Applied Harmonic Analysis and Data Science (hybrid meeting)

Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field

(An overview of) Synergistic reconstruction for multimodality/multichannel imaging methods

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 developments, an ever increasing set of mathematical ideas is appearing regarding how data from different imaging modalities or channels can be synergistically combined in the image reconstruction process, exploiting structural and/or functional correlations between the multiple images. Here we review these developments, give pointers to important challenges and provide an outlook as to how the field may develop in the forthcoming years. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 1'

Extensions of Laplacian Eigenmaps for Manifold Learning

This thesis deals with the theory and practice of manifold learning, especially as they relate to the problem of classification. We begin with a well known algorithm, Laplacian Eigenmaps, and then proceed to extend it in two independent directions. First, we generalize this algorithm to allow for the use of partially labeled data, and establish the theoretical foundation of the resulting semi-supervised learning method. Second, we consider two ways of accelerating the most computationally intensive step of Laplacian Eigenmaps, the construction of an adjacency graph. Both of them produce high quality approximations, and we conclude by showing that they work well together to achieve a dramatic reduction in computational time

Classification of multiple time signals using localized frequency characteristics applied to industrial process monitoring

A general framework for regression modeling using localized frequency characteristics of explanatory variables is proposed. This novel framework can be used in any application where the aim is to model an evolving process sequentially based on multiple time series data. Furthermore, this framework allows time series to be transformed and combined to simultaneously boost important characteristics and reduce noise. A wavelet transform is used to isolate key frequency structure and perform data reduction. The method is highly adaptive, since wavelets are effective at extracting localized information from noisy data. This adaptivity allows rapid identification of changes in the evolving process. Finally, a regression model uses functions of the wavelet coefficients to classify the evolving process into one of a set of states which can then be used for automatic monitoring of the system. As motivation and illustration, industrial process monitoring using electrical tomography measurements is considered. This technique provides useful data without intruding into the industrial process. Statistics derived from the wavelet transform of the tomographic data can be enormously helpful in monitoring and controlling the process. The predictive power of the proposed approach is explored using real and simulated tomographic data. In both cases, the resulting models successfully classify different flow regimes and hence provide the basis for reliable online monitoring and control of industrial processes

Mini-Workshop: Deep Learning and Inverse Problems

Machine learning and in particular deep learning offer several data-driven methods to amend the typical shortcomings of purely analytical approaches. The mathematical research on these combined models is presently exploding on the experimental side but still lacking on the theoretical point of view. This workshop addresses the challenge of developing a solid mathematical theory for analyzing deep neural networks for inverse problems

Electrical Impedance Tomography (EIT): The Establishment of a Dual Current Stimulation EIT System for Improved Image Quality

Electrical Impedance Tomography (EIT) is a noninvasive imaging technique that reproduces images of cross-sections, based on the internal impedance distribution of an object. This Dissertation investigates and confirms the use of a dual current stimulation EIT (DCS EIT) system. The results of this investigation presented a size error of 2.82 % and a position error of 5.93 % in the reconstructed images, when compared to the actual size and position of the anomaly inside a test object. These results confirmed that the DCS EIT system produced images of superior quality (fewer image reconstruction errors) to those produced from reviewed single plane stimulating EIT systems, which confirmed the research hypothesis. This system incorporates two independent current stimulating patterns, which establishes a more even distribution of current in the test object, compared to single plane systems, and is more efficient than 2.5D EIT systems because the DCS EIT system only measures boundary voltages in the center plane, compared to 2.5D EIT systems that measure the boundary voltages in all electrode planes. The system uses 48 compound electrodes, divided into three electrode planes. Current is sourced and sunk perpendicularly in the center plane, to produce a high current density near the center of the test object. Sequentially, current is sourced through an electrode in the top electrode plane and sunk through an electrode in the bottom plane, directly below the source electrode, to produce a high current density near the boundary of the test object, in the center plane. During both injection cycles, boundary potentials are measured in the center plane. Following the measurement of a complete frame, a weighted average is computed from the single and cross plane measured data. The weighted measured voltages, injected currents and Finite Element Model of the object is used to reconstruct an image of the internal impedance distribution along a cross-section of the object. This method is applicable to the biomedical imaging and process monitoring fields

Mathematical Methods in Tomography

This is the seventh Oberwolfach conference on the mathematics of tomography, the first one taking place in 1980. Tomography is the most popular of a series of medical and scientific imaging techniques that have been developed since the mid seventies of the last century

Visualisation of multi-dimensional medical images with application to brain electrical impedance tomography

Medical imaging plays an important role in modem medicine. With the increasing complexity and information presented by medical images, visualisation is vital for medical research and clinical applications to interpret the information presented in these images. The aim of this research is to investigate improvements to medical image visualisation, particularly for multi-dimensional medical image datasets. A recently developed medical imaging technique known as Electrical Impedance Tomography (EIT) is presented as a demonstration. To fulfil the aim, three main efforts are included in this work. First, a novel scheme for the processmg of brain EIT data with SPM (Statistical Parametric Mapping) to detect ROI (Regions of Interest) in the data is proposed based on a theoretical analysis. To evaluate the feasibility of this scheme, two types of experiments are carried out: one is implemented with simulated EIT data, and the other is performed with human brain EIT data under visual stimulation. The experimental results demonstrate that: SPM is able to localise the expected ROI in EIT data correctly; and it is reasonable to use the balloon hemodynamic change model to simulate the impedance change during brain function activity. Secondly, to deal with the absence of human morphology information in EIT visualisation, an innovative landmark-based registration scheme is developed to register brain EIT image with a standard anatomical brain atlas. Finally, a new task typology model is derived for task exploration in medical image visualisation, and a task-based system development methodology is proposed for the visualisation of multi-dimensional medical images. As a case study, a prototype visualisation system, named EIT5DVis, has been developed, following this methodology. to visualise five-dimensional brain EIT data. The EIT5DVis system is able to accept visualisation tasks through a graphical user interface; apply appropriate methods to analyse tasks, which include the ROI detection approach and registration scheme mentioned in the preceding paragraphs; and produce various visualisations