Homepage of the HATA group at DTU

## Research.Research History

22.03.2016, at 09:06 UTC by 130.225.70.147 -
22.03.2016, at 09:00 UTC by 130.225.70.147 -
Deleted lines 5-11:
* [[#finframe| Finite-dimensional frames]]
* [[#genframe|General frames in Hilbert spaces]]
* [[#struc| Structured frames in {$L^2(\mathbb{R}^n)$}]]: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
* [[#lca|Structured frames on LCA groups]]

>>comment<<
Changed lines 10-25 from:
* Structured frames on LCA groups]]
to:
* Structured frames on LCA groups

>>comment<<

* [[#finframe| Finite-dimensional frames
]]
* [[#genframe|General frames in Hilbert spaces]]
* [[#struc| Structured frames in {$L^2(\mathbb{R}^n)$}]]: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
* [[#lca|Structured frames on LCA groups]]

* 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

* [[#finframe| Finite-dimensional frames]]
* [[#genframe|General frames in Hilbert spaces]]
* [[#struc| Structured frames in {$L^2(\mathbb{R}^n)$}]]: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
* [[#lca|Structured frames on LCA groups]]

>>comment<<
Deleted lines 18-23:
>>comment<<

* [[#finframe| Finite-dimensional frames]]
* [[#genframe|General frames in Hilbert spaces]]
* [[#struc| Structured frames in {$L^2(\mathbb{R}^n)$}]]: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
* [[#lca|Structured frames on LCA groups]]
18.10.2015, at 19:49 UTC by 87.48.19.168 -
18.10.2015, at 19:27 UTC by 87.48.19.168 -
18.10.2015, at 19:20 UTC by 87.48.19.168 -
Changed line 8 from:
Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
to:
->Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
18.10.2015, at 19:20 UTC by 87.48.19.168 -
Deleted line 7:
18.10.2015, at 19:20 UTC by 87.48.19.168 -
Changed lines 7-9 from:
* Structured frames in {$L^2(\mathbb{R}^n)$}: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
to:
* Structured frames in {$L^2(\mathbb{R}^n)$}:

Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
18.10.2015, at 19:20 UTC by 87.48.19.168 -
Changed lines 5-12 from:
* Finite-dimensional frames !! [[#finframe| Finite-dimensional frames]]
to:
* 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]]

>>comment<<

*
[[#finframe| Finite-dimensional frames]]
Deleted lines 15-16:

>>comment<<
18.10.2015, at 19:19 UTC by 87.48.19.168 -
Changed line 5 from:
* [[#finframe| Finite-dimensional frames]]
to:
* Finite-dimensional frames !! [[#finframe| Finite-dimensional frames]]
18.10.2015, at 19:19 UTC by 87.48.19.168 -
Changed lines 1-3 from:
! Main research areas

!! Under construction.
to:
!Main research areas

The central research area is %green%frame theory%%, in particular:

* [[#finframe| Finite-dimensional frames]]
* [[#genframe|General frames in Hilbert spaces]]
* [[#struc| Structured frames in {$L^2(\mathbb{R}^n)$}]]: Wavelet (time-scale), Gabor (time-frequency) analysis, directional (shearlets, etc.) systems
* [[#lca|Structured frames on LCA groups]]

>>comment<<

!! Shearlet theory. [[#shearlet]]
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 &ndash; although perfectly suited for isotropic structures &ndash; do not perform equally well when dealing with anisotropic phenomena.

Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | Labate]] (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &ndash; in contrast to previously introduced directional representation systems &ndash; 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.

!! Wavelet and time-frequency analysis. [[#wavelet]]
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. [[#sparse]]
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 [[http://www-stat.stanford.edu/~donoho/ |Donoho]] (Stanford U.) and [[http://www-stat.stanford.edu/~candes/| Candes]] (Stanford U.), [[http://users.ece.gatech.edu/justin/Justin_Romberg.html | Romberg]] (Georgia Tech), and [[http://www.math.ucla.edu/~tao/ |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. [[#frame]]
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 [[http://www.math.missouri.edu/personnel/faculty/casazzap.html |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.

!! Image and signal processing: denoising, geometric separation,... [[#image]]
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.

%rfloat height=140%Attach:test.png"Sinosoid"

!!Shearlets [[#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. %lfloat height=160%Attach: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. 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 [[#sparsity]]

!!!!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 {$y=0$}.
>><<
05.10.2015, at 22:17 UTC by 82.211.223.204 -
Changed lines 3-72 from:
* [[#frame |Frame theory]].
* [[#wavelet| Wavelet (time-scale) and Gabor (time-frequency) analysis]].
* [[#shearlet | Directional representation systems: Shearlets, directional ridge functions & wavepackets, etc. ]].
* [[#sparse| Sparse approximations and recovery]].
* [[#signal | Image and signal processing]].

!! Frame theory [[#frame]]

Frames have been a focus of study in the last two decades in applications where redundancy plays a vital and useful role.

!! Wavelet and Gabor analysis [[#wavelet]]

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.

!! Directional representation systems [[#shearlet]]

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 &ndash; although perfectly suited for isotropic structures &ndash; do not perform equally well when dealing with anisotropic phenomena.

>>comment<<
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | Labate]] (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &ndash; in contrast to previously introduced directional representation systems &ndash; 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.
>><<

!! Sparse approxiamtions and recovery [[#sparse]]

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.

>>comment<<
Compressed Sensing, which was recently introduced by [[http://www-stat.stanford.edu/~donoho/ |Donoho]] (Stanford U.) and [[http://www-stat.stanford.edu/~candes/| Candes]] (Stanford U.), [[http://users.ece.gatech.edu/justin/Justin_Romberg.html | Romberg]] (Georgia Tech), and [[http://www.math.ucla.edu/~tao/ |Tao]] (UCLA), has gained particularly rapid attention by providing methods for measuring sparse signals with an optimally small number of (random) measurements.
>><<

!! Image and signal processing [[#image]]

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.

>>comment<<
%rfloat height=140%Attach:test.png"Sinosoid"

!!Shearlets [[#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. %lfloat height=160%Attach: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. 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 [[#sparsity]]

!!!!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 {$y=0$}.
>><<
to:
!! Under construction.
04.10.2015, at 08:22 UTC by 109.56.235.169 -
Changed line 5 from:
* [[#shearlet | Directional representation systems: Shearlets, rigdelet, wavepackets, etc. ]].
to:
* [[#shearlet | Directional representation systems: Shearlets, directional ridge functions & wavepackets, etc. ]].
04.10.2015, at 08:16 UTC by 109.56.235.169 -
Changed lines 7-9 from:
* [[#signal | Image and signal processing]]: denoising, geometric separation, inpainting, inverse problems...
to:
* [[#signal | Image and signal processing]].
Changed line 37 from:
!! Image and signal processing: denoising, geometric separation,... [[#image]]
to:
!! Image and signal processing [[#image]]
04.10.2015, at 08:12 UTC by 109.56.235.169 -
Changed lines 1-11 from:
>>comment<<
!Project Homepages

*[[http://shearlet.org | Shearlets]]
*[[http://shearlab.org | ShearLab]]
*[[http://www.fusionframe.org/ | Fusion Frame]]
>><<

!
Main research areas
to:
! Main research areas
Changed line 10 from:
!! Frame theory. [[#frame]]
to:
!! Frame theory [[#frame]]
04.10.2015, at 08:10 UTC by 109.56.235.169 -
Changed lines 13-14 from:
* [[#wavelet| Wavelet and time-frequency analysis]].
* [[#shearlet | Shearlet theory
]].
to:
* [[#wavelet| Wavelet (time-scale) and Gabor (time-frequency) analysis]].
* [[#shearlet | Directional representation systems: Shearlets, rigdelet, wavepackets, etc.
]].
Changed lines 23-24 from:
!! Wavelet and time-frequency analysis [[#wavelet]]
to:
!! Wavelet and Gabor analysis [[#wavelet]]
Changed line 30 from:
!! Shearlet theory [[#shearlet]]
to:
!! Directional representation systems [[#shearlet]]
04.10.2015, at 08:06 UTC by 109.56.235.169 -
Deleted lines 9-11:
!DO NOT READ -- THIS WILL UPDATED/REMOVED
----
* [[#frame |Frame theory]].
* [[#wavelet| Wavelet and time-frequency analysis]].
Changed lines 15-22 from:
* [[#wavelet| Wavelet and time-frequency analysis]].
* [[#sparse| Sparse recovery, {$\ell_1$} minimization, and compressed sensing]].
* [[#frame |Frame theory and fusion frame theory]]
.
* [[#signal | Image and signal processing]]: denoising, geometric separation, inpainting, ...

!! Shearlet theory.
[[#shearlet]]
to:
* [[#sparse| Sparse approximations and recovery]].
* [[#signal | Image and signal processing]]: denoising, geometric separation, inpainting, inverse problems...

!! Frame theory. [[#frame]]

Frames have been a focus of study in the last two decades in applications where redundancy plays a vital and useful role.

!! Wavelet and time-frequency analysis
[[#wavelet]]

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.

!! Shearlet theory [[#shearlet]]
>>comment<<
Changed lines 36-44 from:

!! Wavelet and time-frequency analysis. [[#wavelet]]
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. [[#sparse]]
to:
>><<

!! Sparse approxiamtions and recovery
[[#sparse]]
>>comment<<
Changed lines 44-47 from:
!! Frame theory and fusion frame theory. [[#frame]]
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.
to:
>><<
04.10.2015, at 07:19 UTC by 109.56.235.169 -
Deleted lines 41-44:

Fusion frames, which were recently introduced by [[http://www.math.missouri.edu/personnel/faculty/casazzap.html |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.

01.10.2015, at 12:03 UTC by 109.56.87.221 -
30.09.2015, at 13:04 UTC by 130.225.84.229 -

!DO NOT READ -- THIS WILL UPDATED/REMOVED
----
30.09.2015, at 13:03 UTC by 130.225.84.229 -
>>comment<<
>><<
07.01.2010, at 11:11 UTC by 131.173.40.72 -
Changed line 20 from:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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 &ndash; 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.
to:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | Labate]] (U. Houston), and Lim, sparsely encode anisotropic singularities of 2D data in an optimal way, while &ndash; in contrast to previously introduced directional representation systems &ndash; 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.
24.12.2009, at 11:23 UTC by 80.199.188.78 -
Changed line 7 from:
!Current ongoing research projects
to:
!Main research areas
23.12.2009, at 17:40 UTC by 80.199.188.78 -
Changed line 37 from:
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.
to:
Fusion frames, which were recently introduced by [[http://www.math.missouri.edu/personnel/faculty/casazzap.html |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.
23.12.2009, at 17:39 UTC by 80.199.188.78 -
Changed lines 39-40 from:
www.fusionframe.org]].
to:
23.12.2009, at 17:38 UTC by 80.199.188.78 -
Changed lines 32-33 from:
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.
to:
Compressed Sensing, which was recently introduced by [[http://www-stat.stanford.edu/~donoho/ |Donoho]] (Stanford U.) and [[http://www-stat.stanford.edu/~candes/| Candes]] (Stanford U.), [[http://users.ece.gatech.edu/justin/Justin_Romberg.html | Romberg]] (Georgia Tech), and [[http://www.math.ucla.edu/~tao/ |Tao]] (UCLA), has gained particularly rapid attention by providing methods for measuring sparse signals with an optimally small number of (random) measurements.
Changed lines 35-36 from:
to:
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.

www.fusionframe.org]].
Changed lines 43-45 from:
..coming soon..
to:
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.
23.12.2009, at 17:34 UTC by 80.199.188.78 -
Changed lines 20-21 from:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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.
to:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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 &ndash; 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.
Changed lines 25-26 from:
..coming soon..
to:
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
.
Changed lines 30-31 from:
..coming soon..
to:
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
.
Changed line 35 from:
..coming soon..
to:
23.12.2009, at 17:32 UTC by 80.199.188.78 -
23.12.2009, at 17:32 UTC by 80.199.188.78 -
Changed line 22 from:
to:
23.12.2009, at 17:30 UTC by 80.199.188.78 -
Changed lines 18-26 from:
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 &ndash; although perfectly suited for isotropic structures &ndash; do
not perform equally well when dealing with anisotropic phenomena.

Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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.
to:
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 &ndash; although perfectly suited for isotropic structures &ndash; do not perform equally well when dealing with anisotropic phenomena.

Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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.
23.12.2009, at 17:29 UTC by 80.199.188.78 -
Changed lines 25-26 from:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/|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
to:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/ | 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
23.12.2009, at 17:29 UTC by 80.199.188.78 -
Changed line 22 from:
well known that wavelets - although perfectly suited for isotropic structures - do
to:
well known that wavelets &ndash; although perfectly suited for isotropic structures &ndash; do
Changed lines 25-31 from:
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.
to:
Shearlets, which were recently introduced by Kutyniok, [[http://www.math.uh.edu/~dlabate/|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.
23.12.2009, at 17:21 UTC by 80.199.188.78 -
Changed lines 18-33 from:
..coming soon..
to:
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.

.
*[[http://shearlet.org | Shearlets]]
!Project Homepages

*[[http://shearlab.org | ShearLab]]
*[[http://www.fusionframe.org/ | Fusion Frame]]
13.08.2009, at 09:59 UTC by 130.225.72.109 -
Changed line 23 from:
!! Image and signal processing: denoising, geometric separation, inpainting, ... [[#image]]
to:
!! Image and signal processing: denoising, geometric separation,... [[#image]]
13.08.2009, at 09:58 UTC by 130.225.72.109 -
Changed lines 4-24 from:
* 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
[+{$$\psi_{a,s,t}(x)=a^{-3/4} \psi((D_{a,s}^{-1}(x-t)) \qquad \text{where} \quad D_{a,s} = \begin{bmatrix} a & -a^{1/2}s \\ 0 & a^{1/2} \end{bmatrix}.$$}+]

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. %lfloat height=160%Attach: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. 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
to:
* [[#wavelet| Wavelet and time-frequency analysis]].
* [[#sparse| Sparse recovery, {$\ell_1$} minimization, and compressed sensing]].
* [[#frame |Frame theory and fusion frame theory]].
* [[#signal | Image and signal processing]]: denoising, geometric separation, inpainting, ...

Changed lines 14-17 from:
!! 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, ...
to:
!! Wavelet and time-frequency analysis. [[#wavelet]]
..coming soon..

!! Sparse recovery,
{$\ell_1$} minimization, and compressed sensing. [[#sparse]]
..coming soon
..

!! Frame theory and fusion frame theory. [[#frame]]
..coming soon
..

!! Image and signal processing: denoising, geometric separation, inpainting, ... [[#image]]
..coming soon
..
13.08.2009, at 09:56 UTC by 130.225.72.109 -
Changed lines 9-23 from:
to:
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. %lfloat height=160%Attach: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. 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
13.08.2009, at 09:56 UTC by 130.225.72.109 -
Changed line 3 from:
* [[shearlet | Shearlet theory]].
to:
* [[#shearlet | Shearlet theory]].
13.08.2009, at 09:56 UTC by 130.225.72.109 -
Changed lines 3-5 from:
* Shearlets
* Sparse recovery of underdetermined systems, {$\ell_1$}
-minimization
to:
* [[shearlet | 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. [[#shearlet]]
..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, ...

>>comment<<
Changed lines 51-52 from:
* {$\exists r >0$} s.t. if {$y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}$}, then {$y=0$}.
to:
* {$\exists r >0$} s.t. if {$y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}$}, then {$y=0$}.
>><<
Changed line 19 from:
{$$\hat\psi(\xi_1,\xi_2) = \hat\psi_1(\xi_1) \hat\psi_2\Bigl(\frac{\xi_2}{\xi_1}\Bigr),$$}
to:
[+{$$\hat\psi(\xi_1,\xi_2) = \hat\psi_1(\xi_1) \hat\psi_2\Bigl(\frac{\xi_2}{\xi_1}\Bigr),$$}+]
Changed line 16 from:
{$$\psi_{a,s,t}(x)=a^{-3/4} \psi((D_{a,s}^{-1}(x-t)) \qquad \text{where} \quad D_{a,s} = \begin{bmatrix} a & -a^{1/2}s \\ 0 & a^{1/2} \end{bmatrix}.$$}
to:
[+{$$\psi_{a,s,t}(x)=a^{-3/4} \psi((D_{a,s}^{-1}(x-t)) \qquad \text{where} \quad D_{a,s} = \begin{bmatrix} a & -a^{1/2}s \\ 0 & a^{1/2} \end{bmatrix}.$$}+]
%rfloat height=140%Attach:test.png"Sinosoid"

!!Shearlets [[#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. %lfloat height=160%Attach: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. 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 [[#sparsity]]

!!!!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 {$y=0$}.
Deleted lines 5-33:
%rfloat height=140%Attach:test.png"Sinosoid"

!!Shearlets [[#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. %lfloat height=160%Attach: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. 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 [[#sparsity]]

!!!!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 {$y=0$}.
Changed line 8 from:
!!Shearlets
to:
!!Shearlets [[#shearlets]]
!! Sparse recovery of underdetermined systems, {$\ell_1$}-minimization [[#sparsity]]
Changed line 22 from:
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. %lfloat height=160%{$Filesdir}cont_shear.jpg"support in the frequency domain" to: The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. %lfloat height=160%Attach:cont_shear.jpg"support in the frequency domain" Changed line 22 from: The continuous shearlet transform precisely resolves the wavefront set, since the location parameter t does detect the location of singularities, whereas the shear parameter shows the direction perpendicular to the direction of the singularity. %lfloat height=160%{$Imagedir}cont_shear.jpg"support in the frequency domain"
to:
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. %lfloat height=160%{$Filesdir}cont_shear.jpg"support in the frequency domain" Deleted lines 26-41: !!!!Matlab code This is implemented in Matlab by the following code: [@% for loops in Matlab % Set the number of terms n=10; % initialize sum Q=0; % summing over k for k=1:n Q=Q+1/k^2; end sprintf('Approximation with %d terms: Q = %0.10g',n,Q)@] Changed lines 4-5 from: * Sparse recovery of underdetermined systems, {$\ell_1$}-minimization, * Neuro to: * Sparse recovery of underdetermined systems, {$\ell_1$}-minimization Changed line 1 from: !Current ongoing research projects: to: !Current ongoing research projects Changed line 7 from: %rfloat height=160%Attach:test.png"Sinosoid" to: %rfloat height=140%Attach:test.png"Sinosoid" Added lines 6-7: %rfloat height=160%Attach:test.png"Sinosoid" Changed lines 48-50 from: * {$\exists r >0$} s.t. if {$y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}$}, then {$y=0$}. >><< to: * {$\exists r >0$} s.t. if {$y \in \{x \in \Gamma | x_1^2+\dots+x_n^2<r^2 \}$}, then {$y=0$}. Changed lines 1-14 from: !!!Research interest: *In general : Applied Harmonic Analysis, functional analysis, Hilbert space theory, Fourier analysis, numerical analysis, and signal processing. *In particular: {$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$}-minimization, Quasi wavelet frames, oversampling of wavelet frames, shift invariant spaces. !!!Current ongoing research projects: * Generalization of [[Publications#article2| Publ. 2]] to {$\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$}, A-dilative partition of unity. * (with [[{$mb} | M. Bownik]]) 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. [[Attach:quasi-affine.pdf |preprint]]
* (with [[{$mb} | M. Bownik]]) Oversampling of Wavelet Frames for Real Dilations. For any '''real''', expansive dilation matrix {$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)$}. >>comment<< !!!Shearlet to: !Current ongoing research projects: * Shearlets * Sparse recovery of underdetermined systems, {$\ell_1$}-minimization , * Neuro !!Shearlets Added lines 1-57: !!!Research interest: *In general: Applied Harmonic Analysis, functional analysis, Hilbert space theory, Fourier analysis, numerical analysis, and signal processing. *In particular: {$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$}-minimization, Quasi wavelet frames, oversampling of wavelet frames, shift invariant spaces. !!!Current ongoing research projects: * Generalization of [[Publications#article2| Publ. 2]] to {$\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$}, A-dilative partition of unity. * (with [[{$mb} | M. Bownik]]) 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. [[Attach:quasi-affine.pdf |preprint]]
* (with [[{$mb} | M. Bownik]]) Oversampling of Wavelet Frames for Real Dilations. For any '''real''', expansive dilation matrix {$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)$}. >>comment<< !!!Shearlet In the following we will give a short introduction into the theory of shearlets. Unlike the traditional wavelet transform does not posses the ability to detect directionality, since it is merely associated with two parameters, the scaling parameter {$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. %lfloat height=160%{$Imagedir}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. 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.

!!!!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)@]

!!!!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 {$y=0$}.

>><<