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Main research areas

Shearlet theory.

Efficient encoding of anisotropic structures is essential in a variety of areas in applied and pure mathematics such as, for instance, in the analysis of edges in images, when sparsely approximating solutions of particular hyperbolic PDEs, as well as deriving sparse expansions of Fourier Integral Operators. It is well known that wavelets – although perfectly suited for isotropic structures – do not perform equally well when dealing with anisotropic phenomena.

Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while – in contrast to previously introduced directional representation systems – providing a unified treatment of the continuous and digital world. One main idea in the construction is to parametrize directions by slope through shear matrices rather than angle, which greatly supports the treating of the digital setting.

For more information, we refer the interested reader to www.shearlet.org and www.shearlab.org.

Wavelet and time-frequency analysis.

Wavelets are nowadays indispensable as a multiscale encoding system for a wide range of more theoretically to more practically oriented tasks, since they provide optimal approximation rates for smooth 1D data exhibiting singularities. The facts that they provide a unified treatment in both the continuous as well as digital setting and that the digital setting admits a multiresolution analysis leading to a fast spatial domain decomposition were essential for their success.

Time-frequency analysis, manifested through the representation systems called Gabor systems, is particularly suited to sparsely decompose and analyze smooth (sometimes also periodic) data. One main application of Gabor systems is the analysis of audio data.

Sparse recovery, \ell_1 minimization, and compressed sensing.

During the last three years, sparsity has become a key concept in various areas of applied mathematics, computer science, and electrical engineering. Sparsity methodologies explore the fundamental fact that many types of data/signals can be represented by only a few non-vanishing coefficients when choosing a suitable basis or, more generally, a frame. If signals possess such a sparse representation, they can in general be recovered from few measurements using \ell_1 minimization techniques.

Compressed Sensing, which was recently introduced by Donoho (Stanford U.) and Candes (Stanford U.), Romberg (Georgia Tech), and Tao (UCLA), has gained particularly rapid attention by providing methods for measuring sparse signals with an optimally small number of (random) measurements.

Frame theory and fusion frame theory.

Frames have been a focus of study in the last two decades in applications where redundancy plays a vital and useful role. However, recently, a number of new applications have emerged which cannot be modeled naturally by one single frame system. They typically share a common property that requires distributed processing such as sensor networks.

Fusion frames, which were recently introduced by Casazza (U. Missouri) and Kutyniok, extend the notion of a frame and provide exactly the mathematical framework not only to model these applications but also to derive efficient and robust algorithms. In particular, fusion frames generalize frame theory by using subspaces in the place of vectors as signal building blocks. Thus signals can be represented as linear combinations of components that lie in particular, and often overlapping, signal subspaces. Such a representation provides significant flexibility in representing signals of interest compared to classical frame representations.

For more information, we refer the interested reader to www.fusionframe.org.

Image and signal processing: denoising, geometric separation,...

The deluge of data, which we already witness now, will require the development of highly efficient data processing techniques in the future. The previously described novel mathematical methodologies have recently opened a new chapter in data processing, in particular, in image and signal processing, by bringing new ideas to classical tasks such as denoising, edge detection, inpainting, and image registration, but also new tasks such as efficient sensing and geometric separation.

Attach:test.png Δ

Shearlets

In the following we will give a short introduction into the theory of shearlets. Unlike the traditional wavelet transform does not posses the ability to detect directionality, since it is merely associated with two parameters, the scaling parameter a and the the translation parameter t. The idea now is to define a transform, which overcomes this vice, while retaining most aspects of the mathematical framework of wavelets, e.g., the fact that

  • the associated system forms an affine system,
  • the transform can be regarded as matrix coefficients of a unitary representation of a special group,
  • there is an MRA-structure associated with the systems.

The Continuous Theory: The basic idea for the definition of continuous shearlets is the usage of a 2-parameter dilation group, which consists of products of parabolic scaling matrices and shear matrices. Hence the continuous shearlets depend on three parameters, the scaling parameter a > 0, the shear parameter s \in \mathbb{R} and the translation parameter t \in \mathbb{R}^2, and they are defined by

\psi_{a,s,t}(x)=a^{-3/4} \psi((D_{a,s}^{-1}(x-t)) \qquad \text{where} \quad D_{a,s} = \begin{bmatrix} a & -a^{1/2}s \\ 0 & a^{1/2} \end{bmatrix}.

The mother shearlet function \psi is defined almost like a tensor product by

\hat\psi(\xi_1,\xi_2) = \hat\psi_1(\xi_1) \hat\psi_2\Bigl(\frac{\xi_2}{\xi_1}\Bigr),
where \psi_1 is a wavelet and \psi_2\ is a bump function. The figure on the right hand side illustrates the behavior of the continuous shearlets in frequency domain assuming that \psi_1 and \psi_2\ are chosen to be compactly supported in frequency domain.

The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. Attach:cont_shear.jpg Δ The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity.

Sparse recovery of underdetermined systems, \ell_1-minimization

Theorem (Cassels: Intro. to the Geometry of Numbers)

For all subsets \Gamma \subset \mathbb{R}^n is a lattice, if and only if,

  • \Gamma contains n linearily independent vectors
  • x,y \in \Gamma \Rightarrow x \pm y \in \Gamma
  • \exists r >0 s.t. if y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}, then y=0.