Fusion Frames and Distributed Processing

Let {Wi}i∈I be a (redundant) sequence of subspaces each being endowed with a weight vi, and let H be the closed linear span of the Wi’s, a composite Hilbert space. Provided that {(Wi, vi)}i∈I satisfies a certain property which controls the weighted overlaps of the subspaces, it is called a fusion frame. These systems contain conventional frames as a special case, however they go far “beyond frame theory”. In case each subspace Wi is equipped with a frame system {fij}j∈Ji by which it is spanned, we refer to {(Wi, vi, {fij}j∈Ji )}i∈I as a fusion frame system. In this paper, we describe a weighted and distributed processing procedure that fuse together information in all subspaces Wi of a fusion frame system to obtain the global information in H. The weighted and distributed processing technique described in fusion frames is not only a natural fit in distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. We further provide an extensive study of the robustness of fusion frame systems.