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.
By construction, fusion frames easily lend themselves to parallel or distributed processing[1] of sensor networks consisting of arbitrary overlapping sensor fields.
Definition
Given a Hilbert space [math]\displaystyle{ \mathcal{H} }[/math], let [math]\displaystyle{ \{W_i\}_{i \in \mathcal{I}} }[/math] be closed subspaces of [math]\displaystyle{ \mathcal{H} }[/math], where [math]\displaystyle{ \mathcal{I} }[/math] is an index set. Let [math]\displaystyle{ \{ v_i \}_{i \in \mathcal{I}} }[/math] be a set of positive scalar weights. Then [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] is a fusion frame of [math]\displaystyle{ \mathcal{H} }[/math] if there exist constants [math]\displaystyle{ 0 \lt A \leq B\lt \infty }[/math] such that for all [math]\displaystyle{ f\in\mathcal{H} }[/math] we have
[math]\displaystyle{ A\|f\|^2\leq\sum_{i\in\mathcal{I}}v_i^2\big\|P_{W_i}f\big\|^2\leq B\|f\|^2 }[/math],
where [math]\displaystyle{ P_{W_i} }[/math] denotes the orthogonal projection onto the subspace [math]\displaystyle{ W_i }[/math]. The constants [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] becomes a [math]\displaystyle{ A }[/math]-tight fusion frame. Furthermore, if [math]\displaystyle{ A=B=1 }[/math], we can call [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] Parseval fusion frame.[1]
Assume [math]\displaystyle{ \{f_{ij}\}_{i \in \mathcal{I}, j\in J_i} }[/math] is a frame for [math]\displaystyle{ W_i }[/math]. Then [math]\displaystyle{ \{ \left(W_i, v_i, \{f_{ij}\}_{j\in J_i} \right)\}_{i \in \mathcal{I}} }[/math] is called a fusion frame system for [math]\displaystyle{ \mathcal{H} }[/math].[1]
Theorem for the relationship between fusion frames and global frames
Let [math]\displaystyle{ \{W_i\}_{i\in\mathcal{H}} }[/math] be closed subspaces of [math]\displaystyle{ \mathcal{H} }[/math] with positive weights [math]\displaystyle{ \{ v_i \}_{i \in \mathcal{I}} }[/math]. Suppose [math]\displaystyle{ \{f_{ij}\}_{i \in \mathcal{I}, j\in J_i} }[/math] is a frame for [math]\displaystyle{ W_i }[/math] with frame bounds [math]\displaystyle{ C_i }[/math] and [math]\displaystyle{ D_i }[/math]. Let [math]\displaystyle{ C=inf_{i\in\mathcal{I}}C_i }[/math] and [math]\displaystyle{ D=inf_{i\in\mathcal{I}}D_i }[/math], which satisfy that [math]\displaystyle{ 0\lt C\leq D\lt \infty }[/math]. Then [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] is a fusion frame of [math]\displaystyle{ \mathcal{H} }[/math] if and only if [math]\displaystyle{ \{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i} }[/math] is a frame of [math]\displaystyle{ \mathcal{H} }[/math].
Additionally, if [math]\displaystyle{ \{ \left(W_i, v_i, \{f_{ij}\}_{j\in J_i} \right)\}_{i \in \mathcal{I}} }[/math] is called a fusion frame system for [math]\displaystyle{ \mathcal{H} }[/math] with lower and upper bounds [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math], then [math]\displaystyle{ \{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i} }[/math] is a frame of [math]\displaystyle{ \mathcal{H} }[/math] with lower and upper bounds [math]\displaystyle{ AC }[/math] and [math]\displaystyle{ BD }[/math]. And if [math]\displaystyle{ \{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i} }[/math] is a frame of [math]\displaystyle{ \mathcal{H} }[/math] with lower and upper bounds [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math], then [math]\displaystyle{ \{ \left(W_i, v_i, \{f_{ij}\}_{j\in J_i} \right)\}_{i \in \mathcal{I}} }[/math] is called a fusion frame system for [math]\displaystyle{ \mathcal{H} }[/math] with lower and upper bounds [math]\displaystyle{ E/D }[/math] and [math]\displaystyle{ F/C }[/math].[2]
Local frame representation
Let [math]\displaystyle{ W\subset\mathcal{H} }[/math] be a closed subspace, and let [math]\displaystyle{ \{x_n\} }[/math] be an orthonormal basis of [math]\displaystyle{ W }[/math]. Then for all [math]\displaystyle{ f\in\mathcal{H} }[/math], the orthogonal projection of [math]\displaystyle{ f }[/math] onto [math]\displaystyle{ W }[/math] is given by [math]\displaystyle{ P_Wf = \sum\langle f,x_n\rangle x_n }[/math].[3]
We can also express the orthogonal projection of [math]\displaystyle{ f }[/math] onto [math]\displaystyle{ W }[/math] in terms of given local frame [math]\displaystyle{ \{f_k\} }[/math] of [math]\displaystyle{ W }[/math],
[math]\displaystyle{ P_Wf = \sum\langle f,f_k\rangle \tilde{f}_k }[/math],
where [math]\displaystyle{ \{\tilde{f}_k\} }[/math] is a dual frame of the local frame [math]\displaystyle{ \{f_k\} }[/math].[1]
Definition of fusion frame operator
Let [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] be a fusion frame for [math]\displaystyle{ \mathcal{H} }[/math]. Let [math]\displaystyle{ \{\sum\bigoplus W_i\}_{l_2} }[/math] be representation space for projection. The analysis operator [math]\displaystyle{ T_W: \mathcal{H}\rightarrow\{\sum\bigoplus W_i\}_{l_2} }[/math] is defined by
[math]\displaystyle{ T_W\left(f \right)=\{v_iP_{W_i}\left(f \right)\}_{i\in\mathcal{I}} }[/math].
Then The adjoint operator [math]\displaystyle{ T^{\ast}_W: \{\sum\bigoplus W_i\}_{l_2}\rightarrow \mathcal{H} }[/math], which we call the synthesis operator, is given by
[math]\displaystyle{ T^{\ast}_W\left(g \right)=\sum v_if_i }[/math],
where [math]\displaystyle{ g=\{f_i\}_{i\in\mathcal{I}}\in\{\sum\bigoplus W_i\}_{l_2} }[/math].
The fusion frame operator [math]\displaystyle{ S_W: \mathcal{H}\rightarrow\mathcal{H} }[/math] is defined by
[math]\displaystyle{ S_W\left(f \right)=T^{\ast}_WT_W\left(f \right)=\sum v^{2}_iP_{W_i}\left(f \right) }[/math].[2]
Properties of fusion frame operator
Given the lower and upper bounds of the fusion frame [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math], [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math], the fusion frame operator [math]\displaystyle{ S_W }[/math] can be bounded by
[math]\displaystyle{ AI\leq S_W\leq BI }[/math], where [math]\displaystyle{ I }[/math] is the identity operator. Therefore, the fusion frame operator [math]\displaystyle{ S_W }[/math] is positive and invertible.[2]
Representation of fusion frame operator
Given a fusion frame system [math]\displaystyle{ \{ \left(W_i, v_i, \mathcal{F}_i\right)\}_{i \in \mathcal{I}} }[/math] for [math]\displaystyle{ \mathcal{H} }[/math], where [math]\displaystyle{ \mathcal{F}_i=\{f_{ij}\}_{j\in J_i} }[/math], and [math]\displaystyle{ \tilde{\mathcal{F}}_i=\{\tilde{f}_{ij}\}_{j\in J_i} }[/math], which is a dual frame for [math]\displaystyle{ \mathcal{F}_i }[/math], the fusion frame operator [math]\displaystyle{ S_W }[/math] can be expressed as
[math]\displaystyle{ S_W=\sum v^2_iT^{\ast}_{\tilde{\mathcal{F}}_i}T_{\mathcal{F}_i}=\sum v^2_iT^{\ast}_{\mathcal{F}_i}T_{\tilde{\mathcal{F}}_i} }[/math],
where [math]\displaystyle{ T_{\mathcal{F}_i} }[/math], [math]\displaystyle{ T_{\tilde{\mathcal{F}}_i} }[/math] are analysis operators for [math]\displaystyle{ \mathcal{F}_i }[/math] and [math]\displaystyle{ \tilde{\mathcal{F}}_i }[/math] respectively, and [math]\displaystyle{ T^{\ast}_{\mathcal{F}_i} }[/math], [math]\displaystyle{ T^{\ast}_{\tilde{\mathcal{F}}_i} }[/math] are synthesis operators for [math]\displaystyle{ \mathcal{F}_i }[/math] and [math]\displaystyle{ \tilde{\mathcal{F}}_i }[/math] respectively.[1]
For finite frames (i.e., [math]\displaystyle{ \dim\mathcal H =: N \lt \infty }[/math] and [math]\displaystyle{ |\mathcal I|\lt \infty }[/math]), the fusion frame operator can be constructed with a matrix.[1] Let [math]\displaystyle{ \{ W_i, v_i \}_{i \in \mathcal{I}} }[/math] be a fusion frame for [math]\displaystyle{ \mathcal{H}_N }[/math], and let [math]\displaystyle{ \{ f_{ij} \}_{j \in \mathcal{J}_i} }[/math] be a frame for the subspace [math]\displaystyle{ W_i }[/math] and [math]\displaystyle{ J_i }[/math] an index set for each [math]\displaystyle{ i\in\mathcal{I} }[/math]. With
[math]\displaystyle{ F_i = \begin{bmatrix} \vdots & \vdots & & \vdots \\ f_{i1} & f_{i2} & \cdots & f_{i|J_i|} \\ \vdots & \vdots & & \vdots \\\end{bmatrix}_{N \times |J_i|} }[/math]
and
[math]\displaystyle{ \tilde{F}_i = \begin{bmatrix} \vdots & \vdots & & \vdots \\ \tilde{f}_{i1} & \tilde{f}_{i2} & \cdots & \tilde{f}_{i|J_i|} \\ \vdots & \vdots & & \vdots \\\end{bmatrix}_{N \times |J_i|}, }[/math]
where [math]\displaystyle{ \tilde{f}_{ij} }[/math] is the canonical dual frame of [math]\displaystyle{ f_{ij} }[/math], the fusion frame operator [math]\displaystyle{ S: \mathcal{H}\to\mathcal{H} }[/math] is given by
[math]\displaystyle{ S = \sum_{i\in\mathcal{I}}v_i^2 F_i \tilde{F}_i^T }[/math].
The fusion frame operator [math]\displaystyle{ S }[/math] is then given by an [math]\displaystyle{ N\times N }[/math] matrix.
See also
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Casazza, Peter G.; Kutyniok, Gitta; Li, Shidong (2008). "Fusion frames and distributed processing". Applied and Computational Harmonic Analysis 25 (1): 114–132. doi:10.1016/j.acha.2007.10.001.
- ↑ 2.0 2.1 2.2 Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. Contemporary Mathematics. 345. pp. 87–113. doi:10.1090/conm/345/06242. ISBN 9780821833803.
- ↑ Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 978-0817642952.
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