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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.

We present a technique that identifies truly interacting subsystems of a complex system from multichannel data if the recordings are an unknown linear and instantaneous mixture of the true sources. The method is valid for arbitrary noise structure. For this, a blind source separation technique is proposed that diagonalizes antisymmetrized cross- correlation or cross-spectral matrices. The resulting decomposition finds truly interacting subsystems blindly and suppresses any spurious interaction stemming from the mixture. The usefulness of this interacting source analysis is demonstrated in simulations and for real electroencephalography data

A new efficient algorithm is presented for joint diagonalization of several matrices. The algorithm is based on the Frobenius-norm formulation of the joint diagonalization problem, and addresses diagonalization with a general, non- orthogonal transformation. The iterative scheme of the algorithm is based on a multiplicative update which ensures the invertibility of the diagonalizer. The algorithm's efficiency stems from the special approximation of the cost function resulting in a sparse, block-diagonal Hessian to be used in the computation of the quasi-Newton update step. Extensive numerical simulations illustrate the performance of the algorithm and provide a comparison to other leading diagonalization methods. The results of such comparison demonstrate that the proposed algorithm is a viable alternative to existing state-of-the-art joint diagonalization algorithms. The practical use of our algorithm is shown for blind source separation problems

Independent component analysis of noninvasively recorded cortical magnetic DC-fields in humans
(2000)

Unmixing hyperspectral data
(2000)

Recently blind source separation (BSS) methods have been highly successful when applied to biomedical data. This paper reviews the concept of BSS and demonstrates its usefulness in the context of event-related MEG measurements. In a first experiment we apply BSS to artifact identification of raw MEG data and discuss how the quality of the resulting independent component projections can be evaluated. The second part of our study considers averaged data of event-related magnetic fields. Here, it is particularly important to monitor and thus avoid possible overfitting due to limited sample size. A stability assessment of the BSS decomposition allows to solve this task and an additional grouping of the BSS components reveals interesting structure, that could ultimately be used for gaining a better physiological modeling of the data

We propose two methods that reduce the post-nonlinear blind source separation problem (PNL-BSS) to a linear BSS problem. The first method is based on the concept of maximal correlation: we apply the alternating conditional expectation (ACE) algorithm-a powerful technique from nonparametric statistics-to approximately invert the componentwise nonlinear functions. The second method is a Gaussianizing transformation, which is motivated by the fact that linearly mixed signals before nonlinear transformation are approximately Gaussian distributed. This heuristic, but simple and efficient procedure works as good as the ACE method. Using the framework provided by ACE, convergence can be proven. The optimal transformations obtained by ACE coincide with the sought-after inverse functions of the nonlinearitics. After equalizing the nonlinearities, temporal decorrelation separation (TDSEP) allows us to recover the source signals. Numerical simulations testing "ACE-TD" and "Gauss-TD" on realistic examples are performed with excellent results

Phase synchronization is an important phenomenon that occurs in a wide variety of complex oscillatory processes. Measuring phase synchronization can therefore help to gain fundamental insight into nature. In this Letter we point out that synchronization analysis techniques can detect spurious synchronization, if they are fed with a superposition of signals such as in electroencephalography or magnetoencephalography data. We show how techniques from blind source separation can help to nevertheless measure the true synchronization and avoid such pitfalls