@incollection{Lieb2013, author = {Lieb, Florian}, title = {Audio Inpainting Using M-Frames}, series = {Current Trends in Analysis and Its Applications}, volume = {2015}, booktitle = {Current Trends in Analysis and Its Applications}, publisher = {Springer International Publishing}, isbn = {978-3-319-12576-3}, doi = {10.1007/978-3-319-12577-0_77}, pages = {705 -- 713}, year = {2013}, abstract = {Classical short-time Fourier constructions lead to a signal decomposition with a fixed time-frequency resolution. However, having signals with varying features, such time-frequency decompositions are very restrictive. A more flexible and adaptive sampling of the time-frequency plane is achieved by the nonstationary Gabor transform. Here, the resolution can evolve over time or frequency, respectively, by using different windows for the different sampling positions in the time or frequency domain (Multiwindow-frames). This adaptivity in the time-frequency plane leads to a sparser signal representation. In terms of audio inpainting, i.e., filling in blanks of a depleted audio signal, sparsity in some representation space profoundly influences the quality of the reconstructed signal. We will compare this quality using different nonstationary Gabor transforms and the regular Gabor transform with different types of audio signals.}, subject = {Mathematik}, language = {en} } @incollection{Lantzberg2013, author = {Lantzberg, Daniel}, title = {Rank-M Frame Multipliers and Optimality Criterions for Density Operators of Rank M}, series = {Current Trends in Aanalysis and Its Applications}, volume = {2015}, booktitle = {Current Trends in Aanalysis and Its Applications}, publisher = {Springer}, isbn = {978-3-319-12576-3}, doi = {10.1007/978-3-319-12577-0_76}, pages = {695 -- 704}, year = {2013}, abstract = {Ever since the introduction of frames in 1952 [4], the connection between frame theory and decompositions of certain operators, particularly the identity operator, into rank-ones began to be elaborated. Abandoning the idea of restricting to tight frame-like expansions, with respect to systems arising from a single template function, one is led to the concept of resolutions of the identity, with respect to more general systems than the usual rank-one expansions of the identity. In this study, we will investigate various notions of possible generalizations of optimality criterions for rank-M frames and corresponding multipliers. Explicitly, we will lay stress on continuous M-frames, arising from irreducible group representations of locally compact groups, have a look at its connection to time-frequency analysis and comment on adequate notions of optimality.}, subject = {Mathematik}, language = {en} }