## Institut für Informatik und Computational Science

This thesis is concerned with the solution of the blind source separation problem (BSS). The BSS problem occurs frequently in various scientific and technical applications. In essence, it consists in separating meaningful underlying components out of a mixture of a multitude of superimposed signals. In the recent research literature there are two related approaches to the BSS problem: The first is known as Independent Component Analysis (ICA), where the goal is to transform the data such that the components become as independent as possible. The second is based on the notion of diagonality of certain characteristic matrices derived from the data. Here the goal is to transform the matrices such that they become as diagonal as possible. In this thesis we study the latter method of approximate joint diagonalization (AJD) to achieve a solution of the BSS problem. After an introduction to the general setting, the thesis provides an overview on particular choices for the set of target matrices that can be used for BSS by joint diagonalization. As the main contribution of the thesis, new algorithms for approximate joint diagonalization of several matrices with non-orthogonal transformations are developed. These newly developed algorithms will be tested on synthetic benchmark datasets and compared to other previous diagonalization algorithms. Applications of the BSS methods to biomedical signal processing are discussed and exemplified with real-life data sets of multi-channel biomagnetic recordings.

The goal of a Brain-Computer Interface (BCI) consists of the development of a unidirectional interface between a human and a computer to allow control of a device only via brain signals. While the BCI systems of almost all other groups require the user to be trained over several weeks or even months, the group of Prof. Dr. Klaus-Robert Müller in Berlin and Potsdam, which I belong to, was one of the first research groups in this field which used machine learning techniques on a large scale. The adaptivity of the processing system to the individual brain patterns of the subject confers huge advantages for the user. Thus BCI research is considered a hot topic in machine learning and computer science. It requires interdisciplinary cooperation between disparate fields such as neuroscience, since only by combining machine learning and signal processing techniques based on neurophysiological knowledge will the largest progress be made. In this work I particularly deal with my part of this project, which lies mainly in the area of computer science. I have considered the following three main points: <b>Establishing a performance measure based on information theory:</b> I have critically illuminated the assumptions of Shannon's information transfer rate for application in a BCI context. By establishing suitable coding strategies I was able to show that this theoretical measure approximates quite well to what is practically achieveable. <b>Transfer and development of suitable signal processing and machine learning techniques:</b> One substantial component of my work was to develop several machine learning and signal processing algorithms to improve the efficiency of a BCI. Based on the neurophysiological knowledge that several independent EEG features can be observed for some mental states, I have developed a method for combining different and maybe independent features which improved performance. In some cases the performance of the combination algorithm outperforms the best single performance by more than 50 %. Furthermore, I have theoretically and practically addressed via the development of suitable algorithms the question of the optimal number of classes which should be used for a BCI. It transpired that with BCI performances reported so far, three or four different mental states are optimal. For another extension I have combined ideas from signal processing with those of machine learning since a high gain can be achieved if the temporal filtering, i.e., the choice of frequency bands, is automatically adapted to each subject individually. <b>Implementation of the Berlin brain computer interface and realization of suitable experiments:</b> Finally a further substantial component of my work was to realize an online BCI system which includes the developed methods, but is also flexible enough to allow the simple realization of new algorithms and ideas. So far, bitrates of up to 40 bits per minute have been achieved with this system by absolutely untrained users which, compared to results of other groups, is highly successful.