Mathematics
A mathematical background is very important and useful for all physical and engineering sciences. The connection between applied and mathematical research often leads to progress in both directions, due to natural synergy effects. The Acoustic Research Institute considers the investigation of the mathematical background of its numerous research projects, most prominently the signal processing aspects, as an important part of acoustic research.
The fundamental mathematical backbone in the analysis of acoustic signals are time-frequency representations. Our main focus is thus their thorough theoretical investigation. This includes the development and understanding of abstract mathematical results, as well as their implications through discretization and quantization for real world signals and thereby facilitate the way to applications.
The cooperation of the cluster Mathematics with the other clusters of the Institute has been proven to be very fruitful for all partners and will be further strengthened. While the other clusters benefit from our methodologies to solve their relevant problems, well-based in theory, the mathematicians can solve questions relevant for applications while exploring new and interesting theoretical directions. This dialog increases the understanding of other fields enormously.
Frame Theory
Frame Theory
Head: Peter Balazs
A long standing research focus is frame theory, i.e., the mathematical theory of redundant representations, and its connection to other disciplines, like signal processing, physics, numerics and machine learning, and the related applications in psycho- and bioacoustics as well as speech and music signal processing. To that end implementation and optimization concerns become important parts of the research program. We investigate abstract frames, those with special structure (Gabor, wavelet or CNNs) and generalized concepts. We look at the representation not only of elements in Hilbert spaces – “vectors” – but also operators. We go beyond the linear aspect of frame theory and investigate non-linear frames, e.g., in phase retrieval or the layers of Deep Neural Networks. The group covers the range from the purely abstract to applied results, from the continuous theory to the discrete and the finite dimensional one.
Information and Inference
Information and Inference
Head: Günther Koliander
We focus on the integration of concepts from probability theory and Bayesian signal processing into time-frequency analysis and acoustic signal processing. One goal is a fundamental understanding of time-frequency representations of random processes with a current focus on Gaussian processes and the zero set of their short time Fourier transform. In a second line of research, the group establishes methods for the combination of data-driven and model-based inference approaches to utilize the ongoing success of poorly understood deep learning based methods in a strictly Bayesian framework.
Universal Discretization of Frames
Universal Discretization of Frames
Head: Nicki Holighaus
Our central goal is the development of mathematical theory that enables the construction of novel, universal discretization schemes for frames, in particular continuous frames. Like (discrete) frames, continuous frames are function dictionaries that can be used to decompose complex data into simple building blocks, so-called atoms. However, continuous frames often consist of uncountably many atoms, they are highly redundant. Hence, the reduction of the dictionary size is essential for practical use of continuous frames. We investigate reduction schemes that can be applied under mild and rather general conditions on the dictionary, i.e., they are universal. This step is crucial to facilitate the use of more general dictionaries and their efficient implementation across various applications.
Further Information
Further Information
Youtube Talks:
- Nicki Holighaus - Time-Frequency Frames and Applications to Audio Analysis - Part 1
- Peter Balazs - February Fourier Talks 2014
- Peter Balazs - TUForMath Vortrag: Mathematik und Akustik
- Hans G. Feichtinger - Mathematical and Numerical Aspects of Frame Theory - Part 1 (showing the institute own software STx!)
- Georg Tauböck - WWTF Project INSIGHT
Links:
- Dictionary Learning for Sparse Audio Inpainting
- On Dual Frames