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 (a constraint found by limitations of hardware or data throughput), 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[1] of sensor networks consisting of arbitrary overlapping sensor fields.
Given a Hilbert space
be a set of positive scalar weights.
denotes the orthogonal projection onto the subspace
are called lower and upper bound, respectively.
When the lower and upper bounds are equal to each other,
Parseval fusion frame.
is called a fusion frame system for
is a fusion frame system for
with lower and upper bounds
with lower and upper bounds
with lower and upper bounds
is a fusion frame system for
with lower and upper bounds
is given by[3] We can also express the orthogonal projection of
in terms of given local frame
be representation space for projection.
is defined by The adjoint is called the synthesis operator
The fusion frame operator
is defined by[2] Given the lower and upper bounds of the fusion frame
, the fusion frame operator
Therefore, the fusion frame operator
[2] Given a fusion frame system
, the fusion frame operator
), the fusion frame operator can be constructed with a matrix.
Then the fusion frame operator
is the canonical dual frame of