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

Scalable Low-rank Matrix and Tensor Decomposition on Graphs

In many signal processing, machine learning and computer vision applications, one often has to deal with high dimensional and big datasets such as images, videos, web content, etc. The data can come in various forms, such as univariate or multivariate time series, matrices or high dimensional tensors. The goal of the data mining community is to reveal the hidden linear or non-linear structures in the datasets. Over the past couple of decades matrix factorization, owing to its intrinsic association with dimensionality reduction has been adopted as one of the key methods in this context. One can either use a single linear subspace to approximate the data (the standard Principal Component Analysis (PCA) approach) or a union of low dimensional subspaces where each data class belongs to a different subspace. In many cases, however, the low dimensional data follows some additional structure. Knowledge of such structure is beneficial, as we can use it to enhance the representativity of our models by adding structured priors. A nowadays standard way to represent pairwise affinity between objects is by using graphs. The introduction of graph-based priors to enhance matrix factorization models has recently brought them back to the highest attention of the data mining community. Representation of a signal on a graph is well motivated by the emerging field of signal processing on graphs, based on notions of spectral graph theory. The underlying assumption is that high-dimensional data samples lie on or close to a smooth low-dimensional manifold. Interestingly, the underlying manifold can be represented by its discrete proxy, i.e. a graph. A primary limitation of the state-of-the-art low-rank approximation methods is that they do not generalize for the case of non-linear low-rank structures. Furthermore, the standard low-rank extraction methods for many applications, such as low-rank and sparse decomposition, are computationally cumbersome. We argue, that for many machine learning and signal processing applications involving big data, an approximate low-rank recovery suffices. Thus, in this thesis, we present solutions to the above two limitations by presenting a new framework for scalable but approximate low-rank extraction which exploits the hidden structure in the data using the notion of graphs. First, we present a novel signal model, called `Multilinear low-rank tensors on graphs (MLRTG)' which states that a tensor can be encoded as a multilinear combination of the low-frequency graph eigenvectors, where the graphs are constructed along the various modes of the tensor. Since the graph eigenvectors have the interpretation of \textit{non-linear} embedding of a dataset on the low-dimensional manifold, we propose a method called `Graph Multilinear SVD (GMLSVD)' to recover PCA based linear subspaces from these eigenvectors. Finally, we propose a plethora of highly scalable matrix and tensor based problems for low-rank extraction which implicitly or explicitly make use of the GMLSVD framework. The core idea is to replace the expensive iterative SVD operations by updating the linear subspaces from the fixed non-linear ones via low-cost operations. We present applications in low-rank and sparse decomposition and clustering of the low-rank features to evaluate all the proposed methods. Our theoretical analysis shows that the approximation error of the proposed framework depends on the spectral properties of the graph Laplacian