Research.Researchold History
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The central research area is frame theory, in particular:
- Finite-dimensional frames
- General frames in Hilbert spaces
- Structured frames in L^2(\mathbb{R}^n): Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
- Structured frames on LCA groups
Main research areas
Project Homepages
- Shearlets
- ShearLab
- Fusion Frame
DO NOT READ -- THIS WILL UPDATED/REMOVED
Main research areas
- Shearlet theory.
- Wavelet and time-frequency analysis.
- Sparse recovery, \ell_1 minimization, and compressed sensing.
- Frame theory and fusion frame theory.
- Image and signal processing: denoising, geometric separation, inpainting, ...
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &dnash; 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.
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.
Current ongoing research projects
Main research areas
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.
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 [[http://www.fusionframe.org | www.fusionframe.org]].
For more information, we refer the interested reader to www.fusionframe.org.
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.
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.
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 [[http://www.fusionframe.org | www.fusionframe.org]].
..coming soon..
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.
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &dnash; in contrast to previously introduced directional representation systems &dnash; 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.
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &dnash; 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.
..coming soon..
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.
..coming soon..
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.
..coming soon..
For more information, we refer the interested reader to and .
For more information, we refer the interested reader to www.shearlet.org and www.shearlab.org.
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 &dnash; in contrast to previously introduced directional representation systems &dnash; 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.
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 &dnash; in contrast to previously introduced directional representation systems &dnash; 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.
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while
&dnash; in contrast to previously introduced directional representation systems &dnash; providing a unified treatment of the continuous and digital world. One main idea in the construction is to parametrize
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &dnash; in contrast to previously introduced directional representation systems &dnash; providing a unified treatment of the continuous and digital world. One main idea in the construction is to parametrize
well known that wavelets - although perfectly suited for isotropic structures - do
well known that wavelets – although perfectly suited for isotropic structures – do
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.
Shearlets, which were recently introduced by Kutyniok, Labate (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while
&dnash; in contrast to previously introduced directional representation systems &dnash; 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.
..coming soon..
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 and .
Image and signal processing: denoising, geometric separation,...
- Wavelet and time-frequency analysis.
- Sparse recovery, \ell_1 minimization, and compressed sensing.
- Frame theory and fusion frame theory.
- Image and signal processing: denoising, geometric separation, inpainting, ...
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
The mother shearlet function \psi is defined almost like a tensor product by
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 transfor
- Wavelet and time-frequency analysis.
- Sparse recovery, \ell_1 minimization, and compressed sensing.
- Frame theory and fusion frame theory.
- Image and signal processing: denoising, geometric separation, inpainting, ...
Wavelet and time-frequency analysis.
Sparse recovery, \ell_1 minimization, and compressed sensing.
Frame theory and fusion frame theory.
Image and signal processing: denoising, geometric separation, inpainting, ...
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
The mother shearlet function \psi is defined almost like a tensor product by
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 transfor
- Shearlets
- Sparse recovery of underdetermined systems, \ell_1-minimization
- Shearlet theory?.
- Wavelet and time-frequency analysis.
- Sparse recovery, \ell_1 minimization, and compressed sensing.
- Frame theory and fusion frame theory.
- Image and signal processing: denoising, geometric separation, inpainting, ...
Shearlet theory.
..coming soon..
Wavelet and time-frequency analysis.
Sparse recovery, \ell_1 minimization, and compressed sensing.
Frame theory and fusion frame theory.
Image and signal processing: denoising, geometric separation, inpainting, ...
- \exists r >0 s.t. if y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}, then y=0.
- \exists r >0 s.t. if y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}, then y=0.
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
The mother shearlet function \psi is defined almost like a tensor product by
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.
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
The mother shearlet function \psi is defined almost like a tensor product by
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.
Shearlets
Shearlets
Sparse recovery of underdetermined systems, \ell_1-minimization
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. https://hata.compute.dtu.dk/pub/../files/cont_shear.jpg
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. cont_shear.jpg"support in the 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. https://hata.compute.dtu.dk/pub/../files/cont_shear.jpg
Matlab code
This is implemented in Matlab by the following code:
% for loops in Matlab % Set the number of terms n=10; % initialize sum Q=0; % summing over k for k=1:n Q=Q+1/k^2; end sprintf('Approximation with %d terms: Q = %0.10g',n,Q)
- Sparse recovery of underdetermined systems, \ell_1-minimization,
- Neuro
- Sparse recovery of underdetermined systems, \ell_1-minimization
Current ongoing research projects:
Current ongoing research projects
- \exists r >0 s.t. if y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}, then y=0.
- \exists r >0 s.t. if y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}, then y=0.
Research interest:
- In general: Applied Harmonic Analysis, functional analysis, Hilbert space theory, Fourier analysis, numerical analysis, and signal processing.
- In particular: L^2 theory of wavelets and shearlets, constructions and characterizations results, explicit constructions of dual wavelet frames, canonical and alternate duals.
- Also: Sparse recovery of underdetermined systems, \ell_1-minimization, Quasi wavelet frames, oversampling of wavelet frames, shift invariant spaces.
Current ongoing research projects:
- Generalization of Publ. 2 to \mathbb{R}^n. For any given expansive nxn matrix A we want a procedure that allow for constructions of pairs of dual frame wavelets for L^2(\mathbb{R}^n) in a very explicit way. Keywords: Characterizing equations for dual wavelet frames in L^2(\mathbb{R}^n), the unit ball in the adaptive norm associated with A^T, densest regular (lattice) packing of ellipsoids in \mathbb{R}^n, A-dilative partition of unity.
- (with ) Affine and quasi affine frames with adaptive oversampling for rational n \times n dilations. Keywords: (Generalized) shift-invariant systems, integral sublattice and extended integral superlattice in \mathbb{R}^n, oversampling SI-systems, translational averaging wavelet functionals. preprint Δ
- (with ) Oversampling of Wavelet Frames for Real Dilations. For any real, expansive dilation matrix A, we specify condtions on the lattice \Lambda \supset \mathbb{Z}^n such that the frame property carries over from the affine system with \mathbb{Z}^n as translation lattice \mathcal{A}(\Psi,A,\mathbb{Z}^n) to the affine system with \Lambda as translation lattice \mathcal{A}(\Psi,A,\Lambda).
Current ongoing research projects:
- Shearlets
- Sparse recovery of underdetermined systems, \ell_1-minimization,
- Neuro
Shearlets
Research interest:
- In general: Applied Harmonic Analysis, functional analysis, Hilbert space theory, Fourier analysis, numerical analysis, and signal processing.
- In particular: L^2 theory of wavelets and shearlets, constructions and characterizations results, explicit constructions of dual wavelet frames, canonical and alternate duals.
- Also: Sparse recovery of underdetermined systems, \ell_1-minimization, Quasi wavelet frames, oversampling of wavelet frames, shift invariant spaces.
Current ongoing research projects:
- Generalization of Publ. 2 to \mathbb{R}^n. For any given expansive nxn matrix A we want a procedure that allow for constructions of pairs of dual frame wavelets for L^2(\mathbb{R}^n) in a very explicit way. Keywords: Characterizing equations for dual wavelet frames in L^2(\mathbb{R}^n), the unit ball in the adaptive norm associated with A^T, densest regular (lattice) packing of ellipsoids in \mathbb{R}^n, A-dilative partition of unity.
- (with ) Affine and quasi affine frames with adaptive oversampling for rational n \times n dilations. Keywords: (Generalized) shift-invariant systems, integral sublattice and extended integral superlattice in \mathbb{R}^n, oversampling SI-systems, translational averaging wavelet functionals. preprint Δ
- (with ) Oversampling of Wavelet Frames for Real Dilations. For any real, expansive dilation matrix A, we specify condtions on the lattice \Lambda \supset \mathbb{Z}^n such that the frame property carries over from the affine system with \mathbb{Z}^n as translation lattice \mathcal{A}(\Psi,A,\mathbb{Z}^n) to the affine system with \Lambda as translation lattice \mathcal{A}(\Psi,A,\Lambda).