Fusion frame


In mathematics, a fusion frame of a vector space is a natural extension of a frame. It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from the event that a signal can not be acquired by a single sensor alone, rather the partial components of the signal must be collected via a network of sensors, and the partial signal representations are then fused into the complete signal.
By construction, fusion frames easily lend themselves to parallel or distributed processing of sensor networks consisting of arbitrary overlapping sensor fields.

Definition

Given a Hilbert space, let be closed subspaces of, where is an index set. Let be a set of positive scalar weights. Then is a fusion frame of if there exist constants such that
where denotes the orthogonal projection onto the subspace. The constants and are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, becomes a -tight fusion frame. Furthermore, if, we can call Parseval fusion frame.
Assume is a frame for. Then is called a fusion frame system for.

Relation to global frames

Let be closed subspaces of with positive weights. Suppose is a frame for with frame bounds and. Let and, which satisfy that. Then is a fusion frame of if and only if is a frame of.
Additionally, if is a fusion frame system for with lower and upper bounds and, then is a frame of with lower and upper bounds and. And if is a frame of with lower and upper bounds and, then is a fusion frame system for with lower and upper bounds and.

Local frame representation

Let be a closed subspace, and let be an orthonormal basis of. Then the orthogonal projection of onto is given by
We can also express the orthogonal projection of onto in terms of given local frame of
where is a dual frame of the local frame.

Fusion frame operator

Definition

Let be a fusion frame for. Let be representation space for projection. The analysis operator is defined by
The adjoint is called the synthesis operator, defined as
where.
The fusion frame operator is defined by

Properties

Given the lower and upper bounds of the fusion frame, and, the fusion frame operator can be bounded by
where is the identity operator. Therefore, the fusion frame operator is positive and invertible.

Representation

Given a fusion frame system for, where, and, which is a dual frame for, the fusion frame operator can be expressed as
where, are analysis operators for and respectively, and, are synthesis operators for and respectively.
For finite frames, the fusion frame operator can be constructed with a matrix. Let be a fusion frame for, and let be a frame for the subspace and an index set for each. Then the fusion frame operator reduces to an matrix, given by
with
and
where is the canonical dual frame of.