Narrow Results

In this paper, we consider the recovery of fusion frame sparse signals from incomplete structured measurements. This signal has energy only in a few subspaces of the fusion frame. The sufficient condition for uniform recovery of fusion frame sparse signal is a fusion restricted isometry property (FRIP) which is the fusion frame version of restricted isometry property (RIP) in the standard compressed sensing (CS) settings. Different classes of random matrices are well known to satisfy FRIP, but such matrices do not possess any structure that emerged from the practical sensing systems. In this paper, we give the theoretical results to bridge this gap by proving that a random sampling matrix associated with a bounded orthonormal system satisfies the FRIP with high probability. In detail, we established the uniform recovery guarantees for fusion frames and structured measurements, which may be the beginning of practical applications relating to the fusion frame sparse signal model.

After introducing g-frames and fusion frames by Sun and Casazza, respectively, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or g-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and g-fusion frame sequences. Also, the canonical dual g-fusion frames are presented and we introduce a Parseval g-fusion frame.

This paper considers the nonuniform sparse recovery of block signals in a fusion frame, which is a collection of subspaces that provides redundant representation of signal spaces. Combined with specific fusion frame, the sensing mechanism selects block-vector-valued measurements independently at random from a probability distribution $F$. If the probability distribution $F$ obeys a simple incoherence property and an isotropy property, we can faithfully recover approximately block sparse signals via mixed $l^1/l^2$-minimization in ways similar to Compressed Sensing. The number of measurements is significantly reduced by a priori knowledge of a certain incoherence parameter $\lambda$ associated with the angles between the fusion frame subspaces. As an example, the paper shows that an $s$-sparse block signal can be exactly recovered from about $(\lambda s+1)log(Nk)$ Fourier coefficients combined with fusion frame ${\{W_j\}}_{j=1}^N$, where $dim(W_j)=k$.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

In this paper, we study the erasures of submodules of a fusion frame in Hilbert $C^{\ast }$-modules and we present some sufficient conditions under which a sequence remains a standard fusion frame after deletion of some submodules. We also introduce a new perturbation for g-frames and fusion frames in Hilbert $C^{\ast }$-modules and it is shown that under some conditions g-frames and fusion frames are stable under this perturbation. We generalize some of the results obtained for perturbations of g-frames and fusion frames, especially we consider perturbations of direct sums, duals and approximate duals of g-frames.

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving $c$-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about $c$-woven and $c$-fusion woven. Finally, perturbation of these frames will be introduced.

Fusion frame is a generalization of frame, which can analyze signals by projecting them onto multidimensional subspaces. Controlled fusion frame as generalization of fusion frame, it can improve the numerical efficiency of iterative algorithms for inverting the fusion frame operators. In this paper, we first introduce the notion of controlled g-fusion frame, discuss several properties of controlled g-fusion Bessel sequence. Then, we present some sufficient conditions and some characterizations of controlled g-fusion frames. Finally, we study the sum of controlled g-fusion frames.

In this paper, we will present several new results in finite and countable dimensional separable real Hilbert space phase retrieval and norm retrieval by fusion frames. We will characterize of norm retrieval for fusion frames similar norm retrieval for vectors and we will show that only one direction holds for fusion frames. In similar vector case, we will show that every tight fusion frame can do norm retrieval. Also we will show that the unitary operators preserve phase (norm) retrievability of fusion frames. We will make a detailed study of when hyperplanes do norm retrieval and show a general result about it. We will provide numerous examples to show that our results are best possible.

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.