## The Theory of Fusion Frames |

In the following we will give a short introduction into the theory of fusion frames, during which sensor networks will serve as our main example. Our aim will be to develop a mathematical theory which models the structure of sensor networks. Our theory is further used, for instance, to model the behavior of the neural cells of our brain while processing and storing information and to derive efficient processing for large data sets.

In wireless sensor networks, sensors of limited capacity and power are spread in an area sometimes as large as an entire forest to measure the temperature, sound, vibration, pressure, motion and/or pollutants. In some applications, wireless sensors are placed in a geographical area to detect and characterize chemical, biological, radiological, and nuclear material. Such a sensor system is typically redundant, and there is no orthogonality among sensors, therefore each sensor functions as a frame element in the system. Due to practical and cost reasons, most sensors employed in such applications have severe constraints in their processing power and transmission bandwidth. They often have strictly metered power supply as well. Consequently, a typical large sensor network necessarily divides the network into redundant sub-networks -- forming a set of subspaces. The primary goal is to have local measurements transmitted to a local sub-station within a subspace for a subspace combining. An entire sensor system in such applications could have a number of such local processing centers. They function as relay stations, and have the gathered information further submitted to a central processing station for final assembly.

Fusion frame systems are created to model sensor networks perfectly. The sensors in each sub-network are modeled as frame vectors, which form a frame for a subspace in a Hilbert space. The subspaces, i.e., the sub-networks, have to satisfy a certain overlapping property, which ensures that the overlaps are not too large. In practise this condition will always be fulfilled, since we are mostly dealing with finite-dimensional Hilbert spaces, and finite sets of subspaces. The reconstruction in such a system is done in two steps. Inside the subspaces the conventional frame reconstruction is employed. These local reconstructions then serve as the inputs for the fusion frame reconstruction, which reconstructs the initial signal completely.

In the following we will first describe the basic definitions and notations related to fusion frames, and the focus on reconstruction issues.

Let I be some index set, let {W_{i}}_{i ∈ I} be a family of closed subspaces
in H, and let {v_{i}}_{i ∈ I} be a family of weights, i.e., v_{i} > 0 for
all i ∈ I. Further we denote the orthogonal projections onto W_{i} by
P_{i}.
Then {(W_{i},v_{i})}_{i ∈ I} is a *fusion frame*,
if there exist positive, finite constants C and D such that

Let {W

In frame theory an input signal is represented by a collection
of scalar coefficients that measure the projection of that signal onto each
frame vector. The representation space employed in this theory equals *l*^{2}(I).
However, in fusion frame theory an input signal is represented by a collection
of * vector * coefficients that represent the projection (not just the projection
energy) onto each subspace. Therefore the *representation space*
employed in this setting is

Let {(W

It can easily be shown that the

Now we can give the definition of a fusion frame operator. The

This is a positive and invertible operator on H.

**Finite frame setting:**

For computational needs, let us further consider the fusion frame operator in finite frame settings,
where the fusion frame operator will become the sum of (weighted) matrices of each subspace frame
operator.
Let F_{i} be the frame matrices formed by frame vectors
{f_{ij}}_{j ∈ Ji} in the
column-by-column format [F_{i} ≡ (f_{i1},
f_{i2},...,f_{iji})].
Similarly, let G_{i} be defined in the same way by the dual frame
{g_{ij}}_{j ∈ Ji}.
Then the fusion frame operator associated with finite frames has the expression

where M

The first fundamental observation we make consists of the fact that distributed fusion processing is
feasible in an elegant way by employing the inverse fusion frame operator.
Let {(W_{i},v_{i})}_{i ∈ I} be a fusion frame for H
with fusion frame operator S. Then we have the *reconstruction formula*

The fusion frame theory in fact provides two different approaches for distributed fusion procedures. For this, let {(W

One distributed fusion procedure is from the local projections of each subspace:

In this procedure, the local reconstruction takes place first in each subspace W

Another form of distributed fusion actually acts like a global reconstruction if the coefficients of signal/function decompositions are available:

The difference in this fusion procedure compared with global frame reconstruction lies in the fact that the (global) dual frame {S