Fusion frames and distributed processing

Abstract

Let { W i } i ∈ I be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight v i , and let H be the closed linear span of the W i s, a composite Hilbert space. { ( W i , v i ) } 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 W i is equipped with a spanning frame system { f i j } j ∈ J i , we refer to { ( W i , v i , { f i j } j ∈ J i ) } 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.