Tracking Time-Vertex Propagation using Dynamic Graph Wavelets

Graph Signal Processing generalizes classical signal processing to signal or data indexed by the vertices of a weighted graph. So far, the research efforts have been focused on static graph signals. However numerous applications involve graph signals evolving in time, such as spreading or propagation of waves on a network. The analysis of this type of data requires a new set of methods that fully takes into account the time and graph dimensions. We propose a novel class of wavelet frames named Dynamic Graph Wavelets, whose time-vertex evolution follows a dynamic process. We demonstrate that this set of functions can be combined with sparsity based approaches such as compressive sensing to reveal information on the dynamic processes occurring on a graph. Experiments on real seismological data show the efficiency of the technique, allowing to estimate the epicenter of earthquake events recorded by a seismic network

Statistical and Graph-Based Signal Processing: Fundamental Results and Application to Cardiac Electrophysiology

The goal of cardiac electrophysiology is to obtain information about the mechanism, function, and performance of the electrical activities of the heart, the identification of deviation from normal pattern and the design of treatments. Offering a better insight into cardiac arrhythmias comprehension and management, signal processing can help the physician to enhance the treatment strategies, in particular in case of atrial fibrillation (AF), a very common atrial arrhythmia which is associated to significant morbidities, such as increased risk of mortality, heart failure, and thromboembolic events. Catheter ablation of AF is a therapeutic technique which uses radiofrequency energy to destroy atrial tissue involved in the arrhythmia sustenance, typically aiming at the electrical disconnection of the of the pulmonary veins triggers. However, recurrence rate is still very high, showing that the very complex and heterogeneous nature of AF still represents a challenging problem. Leveraging the tools of non-stationary and statistical signal processing, the first part of our work has a twofold focus: firstly, we compare the performance of two different ablation technologies, based on contact force sensing or remote magnetic controlled, using signal-based criteria as surrogates for lesion assessment. Furthermore, we investigate the role of ablation parameters in lesion formation using the late-gadolinium enhanced magnetic resonance imaging. Secondly, we hypothesized that in human atria the frequency content of the bipolar signal is directly related to the local conduction velocity (CV), a key parameter characterizing the substrate abnormality and influencing atrial arrhythmias. Comparing the degree of spectral compression among signals recorded at different points of the endocardial surface in response to decreasing pacing rate, our experimental data demonstrate a significant correlation between CV and the corresponding spectral centroids. However, complex spatio-temporal propagation pattern characterizing AF spurred the need for new signals acquisition and processing methods. Multi-electrode catheters allow whole-chamber panoramic mapping of electrical activity but produce an amount of data which need to be preprocessed and analyzed to provide clinically relevant support to the physician. Graph signal processing has shown its potential on a variety of applications involving high-dimensional data on irregular domains and complex network. Nevertheless, though state-of-the-art graph-based methods have been successful for many tasks, so far they predominantly ignore the time-dimension of data. To address this shortcoming, in the second part of this dissertation, we put forth a Time-Vertex Signal Processing Framework, as a particular case of the multi-dimensional graph signal processing. Linking together the time-domain signal processing techniques with the tools of GSP, the Time-Vertex Signal Processing facilitates the analysis of graph structured data which also evolve in time. We motivate our framework leveraging the notion of partial differential equations on graphs. We introduce joint operators, such as time-vertex localization and we present a novel approach to significantly improve the accuracy of fast joint filtering. We also illustrate how to build time-vertex dictionaries, providing conditions for efficient invertibility and examples of constructions. The experimental results on a variety of datasets suggest that the proposed tools can bring significant benefits in various signal processing and learning tasks involving time-series on graphs. We close the gap between the two parts illustrating the application of graph and time-vertex signal processing to the challenging case of multi-channels intracardiac signals

Novel topological and temporal network analyses for EEG functional connectivity with applications to Alzheimer’s disease

This doctoral thesis outlines several methodological advances in network science aimed towards uncovering rapid, complex interdependencies of electromagnetic brain activity recorded from the Electroencephalogram (EEG). This entails both new analyses and modelling of EEG brain network topologies and a novel approach to analyse rapid dynamics of connectivity. Importantly, we implement these advances to provide novel insights into pathological brain function in Alzheimer’s disease. We introduce the concept of hierarchical complexity of network topology, providing both an index to measure it and a model to simulate it. We then show that the topology of functional connectivity estimated from EEG recordings is hierarchically complex, existing in a scale between random and star-like topologies, this is a paradigm shift from the established understanding that complexity arises between random and regular topologies. We go on to consider the density appropriate for binarisation of EEG functional connectivity, a methodological step recommended to produce compact and unbiased networks, in light of its new-found hierarchical complexity. Through simulations and real EEG data, we show the benefit of going beyond often recommended sparse representations to account for a broader range of hierarchy level interactions. After this, we turn our attention to assessing dynamic changes in connectivity. By constructing a unified framework for multivariate signals and graphs, inspired by network science and graph signal processing, we introduce graph-variate signal analysis which allows us to capture rapid fluctuations in connectivity robust to spurious short-term correlations. We define this for three pertinent brain connectivity estimates- Pearson’s correlation coefficient, coherence and phase-lag index- and show its benefit over standard dynamic connectivity measures in a range of simulations and real data. Applying these novel methods to EEG datasets of the performance of visual short-term memory binding tasks by familial and sporadic Alzheimer’s disease patients, we uncover disorganisation of the topological hierarchy of EEG brain function and abnormalities of transient phase-based activity which paves the way for new interpretations of the disease’s affect on brain function. Hierarchical complexity and graph-variate dynamic connectivity are entirely new methods for analysing EEG brain networks. The former provides new interpretations of complexity in static connectivity patterns while the latter enables robust analysis of transient temporal connectivity patterns, both at the frontiers of analysis. Although designed with EEG functional connectivity in mind, we hope these techniques will be picked up in the broader field, having consequences for research into complex networks in general