Compressed sensing for fusion frames

Abstract

Compressed Sensing (CS) is a new signal acquisition technique that allows sampling of sparse signal using significantly fewer measurements than previously thought possible. On the other hand, a fusion frame is a new signal representation method that uses collections of subspaces instead of vectors to represent signals. This work combines these exciting new fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications.With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it needs not be sparse within each of the subspaces it occupies. We define a mixed 1 / 2 norm for fusion frames. A signal sparse in the subspaces of the fusion frame can thus be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed 1 / 2 norm. The sampling conditions we derive are very similar to the coherence and RIP conditions used in standard CS theory.

Sparse Recovery From Combined Fusion Frame Measurements

“…The rich structure of the fusion frame framework allows us to characterize more complicated signal models than the standard sparse or compressible signals used in compressed sensing techniques. This paper complements and extends our work in [4].…”

“…, N . Definition 3.3: Let A ∈ R n×N and (W j ) N j=1 be a fusion frame for R M and A P as defined in (4). The fusion restricted isometry constant δ k is the smallest constant such that…”

Non-orthogonal Fusion Frames and the Sparsity of Fusion Frame Operators

“…Fusion frames were introduced in [9] (under the name frames of subspaces) and [10], and have quickly turned into an industry (see www.fusionframes.org). Recent developments include applications to sensor networks [12], filter bank fusion frames [13], applications to coding theory [1], compressed sensing [2], construction methods [3,4,7,6,5], sparsity for fusion frames [8], and frame potentials and fusion frames [18]. Until now, most of the work on fusion frames has centered on developing their basic properties and on constructing fusion frames with specific properties.…”

Properties of J-fusion frames in Krein space

“…This concept provides a useful framework in modeling sensor networks [19]. Different aspects and applications of fusion frame can be seen in [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].…”