New methods for anechoic demixing with application to shift invariant feature extraction

Blind source separation problems emerge in many applications, where signals can be modeled as superpositions of multiple sources. Many popular applications of blind source separation are based on linear instantaneous mixture models. If specific invariance properties are known about the sources, e.g. translation or rotation invariance, the simple linear model can be extended by inclusion of the corresponding transformations. When the sources are invariant against translations (i.e. spatial displacements or time shifts) the resulting model is called anechoic mixing model. The main focus of this thesis is the development of new mathematical framework for the solution of the anechoic mixing problem and the successive derivation of concrete algorithms. This framework integrates approaches from many distinct fields of signal processing like stochastic time-frequency analysis, convex optimization, projection onto convex set methods, delay estimation and naturally blind source separation. The developed method is tested on a variety of applications including music recordings, natural two dimensional images, two-dimensional shapes and optic flow. However the main application is the analysis and synthesis of human motion trajectories, which is motivated by the idea in motor control that complex motor behavior can be explained by a superposition of simple basis components, or spatio-temporal primitives. The new anechoic demixing algorithm allows to approximate high-dimensional movement trajectories accurately based on a small number of learned primitives or source signals. It is demonstrated that the new method is significantly more accurate than other common techniques. This allows the modeling of subtle style changes, like the bodily expression of emotion as well as a sufficient synthesis quality for computer animation with only few mixture components.