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 of sensor networks consisting of arbitrary overlapping sensor fields.
Definition Given a Hilbert space H {\displaystyle {\mathcal {H}}} , let { W i } i ∈ I {\displaystyle \{W_{i}\}_{i\in {\mathcal {I}}}} be closed subspaces of H {\displaystyle {\mathcal {H}}} , where I {\displaystyle {\mathcal {I}}} is an index set. Let { v i } i ∈ I {\displaystyle \{v_{i}\}_{i\in {\mathcal {I}}}} be a set of positive scalar weights. Then { W i , v i } i ∈ I {\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}} is a fusion frame of H {\displaystyle {\mathcal {H}}} if there exist constants 0 < A ≤ B < ∞ {\displaystyle 0<A\leq B<\infty } such that
A ‖ f ‖ 2 ≤ ∑ i ∈ I v i 2 ‖ P W i f ‖ 2 ≤ B ‖ f ‖ 2 , ∀ f ∈ H , {\displaystyle A\|f\|^{2}\leq \sum _{i\in {\mathcal {I}}}v_{i}^{2}{\big \|}P_{W_{i}}f{\big \|}^{2}\leq B\|f\|^{2},\quad \forall f\in {\mathcal {H}},}
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