Abstract
Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator \(S_{\mathcal{W}}\), which in turn is heavily dependent on the sparsity of \(S_{\mathcal{W}}\). We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of non-orthogonal fusion frames. We show that for a fusion frame {W i ,v i } i∈I , if dim(W i )=k i , then the matrix of the non-orthogonal fusion frame operator \({\mathcal{S}}_{{\mathcal{W}}}\) has in its corresponding location at most a k i ×k i block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator \({\mathcal{S}}_{{\mathcal{W}}}\) is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of multiple fusion frames whose corresponding fusion frame operator becomes an diagonal operator is also examined.
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Communicated by Hans G. Feichtinger.
The first author is supported by NSF DMS 1008183; the second author is supported by NSF DMS 1008183 DTRA/NSF 1042701, AFOSR F1ATA00183G003; the third author is supported by NSF DMS 0709384 and NSF DMS 1010058.
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Cahill, J., Casazza, P.G. & Li, S. Non-orthogonal Fusion Frames and the Sparsity of Fusion Frame Operators. J Fourier Anal Appl 18, 287–308 (2012). https://doi.org/10.1007/s00041-011-9200-7
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DOI: https://doi.org/10.1007/s00041-011-9200-7
Keywords
- Frames
- Fusion frames
- Sparsity of the fusion frame operator
- Sensor network
- Data fusion
- Distributed processing
- Parallel processing