Fusion frame

Last updated August 15, 2023

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.

Definition

Given a Hilbert space ${\mathcal {H}}$ , let $\{W_{i}\}_{i\in {\mathcal {I}}}$ be closed subspaces of ${\mathcal {H}}$ , where ${\mathcal {I}}$ is an index set. Let $\{v_{i}\}_{i\in {\mathcal {I}}}$ be a set of positive scalar weights. Then $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ is a fusion frame of ${\mathcal {H}}$ if there exist constants $0<A\leq B<\infty$ such that for all $f\in {\mathcal {H}}$ we have

$A\|f\|^{2}\leq \sum _{i\in {\mathcal {I}}}v_{i}^{2}{\big \|}P_{W_{i}}f{\big \|}^{2}\leq B\|f\|^{2}$ ,

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, $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ becomes a $A$ -tight fusion frame. Furthermore, if $A=B=1$ , we can call $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ Parseval fusion frame.^[1]

Assume $\{f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}$ is a frame for $W_{i}$ . Then $\{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}$ is called a fusion frame system for ${\mathcal {H}}$ .^[1]

Theorem for the relationship between fusion frames and global frames

Let $\{W_{i}\}_{i\in {\mathcal {H}}}$ be closed subspaces of ${\mathcal {H}}$ with positive weights $\{v_{i}\}_{i\in {\mathcal {I}}}$ . Suppose $\{f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}$ is a frame for $W_{i}$ with frame bounds $C_{i}$ and $D_{i}$ . Let $C=inf_{i\in {\mathcal {I}}}C_{i}$ and $D=inf_{i\in {\mathcal {I}}}D_{i}$ , which satisfy that $0<C\leq D<\infty$ . Then $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ is a fusion frame of ${\mathcal {H}}$ if and only if $\{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}$ is a frame of ${\mathcal {H}}$ .

Additionally, if $\{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}$ is called a fusion frame system for ${\mathcal {H}}$ with lower and upper bounds $A$ and $B$ , then $\{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}$ is a frame of ${\mathcal {H}}$ with lower and upper bounds $AC$ and $BD$ . And if $\{v_{i}f_{ij}\}_{i\in {\mathcal {I}},j\in J_{i}}$ is a frame of ${\mathcal {H}}$ with lower and upper bounds $E$ and $F$ , then $\{\left(W_{i},v_{i},\{f_{ij}\}_{j\in J_{i}}\right)\}_{i\in {\mathcal {I}}}$ is called a fusion frame system for ${\mathcal {H}}$ with lower and upper bounds $E/D$ and $F/C$ .^[2]

Local frame representation

Let $W\subset {\mathcal {H}}$ be a closed subspace, and let $\{x_{n}\}$ be an orthonormal basis of $W$ . Then for all $f\in {\mathcal {H}}$ , the orthogonal projection of $f$ onto $W$ is given by $P_{W}f=\sum \langle f,x_{n}\rangle x_{n}$ .^[3]

We can also express the orthogonal projection of $f$ onto $W$ in terms of given local frame $\{f_{k}\}$ of $W$ ,

$P_{W}f=\sum \langle f,f_{k}\rangle {\tilde {f}}_{k}$ ,

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

Definition of fusion frame operator

Let $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ be a fusion frame for ${\mathcal {H}}$ . Let $\{\sum \bigoplus W_{i}\}_{l_{2}}$ be representation space for projection. The analysis operator $T_{W}:{\mathcal {H}}\rightarrow \{\sum \bigoplus W_{i}\}_{l_{2}}$ is defined by

$T_{W}\left(f\right)=\{v_{i}P_{W_{i}}\left(f\right)\}_{i\in {\mathcal {I}}}$ .

Then The adjoint operator $T_{W}^{\ast }:\{\sum \bigoplus W_{i}\}_{l_{2}}\rightarrow {\mathcal {H}}$ , which we call the synthesis operator, is given by

$T_{W}^{\ast }\left(g\right)=\sum v_{i}f_{i}$ ,

where $g=\{f_{i}\}_{i\in {\mathcal {I}}}\in \{\sum \bigoplus W_{i}\}_{l_{2}}$ .

The fusion frame operator $S_{W}:{\mathcal {H}}\rightarrow {\mathcal {H}}$ is defined by

$S_{W}\left(f\right)=T_{W}^{\ast }T_{W}\left(f\right)=\sum v_{i}^{2}P_{W_{i}}\left(f\right)$ .^[2]

Properties of fusion frame operator

Given the lower and upper bounds of the fusion frame $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ , $A$ and $B$ , the fusion frame operator $S_{W}$ can be bounded by

$AI\leq S_{W}\leq BI$ , where $I$ is the identity operator. Therefore, the fusion frame operator $S_{W}$ is positive and invertible.^[2]

Representation of fusion frame operator

Given a fusion frame system $\{\left(W_{i},v_{i},{\mathcal {F}}_{i}\right)\}_{i\in {\mathcal {I}}}$ for ${\mathcal {H}}$ , where ${\mathcal {F}}_{i}=\{f_{ij}\}_{j\in J_{i}}$ , and ${\tilde {\mathcal {F}}}_{i}=\{{\tilde {f}}_{ij}\}_{j\in J_{i}}$ , which is a dual frame for ${\mathcal {F}}_{i}$ , the fusion frame operator $S_{W}$ can be expressed as

$S_{W}=\sum v_{i}^{2}T_{{\tilde {\mathcal {F}}}_{i}}^{\ast }T_{{\mathcal {F}}_{i}}=\sum v_{i}^{2}T_{{\mathcal {F}}_{i}}^{\ast }T_{{\tilde {\mathcal {F}}}_{i}}$ ,

where $T_{{\mathcal {F}}_{i}}$ , $T_{{\tilde {\mathcal {F}}}_{i}}$ are analysis operators for ${\mathcal {F}}_{i}$ and ${\tilde {\mathcal {F}}}_{i}$ respectively, and $T_{{\mathcal {F}}_{i}}^{\ast }$ , $T_{{\tilde {\mathcal {F}}}_{i}}^{\ast }$ are synthesis operators for ${\mathcal {F}}_{i}$ and ${\tilde {\mathcal {F}}}_{i}$ respectively.^[1]

For finite frames (i.e., $\dim {\mathcal {H}}=:N<\infty$ and $|{\mathcal {I}}|<\infty$ ), the fusion frame operator can be constructed with a matrix.^[1] Let $\{W_{i},v_{i}\}_{i\in {\mathcal {I}}}$ be a fusion frame for ${\mathcal {H}}_{N}$ , and let $\{f_{ij}\}_{j\in {\mathcal {J}}_{i}}$ be a frame for the subspace $W_{i}$ and $J_{i}$ an index set for each $i\in {\mathcal {I}}$ . With

$F_{i}={\begin{bmatrix}\vdots &\vdots &&\vdots \\f_{i1}&f_{i2}&\cdots &f_{i|J_{i}|}\\\vdots &\vdots &&\vdots \\\end{bmatrix}}_{N\times |J_{i}|}$

and

${\tilde {F}}_{i}={\begin{bmatrix}\vdots &\vdots &&\vdots \\{\tilde {f}}_{i1}&{\tilde {f}}_{i2}&\cdots &{\tilde {f}}_{i|J_{i}|}\\\vdots &\vdots &&\vdots \\\end{bmatrix}}_{N\times |J_{i}|},$

where ${\tilde {f}}_{ij}$ is the canonical dual frame of $f_{ij}$ , the fusion frame operator $S:{\mathcal {H}}\to {\mathcal {H}}$ is given by

$S=\sum _{i\in {\mathcal {I}}}v_{i}^{2}F_{i}{\tilde {F}}_{i}^{T}$ .

The fusion frame operator $S$ is then given by an $N\times N$ matrix.

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References

1 2 3 4 5 6 Casazza, Peter G.; Kutyniok, Gitta; Li, Shidong (2008). "Fusion frames and distributed processing". Applied and Computational Harmonic Analysis. 25 (1): 114–132. arXiv: math/0605374 . doi:10.1016/j.acha.2007.10.001. S2CID 329040.
1 2 3 Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. Contemporary Mathematics. Vol. 345. pp. 87–113. doi:10.1090/conm/345/06242. ISBN 9780821833803. S2CID 16807867.
↑ Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 978-0817642952.

External links

Fusion Frames

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[DistProc-1] 1 2 3 4 5 6 Casazza, Peter G.; Kutyniok, Gitta; Li, Shidong (2008). "Fusion frames and distributed processing". Applied and Computational Harmonic Analysis. 25 (1): 114–132. arXiv: math/0605374 . doi:10.1016/j.acha.2007.10.001. S2CID 329040.

[FrameSubspace-2] 1 2 3 Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. Contemporary Mathematics. Vol. 345. pp. 87–113. doi:10.1090/conm/345/06242. ISBN 9780821833803. S2CID 16807867.

[3] Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 978-0817642952.

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