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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 [1] 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 [2] 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 [3] 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 [4]. Shearlets are to date often used
in combination with compressed sensing [5] techniques for solving
inverse problems [6] or developing efficient numerical solvers for
certain partial differential equations [7].

## 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 [8].
- 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.

Download (PDF, 1,2 MB) [9] - G. Kutyniok and D. Labate.
**Introduction to Shearlets.***Shearlets: Multiscale Analysis for Multivariate Data, 1-38, Birkhäuser Boston, 2012.*

Download (PDF, 401,7 KB) [10] - G. Kutyniok, J.
Lemvig, and W.-Q Lim.

**Shearlets and Optimally Sparse Approximations.**

*Shearlets: Multiscale Analysis for Multivariate Data, 145-198, Birkhäuser Boston, 2012.*Download (PDF, 952,8 KB) [11] - G. Kutyniok, W.-Q Lim,
and X. Zhuang.

**Digital Shearlet Transforms.**

*Shearlets: Multiscale Analysis for Multivariate Data, 239-282, Birkhäuser Boston, 2012.*Download (PDF, 690,0 KB) [12]

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iok/Kutyniok/Papers/GAMM_Shearlets.pdf

niok/Kutyniok/Papers/SHBookIntro.pdf

niok/Kutyniok/Papers/OptimallySparseApproximations.pdf

niok/Kutyniok/Papers/DigitalShearletTransforms.pdf