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Fusion Frames: Existence and Construction

Publication ,  Journal Article
Calderbank, R; Casazza, PG; Heinecke, A; Kutyniok, G; Pezeshki, A
June 30, 2009

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 $M, N, m \in \NN$ and $\{\lambda_j\}_{j=1}^M$, does there exist a fusion frame in $\RR^M$ 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 $M, N, m \in \NN$ 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.

Duke Scholars

Publication Date

June 30, 2009
 

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Calderbank, R., Casazza, P. G., Heinecke, A., Kutyniok, G., & Pezeshki, A. (2009). Fusion Frames: Existence and Construction.
Calderbank, Robert, Peter G. Casazza, Andreas Heinecke, Gitta Kutyniok, and Ali Pezeshki. “Fusion Frames: Existence and Construction,” June 30, 2009.
Calderbank R, Casazza PG, Heinecke A, Kutyniok G, Pezeshki A. Fusion Frames: Existence and Construction. 2009 Jun 30;
Calderbank, Robert, et al. Fusion Frames: Existence and Construction. June 2009.
Calderbank R, Casazza PG, Heinecke A, Kutyniok G, Pezeshki A. Fusion Frames: Existence and Construction. 2009 Jun 30;

Publication Date

June 30, 2009