Using the multi-linear rank-(Lr, Lr, 1) decomposition for the detection of the 200 Hz band activity in somatosensory evoked magnetic fields and somatosensory evoked electrical potentials

Studies of oscillations in the frequency band between 80 Hz and 250 Hz for EEG (Electroencephalogram) and MEG (Magnetoencephalogram) have achieved fruitful results of detecting and interpreting both normal and pathological activities in the brain. This contribution presents a new description of the 200 Hz band activity in somatosensory evoked electrical potentials (SEPs) and somatosensory evoked magnetic fields (SEFs) with the help of tensor decompositions. The SEPs and SEFs elicited by electrical stimulation of the median nerve were measured in eight healthy volunteers. A time-frequency analysis of the SEPs and SEFs produced the time-dependent spectra of the signals that were arranged into three-dimensional EEG and MEG data tensors, respectively. We then propose a novel multi-way component analysis approach by employing a tensor decomposition known as the multi-linear rank-( $L_{r}$ , $L_{r}$ , 1) decomposition. Featuring the ability to extract channel-dependent spectral signatures, this method is able to separate the 200 Hz band activity-related signal components in SEPs and SEFs. Via a coupled version of the multi-linear rank-( $L_{r}$ , $L_{r}$ , 1) decomposition, a joint processing of these simultaneous EEG and MEG recordings has been achieved. The advantages of the joint processing over the separate processing of EEG or MEG alone have been both qualitatively and quantitatively validated in seven out of eight subjects

Advanced tensor based signal processing techniques for wireless communication systems and biomedical signal processing

Many observed signals in signal processing applications including wireless communications, biomedical signal processing, image processing, and machine learning are multi-dimensional. Tensors preserve the multi-dimensional structure and provide a natural representation of these signals/data. Moreover, tensors provide often an improved identifiability. Therefore, we benefit from using tensor algebra in the above mentioned applications and many more. In this thesis, we present the benefits of utilizing tensor algebra in two signal processing areas. These include signal processing for MIMO (Multiple-Input Multiple-Output) wireless communication systems and biomedical signal processing. Moreover, we contribute to the theoretical aspects of tensor algebra by deriving new properties and ways of computing tensor decompositions. Often, we only have an element-wise or a slice-wise description of the signal model. This representation of the signal model does not reveal the explicit tensor structure. Therefore, the derivation of all tensor unfoldings is not always obvious. Consequently, exploiting the multi-dimensional structure of these models is not always straightforward. We propose an alternative representation of the element-wise multiplication or the slice-wise multiplication based on the generalized tensor contraction operator. Later in this thesis, we exploit this novel representation and the properties of the contraction operator such that we derive the final tensor models. There exist a number of different tensor decompositions that describe different signal models such as the HOSVD (Higher Order Singular Value Decomposition), the CP/PARAFAC (Canonical Polyadic / PARallel FACtors) decomposition, the BTD (Block Term Decomposition), the PARATUCK2 (PARAfac and TUCker2) decomposition, and the PARAFAC2 (PARAllel FACtors2) decomposition. Among these decompositions, the CP decomposition is most widely spread and used. Therefore, the development of algorithms for the efficient computation of the CP decomposition is important for many applications. The SECSI (Semi-Algebraic framework for approximate CP decomposition via SImultaneaous matrix diagonalization) framework is an efficient and robust tool for the calculation of the approximate low-rank CP decomposition via simultaneous matrix diagonalizations. In this thesis, we present five extensions of the SECSI framework that reduce the computational complexity of the original framework and/or introduce constraints to the factor matrices. Moreover, the PARAFAC2 decomposition and the PARATUCK2 decomposition are usually described using a slice-wise notation that can be expressed in terms of the generalized tensor contraction as proposed in this thesis. We exploit this novel representation to derive explicit tensor models for the PARAFAC2 decomposition and the PARATUCK2 decomposition. Furthermore, we use the PARAFAC2 model to derive an ALS (Alternating Least-Squares) algorithm for the computation of the PARAFAC2 decomposition. Moreover, we exploit the novel contraction properties for element wise and slice-wise multiplications to model MIMO multi-carrier wireless communication systems. We show that this very general model can be used to derive the tensor model of the received signal for MIMO-OFDM (Multiple-Input Multiple-Output - Orthogonal Frequency Division Multiplexing), Khatri-Rao coded MIMO-OFDM, and randomly coded MIMO-OFDM systems. We propose the transmission techniques Khatri-Rao coding and random coding in order to impose an additional tensor structure of the transmit signal tensor that otherwise does not have a particular structure. Moreover, we show that this model can be extended to other multi-carrier techniques such as GFDM (Generalized Frequency Division Multiplexing). Utilizing these models at the receiver side, we design several types for receivers for these systems that outperform the traditional matrix based solutions in terms of the symbol error rate. In the last part of this thesis, we show the benefits of using tensor algebra in biomedical signal processing by jointly decomposing EEG (ElectroEncephaloGraphy) and MEG (MagnetoEncephaloGraphy) signals. EEG and MEG signals are usually acquired simultaneously, and they capture aspects of the same brain activity. Therefore, EEG and MEG signals can be decomposed using coupled tensor decompositions such as the coupled CP decomposition. We exploit the proposed coupled SECSI framework (one of the proposed extensions of the SECSI framework) for the computation of the coupled CP decomposition to first validate and analyze the photic driving effect. Moreover, we validate the effects of scull defects on the measurement EEG and MEG signals by means of a joint EEG-MEG decomposition using the coupled SECSI framework. Both applications show that we benefit from coupled tensor decompositions and the coupled SECSI framework is a very practical tool for the analysis of biomedical data.Zahlreiche messbare Signale in verschiedenen Bereichen der digitalen Signalverarbeitung, z.B. in der drahtlosen Kommunikation, im Mobilfunk, biomedizinischen Anwendungen, der Bild- oder akustischen Signalverarbeitung und dem maschinellen Lernen sind mehrdimensional. Tensoren erhalten die mehrdimensionale Struktur und stellen eine natürliche Darstellung dieser Signale/Daten dar. Darüber hinaus bieten Tensoren oft eine verbesserte Trennbarkeit von enthaltenen Signalkomponenten. Daher profitieren wir von der Verwendung der Tensor-Algebra in den oben genannten Anwendungen und vielen mehr. In dieser Arbeit stellen wir die Vorteile der Nutzung der Tensor-Algebra in zwei Bereichen der Signalverarbeitung vor: drahtlose MIMO (Multiple-Input Multiple-Output) Kommunikationssysteme und biomedizinische Signalverarbeitung. Darüber hinaus tragen wir zu theoretischen Aspekten der Tensor-Algebra bei, indem wir neue Eigenschaften und Berechnungsmethoden für die Tensor-Zerlegung ableiten. Oftmals verfügen wir lediglich über eine elementweise oder ebenenweise Beschreibung des Signalmodells, welche nicht die explizite Tensorstruktur zeigt. Daher ist die Ableitung aller Tensor-Unfoldings nicht offensichtlich, wodurch die multidimensionale Struktur dieser Modelle nicht trivial nutzbar ist. Wir schlagen eine alternative Darstellung der elementweisen Multiplikation oder der ebenenweisen Multiplikation auf der Grundlage des generalisierten Tensor-Kontraktionsoperators vor. Weiterhin nutzen wir diese neuartige Darstellung und deren Eigenschaften zur Ableitung der letztendlichen Tensor-Modelle. Es existieren eine Vielzahl von Tensor-Zerlegungen, die verschiedene Signalmodelle beschreiben, wie die HOSVD (Higher Order Singular Value Decomposition), CP/PARAFAC (Canonical Polyadic/ PARallel FACtors) Zerlegung, die BTD (Block Term Decomposition), die PARATUCK2-(PARAfac und TUCker2) und die PARAFAC2-Zerlegung (PARAllel FACtors2). Dabei ist die CP-Zerlegung am weitesten verbreitet und wird findet in zahlreichen Gebieten Anwendung. Daher ist die Entwicklung von Algorithmen zur effizienten Berechnung der CP-Zerlegung von besonderer Bedeutung. Das SECSI (Semi-Algebraic Framework for approximate CP decomposition via Simultaneaous matrix diagonalization) Framework ist ein effizientes und robustes Werkzeug zur Berechnung der approximierten Low-Rank CP-Zerlegung durch simultane Matrixdiagonalisierung. In dieser Arbeit stellen wir fünf Erweiterungen des SECSI-Frameworks vor, welche die Rechenkomplexität des ursprünglichen Frameworks reduzieren bzw. Einschränkungen für die Faktormatrizen einführen. Darüber hinaus werden die PARAFAC2- und die PARATUCK2-Zerlegung in der Regel mit einer ebenenweisen Notation beschrieben, die sich in Form der allgemeinen Tensor-Kontraktion, wie sie in dieser Arbeit vorgeschlagen wird, ausdrücken lässt. Wir nutzen diese neuartige Darstellung, um explizite Tensormodelle für diese beiden Zerlegungen abzuleiten. Darüber hinaus verwenden wir das PARAFAC2-Modell, um einen ALS-Algorithmus (Alternating Least-Squares) für die Berechnung der PARAFAC2-Zerlegungen abzuleiten. Weiterhin nutzen wir die neuartigen Kontraktionseigenschaften für elementweise und ebenenweise Multiplikationen, um MIMO Multi-Carrier-Mobilfunksysteme zu modellieren. Wir zeigen, dass dieses sehr allgemeine Modell verwendet werden kann, um das Tensor-Modell des empfangenen Signals für MIMO-OFDM- (Multiple- Input Multiple-Output - Orthogonal Frequency Division Multiplexing), Khatri-Rao codierte MIMO-OFDM- und zufällig codierte MIMO-OFDM-Systeme abzuleiten. Wir schlagen die Übertragungstechniken der Khatri-Rao-Kodierung und zufällige Kodierung vor, um eine zusätzliche Tensor-Struktur des Sendesignal-Tensors einzuführen, welcher gewöhnlich keine bestimmte Struktur aufweist. Darüber hinaus zeigen wir, dass dieses Modell auf andere Multi-Carrier-Techniken wie GFDM (Generalized Frequency Division Multiplexing) erweitert werden kann. Unter Verwendung dieser Modelle auf der Empfängerseite entwerfen wir verschiedene Typen von Empfängern für diese Systeme, die die traditionellen matrixbasierten Lösungen in Bezug auf die Symbolfehlerrate übertreffen. Im letzten Teil dieser Arbeit zeigen wir die Vorteile der Verwendung von Tensor-Algebra in der biomedizinischen Signalverarbeitung durch die gemeinsame Zerlegung von EEG-(ElectroEncephaloGraphy) und MEG- (MagnetoEncephaloGraphy) Signalen. Diese werden in der Regel gleichzeitig erfasst, wobei sie gemeinsame Aspekte derselben Gehirnaktivität beschreiben. Daher können EEG- und MEG-Signale mit gekoppelten Tensor-Zerlegungen wie der gekoppelten CP Zerlegung analysiert werden. Wir nutzen das vorgeschlagene gekoppelte SECSI-Framework (eine der vorgeschlagenen Erweiterungen des SECSI-Frameworks) für die Berechnung der gekoppelten CP Zerlegung, um zunächst den photic driving effect zu validieren und zu analysieren. Darüber hinaus validieren wir die Auswirkungen von Schädeldefekten auf die Messsignale von EEG und MEG durch eine gemeinsame EEG-MEG-Zerlegung mit dem gekoppelten SECSI-Framework. Beide Anwendungen zeigen, dass wir von gekoppelten Tensor-Zerlegungen profitieren, wobei die Methoden des gekoppelten SECSI-Frameworks erfolgreich zur Analyse biomedizinischer Daten genutzt werden können

Applications of multi-way analysis for characterizing paediatric electroencephalogram (EEG) recordings

This doctoral thesis outlines advances in multi-way analysis for characterizing electroencephalogram (EEG) recordings from a paediatric population, with the aim to describe new links between EEG data and changes in the brain. This entails establishing the validity of multi-way analysis as a framework for identifying developmental information at the individual and collective level. Multi-way analysis broadens matrix analysis to a multi-linear algebraic architecture to identify latent structural relationships in naturally occurring higher order (n-way) data, like EEG. We use the canonical polyadic decomposition (CPD) as a multi-way model to efficiently express the complex structures present in paediatric EEG recordings as unique combinations of low-rank matrices, offering new insights into child development. This multi-way CPD framework is explored for both typically developing (TD) children and children with potential developmental delays (DD), e.g. children who suffer from epilepsy or paediatric stroke. Resting-state EEG (rEEG) data serves as an intuitive starting point in analyzing paediatric EEG via multi-way analysis. Here, the CPD model probes the underlying relationships between the spatial, spectral and subject modes of several rEEG datasets. We demonstrate the CPD can reveal distinct population-level features in rEEG that reflect unique developmental traits in varying child populations. These development-affiliated profiles are evaluated with respect to capturing structures well-established in childhood EEG. The identified features are also interrogated for their predictive abilities in anticipating new subjects’ ages. Assessing simulations and real rEEG datasets of TD and DD children establishes the multi-way analysis framework as well suited for identifying developmental profiles from paediatric rEEG. We extend the multi-way analysis scheme to more complex EEG scenarios common in EEG rehabilitation technology, like brain-computer interfaces. We explore the feasibility of multi-way modelling for interventions where developmental changes often pose as barriers. The multi-way CPD model is expanded to include four modes- task, spatial, spectral and subject data, with non-negativity and orthogonality constraints imposed. We analyze a visual attention task that elucidates a steady-state visual evoked potential and present the advantages gained from the extended CPD model. Through direct multi-linear projection, we demonstrate that linear profiles of the CPD can be capitalized upon for rapid task classification sans individual subject classifier calibration. Incorporating concepts from the multi-way analysis scheme with child development measured by psychometric tests, we propose the Joint EEG Development Inference (JEDI) model for inferring development from paediatric EEG. We utilize a common EEG task (button-press) to establish a 4-way CPD model of paediatric EEG data. Structured data fusion of the CPD model and cognitive scores from psychometric evaluations then permits joint decomposition of the two datasets to identify common features associated with each representation of development. Use of grid search optimization and a fully cross-validated design supports the JEDI model as another technique for rapidly discerning the developmental status of a child via EEG. We then briefly turn our attention to associating child development as measured by psychometric tests to markers in the EEG using graph network properties. Using graph networks, we show how the functional connectivity can inform on potential developmental delays in very young epileptic children using routine, clinical rEEG measures. This establishes a potential tool complementary to the JEDI model for identifying and inferring links between the established psychometric evaluation of developing children and functional analysis of the EEG. Multi-way analysis of paediatric EEG data offers a new approach for handling the developmental status and profiles of children. The CPD model offers flexibility in terms of identifying development-related features, and can be integrated into EEG tasks common in rehabilitation paradigms. We aim for the multi-way framework and associated techniques pursued in this thesis to be integrated and adopted as a useful tool clinicians can use for characterizing paediatric development