Fusion frame explained

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

l{H} , let \{W _{i\}} _{i} } be closed subspaces of l{H} , where l{I} is an index set. Let \{ v _{i} \} _{i} } be a set of positive scalar weights. Then \{ W _{i,} v _{i} \} _{i} } is a fusion frame of l{H} if there exist constants 0 < A \leq B< inf ty such thatA\|f\| ^{2 \leq\sum} _{i\in l{I} }v_i^2\big\|P_f\big\|^2\leq B\|f\|^2, \quad \forall f\in\mathcal,where P _{W _{i}} denotes the orthogonal projection onto the subspace W _{i} . The constants A and B are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, \{ W _{i,} v _{i} \} _{i} } becomes a A -tight fusion frame. Furthermore, if A = B =1 , we can call \{ W _{i,} v _{i} \} _{i} } Parseval fusion frame.^[1]

Assume

\{f _{ij} \} _{i} , j\in J _{i}} is a frame for W _{i} . Then \{ \left(W _{i,} v _{i,} \{f _{ij} \} _{j\in J _{i}} \right)\} _{i} } is called a fusion frame system for l{H} .^[1]

Relation to global frames

Let

\{W _{i\}} _{i\inl{H} } be closed subspaces of l{H} with positive weights \{ v _{i} \} _{i} }. Suppose \{f _{ij} \} _{i} , j\in J _{i}} is a frame for W _{i} with frame bounds C _{i} and D _{i} . Let $C=\; \backslash inf\_C\_i$ and $D=\backslash inf\_D\_i$, which satisfy that 0 <C\leq D< inf ty . Then \{ W _{i,} v _{i} \} _{i} } is a fusion frame of l{H} if and only if \{v _{if} _{ij} \} _{i} , j\in J _{i}} is a frame of l{H} .

Additionally, if

\{ \left(W _{i,} v _{i,} \{f _{ij} \} _{j\in J _{i}} \right)\} _{i} } is a fusion frame system for l{H} with lower and upper bounds A and B , then \{v _{if} _{ij} \} _{i} , j\in J _{i}} is a frame of l{H} with lower and upper bounds AC and BD . And if \{v _{if} _{ij} \} _{i} , j\in J _{i}} is a frame of l{H} with lower and upper bounds E and F , then \{ \left(W _{i,} v _{i,} \{f _{ij} \} _{j\in J _{i}} \right)\} _{i} } is a fusion frame system for l{H} with lower and upper bounds E/D and F/C .^[2]

Local frame representation

Let

W\subset l{H} be a closed subspace, and let \{x _{n\}} be an orthonormal basis of W . Then the orthogonal projection of f \in l{H} onto W is given by^[3] P _{Wf} = \sum\langle f,x _{n\rangle} x _{n.} We can also express the orthogonal projection of f onto W in terms of given local frame \{f _{k\}} of W P _{Wf} = \sum\langle f,f _{k\rangle} \tilde{f} _{k,}

where

\{\tilde{f} _{k\}} is a dual frame of the local frame \{f _{k\}} .^[1]

Fusion frame operator

Definition

Let

\{ W _{i,} v _{i} \} _{i} } be a fusion frame for l{H} . Let \{\sum o plus W _{i\}} ^{l _{2}} be representation space for projection. The analysis operator T _{W:} l{H} → \{\sum o plus W _{i\}} ^{l _{2}} is defined byT _{W\left(f} \right) = \{v _{iP} ^{W _{i}} \left(f \right)\} _{i\inl{I} }.

The adjoint is called the synthesis operator

T ^{\ast} _{W:} \{\sum o plus W _{i\}} ^{l _{2}} → l{H} , defined asT ^{\ast} _{W\left(g} \right) = \sum v _{if} _{i,} where g = \{f _{i\}} _{i\inl{I} }\in\_.

The fusion frame operator

S _{W:} l{H} → l{H} is defined by^[2] S _{W\left(f} \right) = T ^{\ast} _{WT} _{W\left(f} \right) = \sum v ^{2} _{iP} ^{W _{i}} \left(f \right).

Properties

Given the lower and upper bounds of the fusion frame

\{ W _{i,} v _{i} \} _{i} }, A and B , the fusion frame operator S _{W} can be bounded byAI\leq S _{W\leq} BI, where I is the identity operator. Therefore, the fusion frame operator S _{W} is positive and invertible.^[2]

Representation

Given a fusion frame system

\{ \left(W _{i,} v _{i,} l{F} _{i\right)\}} _{i} } for l{H} , where l{F} _{i=\{f} _{ij} \} _{j\in J _{i}} , and \tilde{ l{F}} _{i=\{\tilde{f}} _{ij} \} _{j\in J _{i}} , which is a dual frame for l{F} _{i} , the fusion frame operator S _{W} can be expressed asS _{W=\sum} v ^{\ast} _{\tilde{ l{F} }_i}T_=\sum v^2_iT^_T_,

where

T _{l{F} _{i}} , T _{\tilde{l{F} }_i} are analysis operators for l{F} _{i} and \tilde{ l{F}} _{i} respectively, and T ^{\ast} _{l{F} _{i}} , T ^{\ast} _{\tilde{ l{F} }_i} are synthesis operators for l{F} _{i} and \tilde{ l{F}} _{i} respectively.^[1]

For finite frames (i.e.,

\dim l H = : N < inf ty and | l I|< inf ty ), the fusion frame operator can be constructed with a matrix.^[1] Let \{ W _{i,} v _{i} \} _{i} } be a fusion frame for l{H} _{N} , and let \{ f _{ij} \} _{j} _{i}} be a frame for the subspace W _{i} and J _{i} an index set for each i\in l{I} . Then the fusion frame operator S: l{H}\to l{H} reduces to an N x N matrix, given byS = \sum _{i\inl{I} }v_i^2 F_i \tilde_i^T,withF _{i} = \begin{bmatrix} \vdots & \vdots & & \vdots \ f _{i1} & f _{i2} & … & f _{i|J _{i|}} \ \vdots & \vdots & & \vdots \\\end{bmatrix} _{N x |J _{i|}} , and\tilde{F} _{i} = \begin{bmatrix} \vdots & \vdots & & \vdots \ \tilde{f} _{i1} & \tilde{f} _{i2} & … & \tilde{f} _{i|J _{i|}} \ \vdots & \vdots & & \vdots \\\end{bmatrix} _{N x |J _{i|}} ,

where

\tilde{f} _{ij} is the canonical dual frame of f _{ij} .

See also

• Hilbert space
• Frame (linear algebra)

External links

• Fusion Frames

Notes and References

1. Casazza. Peter G.. Kutyniok, Gitta. Gitta Kutyniok. Li, Shidong. Fusion frames and distributed processing. Applied and Computational Harmonic Analysis. 2008. 25. 1. 114–132. 10.1016/j.acha.2007.10.001. math/0605374. 329040.
2. Book: Casazza. P.G.. Kutyniok. Gitta Kutyniok. G. . Wavelets, Frames and Operator Theory . Frames of subspaces . 2004. 345. 87–113. 10.1090/conm/345/06242. Contemporary Mathematics. 9780821833803. 16807867.
3. Book: Christensen, Ole. An introduction to frames and Riesz bases. 2003. Birkhäuser. Boston [u.a.]. 978-0817642952. 8.