Homepage of the HATA group at DTU

## Publications.Publications History

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!! 2019 Publications

bibtexquery:[bibfile.bib][$this->get('YEAR')=='2019' AND$this->get('NOTE')!==preprint][$this->get('AUTHOR')][50] 08.03.2018, at 15:19 UTC by 2.110.50.53 - Added lines 11-16: [[#pub2018]] !! 2018 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2018' AND $this->get('NOTE')!==preprint][$this->get('AUTHOR')][50]
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bibtexquery:[bibfile.bib][!$this->get('NOTE')==preprint] !! 2015 Publications 30.09.2015, at 13:35 UTC by 130.225.84.229 - Changed lines 14-15 from: !! 2007 - 2014 Publications to: !! 2013 & 2014 Publications Deleted lines 17-22: bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] bibtexquery:[bibfile.bib][$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')=='2008'][$this->get('AUTHOR')][50] bibtexquery:[bibfile.bib][$this->get('YEAR')=='2007'][$this->get('AUTHOR')][50] 30.09.2015, at 13:17 UTC by 130.225.84.229 - Changed line 5 from: %%comment * Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions.
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(:comment * Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions. :) 30.09.2015, at 13:15 UTC by 130.225.84.229 - Changed lines 4-5 from: * Click %height=18px%{$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%{$Filesdir}pdf.png to obtain '''preprint''' versions. 30.09.2015, at 13:01 UTC by 130.225.84.229 - 30.09.2015, at 12:35 UTC by 130.225.84.229 - Deleted lines 23-27: [[#phdthesis]] !! Ph.D. Thesis Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008. 21.04.2015, at 09:05 UTC by 109.56.143.201 - Changed lines 9-12 from: !! Preprints &amp; 2014 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2014'][!$this->get('NOTE')==preprint][50] to: !! Preprints &amp; 2015 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2015'][!$this->get('NOTE')==preprint][50] Changed lines 14-15 from: !! 2007 - 2013 Publications to: !! 2007 - 2014 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2014'][$this->get('AUTHOR')][50] 20.05.2014, at 04:15 UTC by 50.194.2.180 - Changed lines 9-12 from: !! Preprints &amp; 2013 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2013'][!$this->get('NOTE')==preprint][50] to: !! Preprints &amp; 2014 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2014'][!$this->get('NOTE')==preprint][50] Changed lines 14-16 from: !! 2007 - 2012 Publications to: !! 2007 - 2013 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2013'][$this->get('AUTHOR')][50] 18.06.2013, at 10:12 UTC by 130.225.84.229 - Changed lines 9-12 from: !! Preprints &amp; 2012 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][!$this->get('NOTE')==preprint][50] to: !! Preprints &amp; 2013 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2013'][!$this->get('NOTE')==preprint][50] Changed lines 14-15 from: !! 2007 - 2011 Publications to: !! 2007 - 2012 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] 02.08.2012, at 20:06 UTC by 90.184.159.203 - Deleted line 15: bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] 02.08.2012, at 20:05 UTC by 90.184.159.203 - Added line 16: bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] 02.08.2012, at 20:04 UTC by 90.184.159.203 - Deleted line 15: bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] 02.08.2012, at 20:02 UTC by 90.184.159.203 - Added line 16: bibtexquery:[bibfile.bib][$this->get('YEAR')=='2012'][$this->get('AUTHOR')][50] Changed lines 25-63 from: Title: '''Wavelet Frames and Their Duals''' ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008. >>comment<< !! 1998 &ndash; 2006 Use the links in the [[#sb|sidebar]]. bibtexquery:[bibfile.bib][strpos($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]].
>><<
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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.
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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.
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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.
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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.
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* Click %height=16px%{$Filesdir}pdf.png to obtain '''preprint''' versions. Changed lines 3-6 from: !! Newest preprint to: !! Quick Guide: * Click %height=18px%{$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 '''
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* Click {$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata. Changed lines 19-23 from: !! Quick Guide: * Click %height=18px%{$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 {$Filesdir}bibtex.gif to retrieve '''BibTeX''' metadata. to: !! Ph.D. Thesis Title : Wavelet Frames and Their Duals ([[Attach:lemvig_phdthesis.pdf|pdf]]). Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2008. Added lines 29-37: !! Preprints %rfloat%[[Attach:FFExistence.pdf | {$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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!! 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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %% Changed lines 13-14 from: bibtexquery:[bibfile.bib][][!$this->get('YEAR')][50]
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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. \\
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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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. %% Changed lines 6-9 from: %maroon%'''Fusion Frames: Existence and Construction.'''%%\\ Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:) >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< %maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods$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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%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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. Changed lines 37-42 from: %rfloat%[[Attach:FFExistence.pdf | {$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][][$this->get('AUTHOR')][50] Changed line 3 from: bibtexquery:[bibfile.bib] to: bibtexquery:[bibfile.bib][50] Changed line 3 from: bibtexquery:[bibfile.bib][$this->get('YEAR')=<'2013'][$this->get('AUTHOR')][50] to: bibtexquery:[bibfile.bib] Changed line 3 from: bibtexquery:[bibfile.bib][$this->get('YEAR')<'2013'][$this->get('AUTHOR')][50] to: bibtexquery:[bibfile.bib][$this->get('YEAR')=<'2013'][$this->get('AUTHOR')][50] Added lines 2-3: bibtexquery:[bibfile.bib][$this->get('YEAR')<'2013'][$this->get('AUTHOR')][50] Changed line 34 from: 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. Changed line 34 from: 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. Changed lines 31-38 from: !! 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<< to: !! 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. Changed lines 43-44 from: >><< to: Changed line 63 from: >><< to: >><< 25.03.2010, at 17:30 UTC by 131.173.40.72 - Changed lines 8-12 from: !! Preprints &amp; 2009 Publications to: !! Preprints &amp; 2010 Publications bibtexquery:[bibfile.bib][$this->get('YEAR')=='2010'][$this->get('AUTHOR')][50] !! 2009 Publications 17.11.2009, at 13:37 UTC by 131.173.202.170 - Deleted line 21: Added line 23: Changed lines 25-26 from: to: >><< Deleted line 32: >><< 20.08.2009, at 19:54 UTC by 82.113.121.137 - Changed line 4 from: * Click %height=16px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]])
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* Click %height=18px%{$Filesdir}world.png to go to the '''journal''' version of the paper ([[http://doi.org | DOI links]]) 20.08.2009, at 19:53 UTC by 82.113.121.137 - Changed lines 4-5 from: * Click {$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.
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* 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]], [[{$Anawww}jlemvig/index.php?n=Mathematics.Publications| Publications:Lemvig]], or [[http://arxiv.org/| arXiv.org]] to obtain '''preprint''' versions. 20.08.2009, at 19:52 UTC by 82.113.121.137 - Changed lines 4-5 from: * Click {$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.
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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 {$Filesdir}bibtex.gif to retrieve BibTeX metadata. to: * Click {$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'''
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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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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%% P. Boufounos, G. Kutyniok, and H. Rauhut. \\ 04.08.2009, at 17:31 UTC by 131.173.40.72 - Changed line 21 from: P. Boufounos, G. Kutyniok, and H. Rauhut. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] \\
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%rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]]%% P. Boufounos, G. Kutyniok, and H. Rauhut. \\ 04.08.2009, at 17:30 UTC by 131.173.40.72 - Changed line 21 from: P. Boufounos, G. Kutyniok, and H. Rauhut.\\ to: P. Boufounos, G. Kutyniok, and H. Rauhut. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] \\
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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.\\
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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$}. Finally, we point out several variations of this approach. Changed lines 30-31 from: >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< %maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$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) = \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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%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$}. Finally, we point out several variations of this approach. Changed line 27 from: %rfloat%[[Attach:Wavefront.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%% to: %rfloat%[[Attach:Wavefront.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"View BibTeX entry"]]%% Changed line 29 from: (:toggle show="Show abstract" hide box1 button=0:) to: (:toggle show="Show abstract" hide box3 button=0:) Changed line 31 from: %maroon%Abstract%%.It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by SH_f(a,s,t) = \ip{f}{\psi_{ast}}. The analyzing elements \psi_{ast} are dilated and translated copies of a single generating function \psi, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements \psi_{ast} form a system of smooth functions at continuous scales a >0, locations t \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. Finally, we point out several variations of this approach. to: %maroon%Abstract%%. It is known that the Continuous Wavelet Transform of a distribution {$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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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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. to: %maroon%Abstract%%.It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the framework of affine systems to construct the so-called Continuous Shearlet Transform, which is defined by SH_f(a,s,t) = \ip{f}{\psi_{ast}}. The analyzing elements \psi_{ast} are dilated and translated copies of a single generating function \psi, where the dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements \psi_{ast} form a system of smooth functions at continuous scales a >0, locations t \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. Finally, we point out several variations of this approach. Deleted lines 33-39: P. Boufounos, G. Kutyniok, and H. Rauhut.\\ %maroon%'''Compressed Sensing for Fusion Frames'''.%%\\ Submitted (2009).\\ Download: Abstract pdf !! 2009 Added lines 20-27: %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%% bibtexsummary:[bibfile.bib,MR2471937]. (:toggle show="Show abstract" hide box1 button=0:) >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< %maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods$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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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"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 | {$Filesdir}pdf2.png"Download article as PDF file"]] [[index.php?n=Publications.Publications?action=bibentry&bibfile=bibfile.bib&bibref=MR2471937#MR2471937Bib |{$Filesdir}bibtex.gif"This link is NOT correct"]]%%\\
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Firstname Lastname: '''What is Web'''
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Firstname Lastname: '''What is Web'''
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!! Talks
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFExistence.pdf | {$Filesdir}pdf.png]]%%\\ Changed line 8 from: %maroon%Abstract%%. Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods$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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. Changed line 6 from: Submitted (2009). (:toggle show=Abstract hide box1 button=0:) to: Submitted (2009). (:toggle show="Show abstract" hide box1 button=0:) Changed line 8 from: %maroon%Abstract%%: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods$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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. Changed line 6 from: Submitted (2009). (:toggle label=Abstract hide box1 button=0:) to: Submitted (2009). (:toggle show=Abstract hide box1 button=0:) Changed line 8 from: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods$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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. Changed lines 4-5 from: R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\ %maroon%'''Fusion Frames: Existence and Construction.'''%% %rfloat%[[Attach:FFexistence.pdf | {$Filesdir}pdf.png]]%%\\
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R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. %rfloat%[[Attach:FFexistence.pdf | {$Filesdir}pdf.png]]%%\\ %maroon%'''Fusion Frames: Existence and Construction.'''%%\\ Changed line 5 from: %maroon%'''Fusion Frames: Existence and Construction.'''%% %rflush%[[Attach:FFexistence.pdf | {$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}$}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. Changed lines 8-13 from: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$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}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. >><< >>id=box1 font-size=100pct bgcolor=#eeeeee margin="0.5em 0em 0.5em 0.5em" padding=0.5em width=95pct border="0.5px solid gray"<< Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$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}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. >><< to: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$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}$} are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m \in N and {\lambda_j}_{j=1,...,M}. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval. >><< Changed lines 11-12 from: >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< to: >>id=box1 font-size=100pct bgcolor=#eeeeee margin="0.5em 0em 0.5em 0.5em" padding=0.5em width=95pct border="0.5px solid gray"<< Changed line 7 from: >>id=box1 border='1px solid red' padding=5px bgcolor=#eee<< to: >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< Added lines 11-14: >>id=box1 border='1px solid maroon' padding=5px bgcolor=#eee<< Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods - the spatial complement and the Naimark complement - for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given {$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. >><< Changed line 6 from: Submitted (2009).(:toggle label=Abstract hide box1 button=0:) to: Submitted (2009). (:toggle label=Abstract hide box1 button=0:) Changed lines 6-7 from: Submitted (2009).(:toggle label=Abstract hide box1 button=1:) >>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<< to: Submitted (2009).(:toggle label=Abstract hide box1 button=0:) >>id=box1 border='1px solid red' padding=5px bgcolor=#eee<< Changed lines 6-7 from: Submitted (2009).\\ (:toggle label=Abstract hide box1 button=1:) to: Submitted (2009).(:toggle label=Abstract hide box1 button=1:) Changed line 7 from: (:toggle labelname=Abstract hide box1 button=1:) to: (:toggle label=Abstract hide box1 button=1:) Changed lines 7-9 from: Download: Abstract pdf (:toggle hide box1 button=1:) to: (:toggle labelname=Abstract hide box1 button=1:) Changed line 11 from: The text in this section can be hidden/shown to: 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. Added lines 8-13: (:toggle hide box1 button=1:) >>id=box1 border='1px solid #999' padding=5px bgcolor=#edf<< The text in this section can be hidden/shown >><< Changed line 5 from: %maroon%Fusion Frames: Existence and Construction%%.\\ to: %maroon%'''Fusion Frames: Existence and Construction.'''%%\\ Changed line 10 from: %maroon%Compressed Sensing for Fusion Frames.%%\\ to: %maroon%'''Compressed Sensing for Fusion Frames'''.%%\\ Changed line 20 from: %comment% to: >>comment<< Changed line 23 from: %% to: >><< Changed lines 9-13 from: # P. Boufounos, G. Kutyniok, and H. Rauhut. Compressed Sensing for Fusion Frames. Submitted (2009). Abstract pdf to: P. Boufounos, G. Kutyniok, and H. Rauhut.\\ %maroon% Compressed Sensing for Fusion Frames.%%\\ Submitted (2009).\\ Download: Abstract pdf Changed lines 15-23 from: # 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 to: 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 Changed line 5 from: %green%Fusion Frames: Existence and Construction%%.\\ to: %maroon%Fusion Frames: Existence and Construction%%.\\ Changed line 7 from: Abstract pdf to: Download: Abstract pdf Changed line 5 from: %green%Fusion Frames: Existence and Construction.\\ to: %green%Fusion Frames: Existence and Construction%%.\\ Changed line 5 from: Fusion Frames: Existence and Construction.\\ to: %green%Fusion Frames: Existence and Construction.\\ Changed lines 4-8 from: # R . Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. Fusion Frames: Existence and Construction. Submitted (2009). to: R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki.\\ Fusion Frames: Existence and Construction.\\ Submitted (2009).\\ Added lines 1-2: ! AAG Publications Deleted lines 4-6: * 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.
* [[{$mb} | Bownik,M.]]; Lemvig, J.: Affine and quasi-affine frames for rational dilations. Preprint: [[Attach:quasi-affine.pdf |a4]] Changed lines 6-7 from: !! Published papers to: # R. Calderbank, P. G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki. Fusion Frames: Existence and Construction. Submitted (2009). Abstract pdf # P. Boufounos, G. Kutyniok, and H. Rauhut. Compressed Sensing for Fusion Frames. Submitted (2009). Abstract pdf !! 2009 # B. G. Bodmann and G. Kutyniok. Erasure-Proof Transmissions: Fusion Frames meet Coding Theory. Submitted (2009). Abstract pdf # B. G. Bodmann, P. G. Casazza, G. Kutyniok, and S. Senger. A Low Complexity Replacement Scheme for Erased Frame Coefficients. Submitted (2009). Abstract pdf %comment% Changed lines 30-35 from: !!! Ph.D. Thesis -> Title: [[Attach:lemvig_phdthesis_20081007.pdf|Wavelet Frames and Their Duals]]. Supervisor: Ole Christensen. Thesis examiners: Prof. Per Christian Hansen (chairman, DTU), Prof. Say Song Goh (National University of Singapore), Prof. Richard S. Laugesen (University of Illinois, U.S.A.). Thesis defence: Friday September 26, 2009. !!! Nonmathematical Conference paper -> Lemvig, J.; Rasmussen, J.; Jensen, J.F.; Olsson, M.J.; Schultz, J.P.; Hansen, K.K.; Grelk, B.: Natural stone panels for building facades. Loss of strength caused by temperature cycles. Proceedings of the 6th Symposium on Building Physics in the Nordic Countries, Trondheim, Norway, 2002. (pdf). to: %% Added lines 1-16: !! Preprints * Lemvig, J.: Constructing pairs of dual bandlimited frame wavelets in {$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).