Homepage of the HATA group at DTU

Publications.Publications History

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09.01.2019, at 10:36 UTC by 10.16.172.129 -
09.01.2019, at 07:56 UTC by 10.16.172.129 -
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[[#pub2019]]
!! 2019 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2019' AND $this->get('NOTE')!==preprint][$this->get('AUTHOR')][50]
08.03.2018, at 15:19 UTC by 2.110.50.53 -
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[[#pub2018]]
!! 2018 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2018' AND $this->get('NOTE')!==preprint][$this->get('AUTHOR')][50]
09.03.2017, at 11:04 UTC by 94.191.186.12 -
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[[#pub2017]]
!! 2017 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2017' AND $this->get('NOTE')!==preprint][$this->get('AUTHOR')][50]
21.03.2016, at 10:23 UTC by 176.23.45.186 -
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[[#pub2016]]
!! 2016 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2016' AND $this->get('NOTE')!==preprint][$this->get('AUTHOR')][50]
23.02.2016, at 08:52 UTC by 188.183.155.47 -
05.11.2015, at 14:47 UTC by 130.225.70.147 -
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05.10.2015, at 07:03 UTC by 87.48.19.168 -
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(: [[#2014]] :)
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[[#2014]]
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01.10.2015, at 14:46 UTC by 109.56.87.221 -
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01.10.2015, at 14:44 UTC by 109.56.87.221 -
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01.10.2015, at 14:42 UTC by 109.56.87.221 -
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01.10.2015, at 14:40 UTC by 109.56.87.221 -
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01.10.2015, at 14:39 UTC by 109.56.87.221 -
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01.10.2015, at 14:37 UTC by 109.56.87.221 -
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01.10.2015, at 14:35 UTC by 109.56.87.221 -
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!! 2013 & 2014 Publications
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!! 2014 Publications
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!! 2013 Publications
01.10.2015, at 14:34 UTC by 109.56.87.221 -
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01.10.2015, at 14:33 UTC by 109.56.87.221 -
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!! Preprints & 2015 Publications
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!! Preprints
bibtexquery:[bibfile.bib][!$this->get('NOTE')==preprint]


!!
2015 Publications
30.09.2015, at 13:35 UTC by 130.225.84.229 -
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2010'][$this->get('AUTHOR')][50]
bibtexquery:[bibfile.bib][$this->get('YEAR')=='2009'][$this->get('AUTHOR')][50]
bibtexquery:[bibfile.bib][$this->get('YEAR')=='2008'][$this->get('AUTHOR')][50]
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30.09.2015, at 13:17 UTC by 130.225.84.229 -
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%%comment * Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions.
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(:comment * Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions. :)
30.09.2015, at 13:15 UTC by 130.225.84.229 -
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* Click %height=18px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions.
to:
* Click %height=18px%{$Filesdir}world.png to go to the '''arXiv''' or, if available, the '''journal''' version of the paper ([[http://doi.org | DOI links]])
%%comment * Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions.
30.09.2015, at 13:01 UTC by 130.225.84.229 -
30.09.2015, at 12:35 UTC by 130.225.84.229 -
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[[#phdthesis]]
!! Ph.D. Thesis

Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
21.04.2015, at 09:05 UTC by 109.56.143.201 -
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!! Preprints & 2014 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2014'][!$this->get('NOTE')==preprint][50]
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!! Preprints & 2015 Publications

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!! 2007 - 2013 Publications
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!! 2007 - 2014 Publications

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20.05.2014, at 04:15 UTC by 50.194.2.180 -
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!! Preprints & 2013 Publications

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!! Preprints & 2014 Publications

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!! 2007 - 2012 Publications
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!! 2007 - 2013 Publications

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18.06.2013, at 10:12 UTC by 130.225.84.229 -
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!! Preprints & 2012 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][!$this->get('NOTE')==preprint][50]
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!! Preprints & 2013 Publications

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!! 2007 - 2011 Publications
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!! 2007 - 2012 Publications


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02.08.2012, at 20:06 UTC by 90.184.159.203 -
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Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.

>>comment<<
!! 1998 &ndash; 2006
Use the links in the [[#sb|sidebar]].

bibtexquery:[bibfile.bib][strpos($this->get('AUTHOR'),'Lim')!==FALSE || strpos($this->get('AUTHOR'),'Lemvig')!==FALSE || strpos($this->get('AUTHOR'),'Kutyniok')!==FALSE][!$this->get('YEAR')][50]


!! Preprints
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
>><<


>>comment<<
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%%
P. Boufounos, G. Kutyniok, and H. Rauhut. \\
%maroon%'''Compressed Sensing for Fusion Frames'''.%%\\
Submitted (2009).\\

>>comment<<
!! 2009
%rfloat%[[Attach:Wavefront.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"View BibTeX entry"]]%%
bibtexsummary:[bibfile.bib,MR2471937].
(:toggle show="Show abstract" hide box3 button=0:)
>>id=box3 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = <f,\psi_{ast}>$}. The analyzing elements {$\psi_{ast}$} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
>><<
>><<

>>comment<<
#%item value=2% [[#article2]] Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization, Adv. Comput. Math. '''30''' (2009) 231-247, [[Attach:lemvig_pairs-dual-framelets_A4.pdf |a4]] or [[Attach:lemvig_pairs-dual-framelets_LETTER.pdf |letter]].
#%item value=1% [[{$mb} | Bownik,M.]]; Lemvig, J.: The canonical and alternate duals of a wavelet frame, Appl. Comput. Harmon. Anal. '''23''' (2007) 263–272. [[Attach:cadwf_A4.pdf |a4]] or [[Attach:cadwf_LETTER.pdf |letter]].
>><<
to:
Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('NOTE')==preprint][50]
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Title: Wavelet Frames and Their Duals ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
to:
Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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!! 2008 - 2011 Publications
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!! 2007 - 2011 Publications
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2007'][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2018'][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2008'][$this->get('AUTHOR')][50]
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!! Quick Guide:
to:
'''Quick Guide:'''
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* Click {$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata.
to:
* Click %height=16px%{$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata.
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* Click %height=16px%{$Filesdir}pdf.png or use external links [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
to:
* Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions.
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!! Newest preprint
to:
!! Quick Guide:
* Click %height=18px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click %height=16px%{$Filesdir}pdf.png or use external links [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''
preprint''' versions.
* Click {$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata.
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!! Quick Guide:
* Click %height=18px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click %height=16px%{$Filesdir}pdf.png or use external links [[http://www
.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
* Click {$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata.
to:
!! Ph.D. Thesis

Title
: Wavelet Frames and Their Duals ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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!! Preprints
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
Deleted lines 39-50:

!! Ph.D. Thesis

Title: Wavelet Frames and Their Duals ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.

!! Preprints
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='2011'][$this->get('AUTHOR')][50]
bibtexquery:[bibfile.bib][$this->get('YEAR')=='2010'][$this->get('AUTHOR')][50]
bibtexquery:[bibfile.bib][$this->get('YEAR')=='2009'][$this->get('AUTHOR')][50]
bibtexquery:[bibfile.bib][$this->get('YEAR')=='2018'][$this->get('AUTHOR
')][50]
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>>comment<<
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
to:
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
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%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
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%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
%%
to:
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %%
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%%
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!! Newest preprint
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:)
>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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!! Preprints
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! Selected AAG Publications
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! Publications

!! Preprints

!! Preprints &amp; 2012 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50]

!! 2008 - 2011
Publications
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!! Quick Guide:
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!! Preprints &amp; 2010 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2010'][$this->get('AUTHOR')][50]

!! 2009 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2009'][$this->get('AUTHOR')][50]

!! 2008 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2008'][$this->get('AUTHOR')][50]

!! 2007 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2007'][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][][!$this->get('YEAR')][50]
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bibtexquery:[bibfile.bib]
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Title: Wavelet Frames and Their Duals ([[Attach:phd-thesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
to:
Title: Wavelet Frames and Their Duals ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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Title: Wavelet Frames and Their Duals (Attach:PDF). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
to:
Title: Wavelet Frames and Their Duals ([[Attach:phd-thesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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!! Talks

Firstname Lastname
: '''What is Web 2.0?'''
(:toggle show="(Show slideshow)" hide box2 button=0:)
>>id=box2<<
(:slideshare what-is-web-20-1194363300579044-4:
)

>>comment<<
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!! Ph.D. Thesis

Title
: Wavelet Frames and Their Duals (Attach:PDF). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008.
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25.03.2010, at 17:30 UTC by 131.173.40.72 -
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!! Preprints &amp; 2009 Publications
to:
!! Preprints &amp; 2010 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2010'][$this->get('AUTHOR')][50]

!!
2009 Publications
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* Click %height=16px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
to:
* Click %height=18px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
20.08.2009, at 19:53 UTC by 82.113.121.137 -
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* Click {$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click {$Filesdir}pdf.png or use external links [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
to:
* Click %height=16px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click %height=16px%{$Filesdir}pdf.png or use external links [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
20.08.2009, at 19:52 UTC by 82.113.121.137 -
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* Click {$Filesdir}pdf.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Use [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
to:
* Click {$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
* Click {$Filesdir}pdf.png or use external links [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
20.08.2009, at 09:45 UTC by 131.173.40.72 -
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* Use [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
to:
* Use [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Publications:Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
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* Click {$Filesdir}pdf.png to go to the '''journal''' version of the paper
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* Click {$Filesdir}pdf.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
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* Click {$Filesdir}pdf.png to link the '''journal''' version of the paper
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* Click {$Filesdir}pdf.png to go to the '''journal''' version of the paper
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* Click {$Filesdir}pdf.png to link the '''journal''' version of the paper; preprint versions can be obtained from: [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]], or [[http://arxiv.org/| arXiv.org]].
* Click {$Filesdir}bibtex.gif to retrieve BibTeX
metadata.
to:
* Click {$Filesdir}pdf.png to link the '''journal''' version of the paper
* Use [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions.
* Click {$Filesdir}bibtex.gif to retrieve '''BibTeX'''
metadata.
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!! 2004 &ndash; 2006
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!! 1998 &ndash; 2006
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='1998' || $this->get('YEAR')=='2003' || $this->get('YEAR')=='2002' || $this->get('YEAR')=='2001' || $this->get('YEAR')=='2000' || $this->get('YEAR')=='1999'][$this->get('YEAR')][50]
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to:
bibtexquery:[bibfile.bib][$this->get('YEAR')=='1998' || $this->get('YEAR')=='2003' || $this->get('YEAR')=='2002' || $this->get('YEAR')=='2001' || $this->get('YEAR')=='2000' || $this->get('YEAR')=='1999'][$this->get('YEAR')][50]
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='1998' || $this->get('YEAR')=='2003' $this->get('YEAR')=='2002' || $this->get('YEAR')=='2001' $this->get('YEAR')=='2000' || $this->get('YEAR')=='1999'][$this->get('AUTHOR')][50]
to:
bibtexquery:[bibfile.bib][$this->get('YEAR')=='1998' || $this->get('YEAR')=='2003' || $this->get('YEAR')=='2002' || $this->get('YEAR')=='2001' || $this->get('YEAR')=='2000' || $this->get('YEAR')=='1999'][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][$this->get('YEAR')=='1998' || $this->get('YEAR')=='2003' $this->get('YEAR')=='2002' || $this->get('YEAR')=='2001' $this->get('YEAR')=='2000' || $this->get('YEAR')=='1999'][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][$this->get('YEAR')<='2006'][$this->get('AUTHOR')][50]
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Use the links in the sidebar.
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Use the links in the [[#sb|sidebar]].
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!! 2009 Publications and preprints
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!! Preprints &amp; 2009 Publications
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!! 2008 Publications and preprints
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!! 2008 Publications
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!! 2007 Publications and preprints
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!! 2007 Publications
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!! 2004 &ndash; 2006
Use the links in the sidebar.
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!! 2008 Publications and preprints

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2008'][$this->get('AUTHOR')][50]

!! 2007 Publications and preprints

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2007'][$this->get('AUTHOR')][50]
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! Selected AAG Publications (%red%Under construction%%; will be updated soon)
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! Selected AAG Publications
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!! 2009 Publications
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!! 2009 Publications and preprints
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Click {$Filesdir}pdf.png to link the journal version of the paper; preprint version of the papers can be obtained from: [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]]
Click {$Filesdir}bibtex.gif to retrieve BibTeX metadata.
to:
* Click {$Filesdir}pdf.png to link the '''journal''' version of the paper; preprint versions can be obtained from: [[http://www.home.uni-osnabrueck.de/kutyniok/publications.html| Kutyniok]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]], or [[http://arxiv.org/| arXiv.org]].
*
Click {$Filesdir}bibtex.gif to retrieve BibTeX metadata.
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Click {$Filesdir}pdf.png to link the journal version of the paper; preprint version of the papers can be obtained from: [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Lemvig]]
Click {$Filesdir}bibtex.gif to retrieve BibTeX metadata.
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!! 2009 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2009'][$this->get('AUTHOR')][50]


>>comment<<
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to:
bibtexquery:[bibfile.bib][strpos($this->get('AUTHOR'),'Lim')!==FALSE || strpos($this->get('AUTHOR'),'Lemvig')!==FALSE || strpos($this->get('AUTHOR'),'Kutyniok')!==FALSE][!$this->get('YEAR')][50]
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! Selected AAG Publications (Under construction; will be updated soon)
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! Selected AAG Publications (%red%Under construction%%; will be updated soon)
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! Selected AAG Publications (Only a test page; will be updated soon)
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! Selected AAG Publications
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! Selected AAG Publications (Only a test page; will be updated soon)

>>comment<<
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04.08.2009, at 17:33 UTC by 131.173.40.72 -
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\ %maroon%'''Fusion Frames: Existence and Construction.'''%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. \\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
04.08.2009, at 17:33 UTC by 131.173.40.72 -
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %maroon%'''Fusion Frames: Existence and Construction.'''%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\ %maroon%'''Fusion Frames: Existence and Construction.'''%%\\
04.08.2009, at 17:33 UTC by 131.173.40.72 -
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
%maroon%'''Fusion Frames: Existence and Construction.'''%%\\
to:
%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
R. Calderbank, P
. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %maroon%'''Fusion Frames: Existence and Construction.'''%%\\
04.08.2009, at 17:32 UTC by 131.173.40.72 -
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%% P. Boufounos, G. Kutyniok, and H. Rauhut. \\
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%%
P. Boufounos, G. Kutyniok, and H. Rauhut. \\
04.08.2009, at 17:31 UTC by 131.173.40.72 -
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%% P. Boufounos, G. Kutyniok, and H. Rauhut. \\
04.08.2009, at 17:30 UTC by 131.173.40.72 -
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G. Kutyniok and D. Labate.\\
%maroon%'''Resolution of the Wavefront Set using Continuous Shearlets'''.%%\\
Trans. Amer. Math. Soc. 361 (2009), 2719-2754.\\
Download: Abstract pdf
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%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = <f,\psi_{ast}>$}. The analyzing elements {$\psi_{ast} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
to:
%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = <f,\psi_{ast}>$}. The analyzing elements {$\psi_{ast}$} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
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%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = \ip{f}{\psi_{ast}}$}. The analyzing elements {$\psi_{ast} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
to:
>>id=box3 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = <f,\psi_{ast}>$}. The analyzing elements {$\psi_{ast} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
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%rfloat%[[Attach:Wavefront.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
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%rfloat%[[Attach:Wavefront.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"View BibTeX entry"]]%%
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%maroon%Abstract%%.It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by SH_f(a,s,t) = \ip{f}{\psi_{ast}}. The analyzing elements \psi_{ast} are dilated and translated copies of a single generating function \psi, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements \psi_{ast} form a system of smooth functions at continuous scales a >0, locations t \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. Finally, we point out several variations of this approach.
to:
%maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$f$} decays rapidly near the points where {$f$} is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of {$f$}. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by {$SH_f(a,s,t) = \ip{f}{\psi_{ast}}$}. The analyzing elements {$\psi_{ast} are dilated and translated copies of a single generating function {$\psi$}, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {$\psi_{ast}$} form a system of smooth functions at continuous scales {$a >0$}, locations {$t \in R^2$}, and oriented along lines of slope {$s \in R$} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution {$f$}. Finally, we point out several variations of this approach.
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
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P. Boufounos, G. Kutyniok, and H. Rauhut.\\
%maroon%'''Compressed Sensing for Fusion Frames'''.%%\\
Submitted (2009).\\
Download: Abstract pdf

!! 2009
%rfloat%[[Attach:Wavefront
.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
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%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
to:
%maroon%Abstract%%.It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by SH_f(a,s,t) = \ip{f}{\psi_{ast}}. The analyzing elements \psi_{ast} are dilated and translated copies of a single generating function \psi, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements \psi_{ast} form a system of smooth functions at continuous scales a >0, locations t \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. Finally, we point out several variations of this approach.
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P. Boufounos, G. Kutyniok, and H. Rauhut.\\
%maroon%'''Compressed Sensing for Fusion Frames'''.%%\\
Submitted (2009).\\
Download: Abstract pdf

!! 2009
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%
bibtexsummary:[bibfile.bib,MR2471937].
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>>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<<
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
>><<
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"Here should be link to bibtex info"]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] {$Filesdir}bibtex.gif"Here should be link to bibtex info"%%\\
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Firstname Lastname: '''What is Web'''
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Firstname Lastname: '''What is Web'''
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!! Talks
(:slideshare what-is-web-20-1194363300579044-4:)
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%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
to:
%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {$\{\lambda_j\}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {$\{\lambda_j\}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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%maroon%Abstract%%: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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%maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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%maroon%Abstract%%: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFexistence.pdf | {$Filesdir}pdf.png]]%%\\
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods $ndash; the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in {$\mathbb{R}^M$} with {$N$} subspaces of dimension {$m$} for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in N$} {$and {\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in N$} {$and {\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in \mathbb{N}$} and {${\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in N$} {$and {\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$M, N, m \in N$} {$and {\lambda_j}_{j=1,...,M}$}, does there exist a fusion frame in R^M with N subspaces of dimension m for which {${\lambda_j}_{j=1,...,M}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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# P. Boufounos, G. Kutyniok, and H. Rauhut.
Compressed Sensing for Fusion Frames.
Submitted (2009).
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P. Boufounos, G. Kutyniok, and H. Rauhut.\\
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Compressed Sensing for Fusion Frames.%%\\
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# B. G. Bodmann and G. Kutyniok.
Erasure-Proof Transmissions: Fusion Frames meet Coding Theory.
Submitted (2009)
.
Abstract pdf

# B
. G. Bodmann, P. G. Casazza, G. Kutyniok, and S. Senger.
A Low Complexity Replacement Scheme for Erased Frame Coefficients.
Submitted (2009).
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G. Kutyniok and D. Labate.\\
%maroon%'''Resolution of the Wavefront Set using Continuous Shearlets'''
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Trans. Amer. Math. Soc. 361 (2009), 2719-2754
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# R
. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.
Fusion Frames: Existence and Construction.
Submitted (2009).
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\
Fusion Frames: Existence and Construction.\\
Submitted (2009).\\
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! AAG Publications
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* Lemvig, J.: Constructing pairs of dual bandlimited frame wavelets in {$L^2(\mathbb{R}^n)$}.
* [[{$mb} | Bownik,M.]]; Lemvig, J.: Oversampling of wavelet frames for real dilations.
* [[{$mb} | Bownik,M.]]; Lemvig, J.: Affine and quasi-affine frames for rational dilations. Preprint: [[Attach:quasi-affine.pdf |a4]]
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!! Published papers
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# R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.
Fusion Frames: Existence and Construction.
Submitted (2009).
Abstract pdf

# P. Boufounos, G. Kutyniok, and H. Rauhut.
Compressed Sensing for Fusion Frames.
Submitted (2009).
Abstract pdf

!! 2009
# B. G. Bodmann and G. Kutyniok.
Erasure-Proof Transmissions: Fusion Frames meet Coding Theory.
Submitted (2009).
Abstract pdf

# B. G. Bodmann, P. G. Casazza, G. Kutyniok, and S. Senger.
A Low Complexity Replacement Scheme for Erased Frame Coefficients.
Submitted (2009).
Abstract pdf

%comment%
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!!! Ph.D. Thesis
-> Title: [[Attach:lemvig_phdthesis_20081007.pdf|Wavelet Frames and Their Duals]]. Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2009.

!!! Nonmathematical Conference paper
-> Lemvig, J.; Rasmussen, J.; Jensen, J.F.; Olsson, M.J.; Schultz, J.P.; Hansen, K.K.; Grelk, B.: Natural stone panels for building facades. Loss of strength caused by temperature cycles. Proceedings of the 6th Symposium on Building Physics in the Nordic Countries, Trondheim, Norway, 2002. (pdf).
to:
%%
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!! Preprints

* Lemvig, J.: Constructing pairs of dual bandlimited frame wavelets in {$L^2(\mathbb{R}^n)$}.
* [[{$mb} | Bownik,M.]]; Lemvig, J.: Oversampling of wavelet frames for real dilations.
* [[{$mb} | Bownik,M.]]; Lemvig, J.: Affine and quasi-affine frames for rational dilations. Preprint: [[Attach:quasi-affine.pdf |a4]]

!! Published papers

#%item value=2% [[#article2]] Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization, Adv. Comput. Math. '''30''' (2009) 231-247, [[Attach:lemvig_pairs-dual-framelets_A4.pdf |a4]] or [[Attach:lemvig_pairs-dual-framelets_LETTER.pdf |letter]].
#%item value=1% [[{$mb} | Bownik,M.]]; Lemvig, J.: The canonical and alternate duals of a wavelet frame, Appl. Comput. Harmon. Anal. '''23''' (2007) 263–272. [[Attach:cadwf_A4.pdf |a4]] or [[Attach:cadwf_LETTER.pdf |letter]].

!!! Ph.D. Thesis
-> Title: [[Attach:lemvig_phdthesis_20081007.pdf|Wavelet Frames and Their Duals]]. Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2009.

!!! Nonmathematical Conference paper
-> Lemvig, J.; Rasmussen, J.; Jensen, J.F.; Olsson, M.J.; Schultz, J.P.; Hansen, K.K.; Grelk, B.: Natural stone panels for building facades. Loss of strength caused by temperature cycles. Proceedings of the 6th Symposium on Building Physics in the Nordic Countries, Trondheim, Norway, 2002. (pdf).