Английская Википедия:Fusion frame
Шаблон:Short description Шаблон:Multiple issues 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>\mathcal{H}</math>, let <math>\{W_i\}_{i \in \mathcal{I}}</math> be closed subspaces of <math>\mathcal{H}</math>, where <math>\mathcal{I}</math> is an index set. Let <math>\{ v_i \}_{i \in \mathcal{I}}</math> be a set of positive scalar weights. Then <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> is a fusion frame of <math>\mathcal{H}</math> if there exist constants <math>0 < A \leq B<\infty</math> such that
- <math>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},</math>
where <math>P_{W_i}</math> denotes the orthogonal projection onto the subspace <math>W_i</math>. The constants <math>A</math> and <math>B</math> are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> becomes a <math>A</math>-tight fusion frame. Furthermore, if <math>A=B=1</math>, we can call <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> Parseval fusion frame.[1]
Assume <math>\{f_{ij}\}_{i \in \mathcal{I}, j\in J_i}</math> is a frame for <math>W_i</math>. Then <math>\{ \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>\mathcal{H}</math>.[1]
Relation to global frames
Let <math>\{W_i\}_{i\in\mathcal{H}}</math> be closed subspaces of <math>\mathcal{H}</math> with positive weights <math>\{ v_i \}_{i \in \mathcal{I}}</math>. Suppose <math>\{f_{ij}\}_{i \in \mathcal{I}, j\in J_i}</math> is a frame for <math>W_i</math> with frame bounds <math>C_i</math> and <math>D_i</math>. Let <math display="inline">C= \inf_{i\in\mathcal{I}}C_i</math> and <math display="inline">D=\inf_{i\in\mathcal{I}}D_i</math>, which satisfy that <math>0<C\leq D<\infty</math>. Then <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> is a fusion frame of <math>\mathcal{H}</math> if and only if <math>\{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i}</math> is a frame of <math>\mathcal{H}</math>.
Additionally, if <math>\{ \left(W_i, v_i, \{f_{ij}\}_{j\in J_i} \right)\}_{i \in \mathcal{I}}</math> is a fusion frame system for <math>\mathcal{H}</math> with lower and upper bounds <math>A</math> and <math>B</math>, then <math>\{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i}</math> is a frame of <math>\mathcal{H}</math> with lower and upper bounds <math>AC</math> and <math>BD</math>. And if <math>\{v_if_{ij}\}_{i \in \mathcal{I}, j\in J_i}</math> is a frame of <math>\mathcal{H}</math> with lower and upper bounds <math>E</math> and <math>F</math>, then <math>\{ \left(W_i, v_i, \{f_{ij}\}_{j\in J_i} \right)\}_{i \in \mathcal{I}}</math> is a fusion frame system for <math>\mathcal{H}</math> with lower and upper bounds <math>E/D</math> and <math>F/C</math>.[2]
Local frame representation
Let <math>W\subset\mathcal{H}</math> be a closed subspace, and let <math>\{x_n\}</math> be an orthonormal basis of <math>W</math>. Then the orthogonal projection of <math>f \in \mathcal{H}</math> onto <math>W</math> is given by[3]
- <math>P_Wf = \sum\langle f,x_n\rangle x_n.</math>
We can also express the orthogonal projection of <math>f</math> onto <math>W</math> in terms of given local frame <math>\{f_k\}</math> of <math>W</math>
- <math>P_Wf = \sum\langle f,f_k\rangle \tilde{f}_k,</math>
where <math>\{\tilde{f}_k\}</math> is a dual frame of the local frame <math>\{f_k\}</math>.[1]
Fusion frame operator
Definition
Let <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> be a fusion frame for <math>\mathcal{H}</math>. Let <math>\{\sum\bigoplus W_i\}_{l_2}</math> be representation space for projection. The analysis operator <math>T_W: \mathcal{H}\rightarrow\{\sum\bigoplus W_i\}_{l_2}</math> is defined by
- <math>T_W\left(f \right)=\{v_iP_{W_i}\left(f \right)\}_{i\in\mathcal{I}}.</math>
The adjoint is called the synthesis operator <math>T^{\ast}_W: \{\sum\bigoplus W_i\}_{l_2}\rightarrow \mathcal{H}</math>, defined as
- <math>T^{\ast}_W\left(g \right)=\sum v_if_i,</math>
where <math>g=\{f_i\}_{i\in\mathcal{I}}\in\{\sum\bigoplus W_i\}_{l_2}</math>.
The fusion frame operator <math>S_W: \mathcal{H}\rightarrow\mathcal{H}</math> is defined by[2]
- <math>S_W\left(f \right)=T^{\ast}_WT_W\left(f \right)=\sum v^{2}_iP_{W_i}\left(f \right).</math>
Properties
Given the lower and upper bounds of the fusion frame <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math>, <math>A</math> and <math>B</math>, the fusion frame operator <math>S_W</math> can be bounded by
- <math>AI\leq S_W\leq BI,</math>
where <math>I</math> is the identity operator. Therefore, the fusion frame operator <math>S_W</math> is positive and invertible.[2]
Representation
Given a fusion frame system <math>\{ \left(W_i, v_i, \mathcal{F}_i\right)\}_{i \in \mathcal{I}}</math> for <math>\mathcal{H}</math>, where <math>\mathcal{F}_i=\{f_{ij}\}_{j\in J_i} </math>, and <math>\tilde{\mathcal{F}}_i=\{\tilde{f}_{ij}\}_{j\in J_i} </math>, which is a dual frame for <math>\mathcal{F}_i</math>, the fusion frame operator <math>S_W</math> can be expressed as
- <math>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>T_{\mathcal{F}_i}</math>, <math>T_{\tilde{\mathcal{F}}_i}</math> are analysis operators for <math>\mathcal{F}_i</math> and <math>\tilde{\mathcal{F}}_i</math> respectively, and <math>T^{\ast}_{\mathcal{F}_i}</math>, <math>T^{\ast}_{\tilde{\mathcal{F}}_i}</math> are synthesis operators for <math>\mathcal{F}_i</math> and <math>\tilde{\mathcal{F}}_i</math> respectively.[1]
For finite frames (i.e., <math>\dim\mathcal H =: N < \infty</math> and <math>|\mathcal I|<\infty</math>), the fusion frame operator can be constructed with a matrix.[1] Let <math>\{ W_i, v_i \}_{i \in \mathcal{I}}</math> be a fusion frame for <math>\mathcal{H}_N</math>, and let <math>\{ f_{ij} \}_{j \in \mathcal{J}_i}</math> be a frame for the subspace <math>W_i</math> and <math>J_i</math> an index set for each <math>i\in\mathcal{I}</math>. Then the fusion frame operator <math>S: \mathcal{H}\to\mathcal{H}</math> reduces to an <math>N\times N</math> matrix, given by
- <math>S = \sum_{i\in\mathcal{I}}v_i^2 F_i \tilde{F}_i^T,</math>
with
- <math>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>\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>\tilde{f}_{ij}</math> is the canonical dual frame of <math>f_{ij}</math>.
See also
References
External links
