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

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

where ${\displaystyle P_{W_{i}}}$ denotes the orthogonal projection onto the subspace ${\displaystyle W_{i}}$. The constants ${\displaystyle A}$ and ${\displaystyle B}$ are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, ${\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}$ becomes a ${\displaystyle A}$-tight fusion frame. Furthermore, if ${\displaystyle A=B=1}$, we can call ${\displaystyle \{W_{i},v_{i}\}_{i\in {\mathcal {I}}}}$ Parseval fusion frame.[1]

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

## Theorem for the relationship between fusion frames and global frames

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

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

## Local frame representation

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

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

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

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

## Definition of fusion frame operator

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

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

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

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

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

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

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

## Properties of fusion frame operator

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

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

## Representation of fusion frame operator

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

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

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

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

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

and

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

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

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

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

## References

1. 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. ^ a b c Casazza, P.G.; Kutyniok, G. (2004). "Frames of subspaces". Wavelets, Frames and Operator Theory. 345: 87–113.
3. ^ Christensen, Ole (2003). An introduction to frames and Riesz bases. Boston [u.a.]: Birkhäuser. p. 8. ISBN 0817642951.