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 (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. 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. Given a Hilbert space H {displaystyle {mathcal {H}}} , let { W i } i ∈ I {displaystyle {W_{i}}_{iin {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}}_{iin {mathcal {I}}}} be a set of positive scalar weights. Then { W i , v i } i ∈ I {displaystyle {W_{i},v_{i}}_{iin {mathcal {I}}}} is a fusion frame of H {displaystyle {mathcal {H}}} if there exist constants 0 < A ≤ B < ∞ {displaystyle 0<Aleq B<infty } such that for all f ∈ H {displaystyle fin {mathcal {H}}} we have A ‖ f ‖ 2 ≤ ∑ i ∈ I v i 2 ‖ P W i f ‖ 2 ≤ B ‖ f ‖ 2 {displaystyle A|f|^{2}leq sum _{iin {mathcal {I}}}v_{i}^{2}{ig |}P_{W_{i}}f{ig |}^{2}leq B|f|^{2}} , where P W i {displaystyle P_{W_{i}}} denotes the orthogonal projection onto the subspace W i {displaystyle W_{i}} . The constants A {displaystyle A} and B {displaystyle B} are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, { W i , v i } i ∈ I {displaystyle {W_{i},v_{i}}_{iin {mathcal {I}}}} becomes a A {displaystyle A} -tight fusion frame. Furthermore, if A = B = 1 {displaystyle A=B=1} , we can call { W i , v i } i ∈ I {displaystyle {W_{i},v_{i}}_{iin {mathcal {I}}}} Parseval fusion frame. Assume { f i j } i ∈ I , j ∈ J i {displaystyle {f_{ij}}_{iin {mathcal {I}},jin J_{i}}} is a frame for W i {displaystyle W_{i}} . Then { ( W i , v i , { f i j } j ∈ J i ) } i ∈ I {displaystyle {left(W_{i},v_{i},{f_{ij}}_{jin J_{i}} ight)}_{iin {mathcal {I}}}} is called a fusion frame system for H {displaystyle {mathcal {H}}} . Let { W i } i ∈ H {displaystyle {W_{i}}_{iin {mathcal {H}}}} be closed subspaces of H {displaystyle {mathcal {H}}} with positive weights { v i } i ∈ I {displaystyle {v_{i}}_{iin {mathcal {I}}}} . Suppose { f i j } i ∈ I , j ∈ J i {displaystyle {f_{ij}}_{iin {mathcal {I}},jin J_{i}}} is a frame for W i {displaystyle W_{i}} with frame bounds C i {displaystyle C_{i}} and D i {displaystyle D_{i}} . Let C = i n f i ∈ I C i {displaystyle C=inf_{iin {mathcal {I}}}C_{i}} and D = i n f i ∈ I D i {displaystyle D=inf_{iin {mathcal {I}}}D_{i}} , which satisfy that 0 < C ≤ D < ∞ {displaystyle 0<Cleq D<infty } . Then { W i , v i } i ∈ I {displaystyle {W_{i},v_{i}}_{iin {mathcal {I}}}} is a fusion frame of H {displaystyle {mathcal {H}}} if and only if { v i f i j } i ∈ I , j ∈ J i {displaystyle {v_{i}f_{ij}}_{iin {mathcal {I}},jin J_{i}}} is a frame of H {displaystyle {mathcal {H}}} . Additionally, if { ( W i , v i , { f i j } j ∈ J i ) } i ∈ I {displaystyle {left(W_{i},v_{i},{f_{ij}}_{jin J_{i}} ight)}_{iin {mathcal {I}}}} is called a fusion frame system for H {displaystyle {mathcal {H}}} with lower and upper bounds A {displaystyle A} and B {displaystyle B} , then { v i f i j } i ∈ I , j ∈ J i {displaystyle {v_{i}f_{ij}}_{iin {mathcal {I}},jin J_{i}}} is a frame of H {displaystyle {mathcal {H}}} with lower and upper bounds A C {displaystyle AC} and B D {displaystyle BD} . And if { v i f i j } i ∈ I , j ∈ J i {displaystyle {v_{i}f_{ij}}_{iin {mathcal {I}},jin J_{i}}} is a frame of H {displaystyle {mathcal {H}}} with lower and upper bounds E {displaystyle E} and F {displaystyle F} , then { ( W i , v i , { f i j } j ∈ J i ) } i ∈ I {displaystyle {left(W_{i},v_{i},{f_{ij}}_{jin J_{i}} ight)}_{iin {mathcal {I}}}} is called a fusion frame system for H {displaystyle {mathcal {H}}} with lower and upper bounds E / D {displaystyle E/D} and F / C {displaystyle F/C} . Let W ⊂ H {displaystyle Wsubset {mathcal {H}}} be a closed subspace, and let { x n } {displaystyle {x_{n}}} be an orthonormal basis of W {displaystyle W} . Then for all f ∈ H {displaystyle fin {mathcal {H}}} , the orthogonal projection of f {displaystyle f} onto W {displaystyle W} is given by P W f = ∑ ⟨ f , x n ⟩ x n {displaystyle P_{W}f=sum langle f,x_{n} angle x_{n}} .