## AWG Computational Data Analysis

Leitung: Prof. Dr. Emily King

The research group Computational Data Analysis was established in April 2014 and is supported by the Excellence Initiative. This group is officially associated with the Institute for Industrial Mathematics (ZeTeM) and the Institute for Algebra, Geometry, Topology und their Applications (ALTA). Additionally, we are working on establishing a long-term partnership with the Institute for Environmental Physics (IUP).

The mathematical focus of the research group Computational Data Analysis is algebraic and applied harmonic analysis. In this case, harmonic analysis includes in addition to classical Fourier analysis and the more modern time-frequency analysis or wavelet analysis all methods in which an object is analyzed with a change-of-basis-like transformation and then processed in the transformation domain. This additional level of abstraction allows one to connect certain areas of research, for example wavelet analysis over a p-adic domain and structured dictionary learning. So-called sparsity-based methods, which have only recently received a lot of attention in mathematical research, also play a large role in the work of the research group Computational Data Analysis. Central to this is the assumption that representations of certain types of data (i.e. natural images or pieces of music) can always be represented by a linear combination of a small number of corresponding atomic functions associated with a transformation. If one chooses the correct transformation for the data, which is able to represent the data with only a few significant coefficients, the sparsity prerequisite can be used to regularize otherwise under-determined problems.

A further specialty of the research group Computational Data Analysis is shearlet theory, which can be understood as a further development or generalization of wavelet analysis. Shearlet systems are particularly suitable for the representation of geometric characteristics of two- or more dimensional data, which occur in natural images or seismic measurements. This is achieved in the construction of shearlet systems by an anisotropic scaling of the underlying generating function in a particular dimension (in contrast to multi-dimensional wavelet bases in which the generated wavelet is scaled isotropic or uniformly), which is then combined with shearing. As a result of this process, highly directional atoms are created which are capable of sparsely representing directional and curvilinear features.

Currently the research group Computational Data Analysis is collaborating with the research group Technical Thermodynamics of the Production Engineering / Industrial Engineering & Management department and the research group Physical Analysis of Remote Sensing Images of the Physics department to develop better image processing methods using shearlet systems for data of the respective subjects.

(Hilbert space) frames generalize orthonormal bases but allow redundancy, making them more powerful in a number of applications as well as an important tool in modern harmonic analysis. The connections between frames and certain algebro- and discrete geometric structures has only recently begun to be explored. The further development of this connection both theoretically and to construct new and useful frames is the goal of the Exploratory Project ''Hilbert space frames and algebraic geometry'' started in October in cooperation with ALTA.