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 ${\displaystyle 0<A\leq B<\infty }$ such that

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

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]

Relation to 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 ${\textstyle C=\inf _{i\in {\mathcal {I}}}C_{i}}$ and ${\textstyle D=\inf _{i\in {\mathcal {I}}}D_{i}}$, which satisfy that ${\displaystyle 0<C\leq D<\infty }$. 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 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 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 the orthogonal projection of ${\displaystyle f\in {\mathcal {H}}}$ onto ${\displaystyle W}$ is given by^[3]

${\displaystyle P_{W}f=\sum \langle f,x_{n}\rangle x_{n}.}$

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]

Fusion frame operator

Definition

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

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

The adjoint is called the synthesis operator ${\displaystyle T_{W}^{\ast }:\{\sum \bigoplus W_{i}\}_{l_{2}}\rightarrow {\mathcal {H}}}$, defined as

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

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^[2]

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

Properties

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

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 ${\displaystyle i\in {\mathcal {I}}}$. Then the fusion frame operator ${\displaystyle S:{\mathcal {H}}\to {\mathcal {H}}}$ reduces to an ${\displaystyle N\times N}$ matrix, given by

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

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 ${\displaystyle f_{ij}}$.

See also

• Hilbert space
• Frame (linear algebra)