discrete gradient methods. Inverse scale space flow generalises gradient flows by incorporating a generalised Bregman distance as the underlying metric. Its discrete-time counterparts, Bregman iterations and linearised Bregman iterations, are popular regularisation schemes for inverse problems that incorporate a priori information without loss of contrast. Discrete gradient methods are tools from geometric numerical integration for preserving energy dissipation of dissipative differential systems. The resultant Bregman discrete gradient methods are unconditionally dissipative, and achieve rapid convergence rates by exploiting structures of the problem such as sparsity. Building on previous work on discrete gradients for non-smooth, non-convex optimisation, we prove convergence guarantees for these methods in a Clarke subdifferential framework. Numerical results for convex and non-convex examples are presented.

math.OC↗

arXiv2019

Deep learning as optimal control problems: models and numerical methods

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 discretisation. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. The differential equation setting lends itself to learning additional parameters such as the time discretisation. We explore this extension alongside natural constraints (e.g. time steps lie in a simplex). We compare these deep learning algorithms numerically in terms of induced flow and generalisation ability.

math.OC↗

arXiv2019

Joint phase reconstruction and magnitude segmentation from velocity-encoded MRI data

Velocity-encoded MRI is an imaging technique used in different areas to assess flow motion. Some applications include medical imaging such as cardiovascular blood flow studies, and industrial settings in the areas of rheology, pipe flows, and reactor hydrodynamics, where the goal is to characterise dynamic components of some quantity of interest. The problem of estimating velocities from such measurements is a nonlinear dynamic inverse problem. To retrieve time-dependent velocity information, careful mathematical modelling and appropriate regularisation is required. In this work, we propose an optimisation algorithm based on non-convex Bregman iteration to jointly estimate velocity-, magnitude- and segmentation-information for the application of bubbly flow imaging. Furthermore, we demonstrate through numerical experiments on synthetic and real data that the joint model improves velocity, magnitude and segmentation over a classical sequential approach.

eess.IV↗

arXiv2019

An entropic Landweber method for linear ill-posed problems

The aim of this paper is to investigate the use of an entropic projection method for the iterative regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering probability distributions. Moreover, we discuss several variants of the algorithm and relations to other methods in the literature. The effectiveness of the approach is studied numerically in several examples.

math.OC↗

arXiv2019

Enhancing joint reconstruction and segmentation with non-convex Bregman iteration

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 that combines reconstruction and segmentation in a unified framework. We derive a variational model that consists of a total variation regularised reconstruction from undersampled measurements and a Chan-Vese based segmentation. We extend the variational regularisation scheme to a Bregman iteration framework to improve the reconstruction and therefore the segmentation. We develop a novel alternating minimisation scheme that solves the non-convex optimisation problem with provable convergence guarantees. Our results for synthetic and real data show that both reconstruction and segmentation are improved compared to the classical sequential approach.

math.NA↗

arXiv2018

Directional Sinogram Inpainting for Limited Angle Tomography

In this paper we propose a new joint model for the reconstruction of tomography data under limited angle sampling regimes. In many applications of Tomography, e.g. Electron Microscopy and Mammography, physical limitations on acquisition lead to regions of data which cannot be sampled. Depending on the severity of the restriction, reconstructions can contain severe, characteristic, artefacts. Our model aims to address these artefacts by inpainting the missing data simultaneously with the reconstruction. Numerically, this problem naturally evolves to require the minimisation of a non-convex and non-smooth functional so we review recent work in this topic and extend results to fit an alternating (block) descent framework. We illustrate the effectiveness of this approach with numerical experiments on two synthetic datasets and one Electron Microscopy dataset.

math.OC↗

arXiv2018

Modern Regularization Methods for Inverse Problems

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research. In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.

math.NA↗

arXiv2017

Learning Filter Functions in Regularisers by Minimising Quotients

Learning approaches have recently become very popular in the field of inverse problems. A large variety of methods has been established in recent years, ranging from bi-level learning to high-dimensional machine learning techniques. Most learning approaches, however, only aim at fitting parametrised models to favourable training data whilst ignoring misfit training data completely. In this paper, we follow up on the idea of learning parametrised regularisation functions by quotient minimisation as established in [3]. We extend the model therein to include higher-dimensional filter functions to be learned and allow for fit- and misfit-training data consisting of multiple functions. We first present results resembling behaviour of well-established derivative-based sparse regularisers like total variation or higher-order total variation in one-dimension. Our second and main contribution is the introduction of novel families of non-derivative-based regularisers. This is accomplished by learning favourable scales and geometric properties while at the same time avoiding unfavourable ones.

math.OC↗

arXiv2017

Nonlinear Spectral Image Fusion

In this paper we demonstrate that the framework of nonlinear spectral decompositions based on total variation (TV) regularization is very well suited for image fusion as well as more general image manipulation tasks. The well-localized and edge-preserving spectral TV decomposition allows to select frequencies of a certain image to transfer particular features, such as wrinkles in a face, from one image to another. We illustrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing, linear osmosis, wavelet fusion and Laplacian pyramid fusion. We conclude that the proposed spectral TV image decomposition framework is a valuable tool for semi- and fully-automatic image editing and fusion.

cs.CV↗

arXiv2016

Inverse Scale Space Decomposition

We investigate the inverse scale space flow as a decomposition method for decomposing data into generalised singular vectors. We show that the inverse scale space flow, based on convex and absolutely one-homogeneous regularisation functionals, can decompose data represented by the application of a forward operator to a linear combination of generalised singular vectors into its individual singular vectors. We verify that for this decomposition to hold true, two additional conditions on the singular vectors are sufficient: orthogonality in the data space and inclusion of partial sums of the subgradients of the singular vectors in the subdifferential of the regularisation functional at zero. We also address the converse question of when the inverse scale space flow returns a generalised singular vector given that the initial data is arbitrary (and therefore not necessarily in the range of the forward operator). We prove that the inverse scale space flow is guaranteed to return a singular vector if the data satisfies a novel dual singular vector condition. We conclude the paper with numerical results that validate the theoretical results and that demonstrate the importance of the additional conditions required to guarantee the decomposition result.

math.NA↗

arXiv2016

Gradient descent in a generalised Bregman distance framework

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 descent as well as the entropic mirror descent by Nemirovsky and Yudin are special cases, as is a specific form of non-linear Landweber iteration introduced by Bachmayr and Burger. We are going to analyse the linearised Bregman iteration in a setting where the functional we want to minimise is neither necessarily Lipschitz-continuous (in the classical sense) nor necessarily convex, and establish a global convergence result under the additional assumption that the functional we wish to minimise satisfies the so-called Kurdyka-Łojasiewicz property.

math.OC↗

arXiv2016

Explorations on anisotropic regularisation of dynamic inverse problems by bilevel optimisation

We explore anisotropic regularisation methods in the spirit of [Holler & Kunisch, 14]. Based on ground truth data, we propose a bilevel optimisation strategy to compute the optimal regularisation parameters of such a model for the application of video denoising. The optimisation poses a challenge in itself, as the dependency on one of the regularisation parameters is non-linear such that the standard existence and convergence theory does not apply. Moreover, we analyse numerical results of the proposed parameter learning strategy based on three exemplary video sequences and discuss the impact of these results on the actual modelling of dynamic inverse problems.

math.NA↗

arXiv2015

Preconditioned ADMM with nonlinear operator constraint

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).

math.NA↗