Fusion frames and distributed processing

By Peter G. Casazza, Gitta Kutyniok, Shidong Li. 2006

379 citations recorded by the source corpus.

Let {Wi}i∈I be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight vi, and let H be the closed linear span of the Wis, a composite Hilbert space. {(Wi,vi)}i∈I is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory.” In case each subspace Wi is equipped with a spanning frame system {fij}j∈Ji, we refer to {(Wi,vi,{fij}j∈Ji)}i∈I as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geophysics

View Paper: https://www.semanticscholar.org/paper/dbb5721be4667b434b1de31b47886b9c7dbab7d0

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Citations in the corpus

The Structure of Minimizers of the Frame Potential on Fusion Frames (2008, background): “We refer the reader to [11] and the references therein, for a detailed treatment of the fusion frame theory.”
Dynamical Sampling and Systems from Iterative Actions of Operators (2016, background): “The dynamical sampling problem has connections and applications to other areas of mathematics including, Banach algebras, C∗-algebras, spectral theory of normal operators, and frame theory [2, 8, 10, 13, 14, 15, 26, 32].”
Compressed Sensing (2012, background): “j=1 ‖PWj (x)‖2 ≤ B‖x‖2 for all x ∈ R, where PWj denotes the orthogonal projection onto the subspace Wj, see also [13] and [12, Chapter 13].”
Dynamical sampling (2014, methodology): “The methods that we develop for studying the dynamical sampling problem are related to methods in spectral theory, operator algebras, and frame theory [2, 10, 13, 15, 18, 20, 21, 22, 45].”
Operator-valued frames (2007, result): “cerning norm path-connectedness. It has been brought to our attention that a few other recent papersin the literatureoverlapto some extent with our approach, notably works of Casazza, Kutyniok and Li [6] on ”fusion frames”, and also recent work of Bodmann [1] on quantum computing and work of W. Sun ”