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# Applied Harmonic Analysis

Applied harmonic analysis has established itself as the main area in applied mathematics focused on the efficient representation, analysis, and encoding of data. The primary object of this discipline is the process of ‘breaking into pieces’ – from the Greek word analysis –, to gain insight into an object.

The Fourier basis is historically the first representation system, which significantly impacted various areas within mathematics but also in applications. The associated Fast Fourier Transform (FFT) is to date in fact one of the most often utilized algorithm in numerical computations. However, the Fourier basis has certain disadvantages such as non-locality and also does not sparsely approximates singularities, which are the most distinct and hence important features of functions/signals.

Historically, the introduction of wavelets about 20 years ago represents a milestone in the development of efficient encoding of piecewise regular signals. The major reason for the spectacular success of wavelets consists not only in their ability to provide optimally sparse approximations of a large class of frequently occurring signals and to represent singularities much more efficiently than traditional Fourier methods, but also in the existence of fast algorithmic implementations which precisely digitalize the continuum domain transforms. Wavelets are nowadays widely used both for more theoretical tasks such as for elliptic partial differential equations or for more practical tasks such as for the image compression standard JPEG2000.

Despite their success, wavelets are not very effective when dealing with multivariate data.The reason for this failure is that multivariate data is typically governed by anisotopic features such as edges in images, whereas wavelets are isotropic objects, hence not optimally adapted to those. The limitations of wavelets and traditional multiscale systems have stimulated a flurry of activity involving mathematicians, engineers, and applied scientists. Many systems were suggested such as ridgelets and curvelets.

In 2006 the system of shearlets was introduced by Kutyniok and Labate and is by now the first anisotropic system, which optimally sparsely approximates anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for which a compactly supported version is available for high spatial localization, and which admits a unified treatment of the continuum and digital world to ensure faithful implementations. As most of the other anisotropic representation systems, they do not form an orthonormal basis, but a frame. Shearlets are to date often used in combination with compressed sensing techniques for solving inverse problems or developing efficient numerical solvers for certain partial differential equations.

## Some of our Research Topics

• Development of more general frameworks including shearlets such as parabolic molecules and alpha-molecules.
• Introducing novel shearlet systems for dealing with, for instance, data on bounded domains.
• Development of the software package ShearLab.
• Development and analysis of approaches using compressed sensing in combination with shearlets for solving inverse problems such as the inverse scattering problem.
• Development and analysis of approaches to numerically solve certain partial differential equations such as transport equations using shearlets as a trial basis for sparsely approximating the solution.
• Application of shearlets to speed up the data acquisition in Magnetic Resoance Imaging.
• Application of shearlets to solve imaging problems such as inpainting or feature extraction, also in real-world problems such as electron microscopy.
• Parametrized shearlet systems and learning of their parameters.

## Survey Papers

• G. Kutyniok, W.-Q Lim, and G. Steidl.
Shearlets: Theory and Applications.
GAMM-Mitteilungen 37 (2014), 259-280.
• G. Kutyniok and D. Labate.
Introduction to Shearlets.
Shearlets: Multiscale Analysis for Multivariate Data, 1-38, Birkhäuser Boston, 2012.
• G. Kutyniok, J. Lemvig, and W.-Q Lim.
Shearlets and Optimally Sparse Approximations.
Shearlets: Multiscale Analysis for Multivariate Data, 145-198, Birkhäuser Boston, 2012.