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 of sensor networks consisting of arbitrary overlapping sensor fields.

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.

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 ${\displaystyle {\mathcal {H}}}$.

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 ${\displaystyle F/C}$.

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

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

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 ${\displaystyle \{f_{k}\}}$.

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

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

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.

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.

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. 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 ${\displaystyle f_{ij}}$.

• Hilbert space
• Frame (linear algebra)

• Fusion Frames