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.
Definition.
Given a Hilbert space formula_1, let formula_2 be closed subspaces of formula_1, where formula_4 is an index set. Let formula_5 be a set of positive scalar weights. Then formula_6 is a fusion frame of formula_1 if there exist constants formula_8 such that for all formula_9 we have
formula_10,
where formula_11 denotes the orthogonal projection onto the subspace formula_12. The constants formula_13 and formula_14 are called lower and upper bound, respectively. When the lower and upper bounds are equal to each other, formula_6 becomes a formula_13-tight fusion frame. Furthermore, if formula_17, we can call formula_6 Parseval fusion frame.
Assume formula_19 is a frame for formula_12. Then formula_21 is called a fusion frame system for formula_1.
Theorem for the relationship between fusion frames and global frames.
Let formula_23 be closed subspaces of formula_1 with positive weights formula_5. Suppose formula_19 is a frame for formula_12 with frame bounds formula_28 and formula_29. Let formula_30 and formula_31, which satisfy that formula_32. Then formula_6 is a fusion frame of formula_1 if and only if formula_35 is a frame of formula_1.
Additionally, if formula_21 is called a fusion frame system for formula_1 with lower and upper bounds formula_13 and formula_14, then formula_35 is a frame of formula_1 with lower and upper bounds formula_43 and formula_44. And if formula_35 is a frame of formula_1 with lower and upper bounds formula_47 and formula_48, then formula_21 is called a fusion frame system for formula_1 with lower and upper bounds formula_51 and formula_52.
Local frame representation.
Let formula_53 be a closed subspace, and let formula_54 be an orthonormal basis of formula_55. Then for all formula_9, the orthogonal projection of formula_57 onto formula_55 is given by formula_59.
We can also express the orthogonal projection of formula_57 onto formula_55 in terms of given local frame formula_62 of formula_55,
formula_64,
where formula_65 is a dual frame of the local frame formula_62.
Definition of fusion frame operator.
Let formula_6 be a fusion frame for formula_1. Let formula_69 be representation space for projection. The analysis operator formula_70 is defined by
formula_71.
Then The adjoint operator formula_72, which we call the synthesis operator, is given by
formula_73,
where formula_74.
The fusion frame operator formula_75 is defined by
formula_76.
Properties of fusion frame operator.
Given the lower and upper bounds of the fusion frame formula_6, formula_13 and formula_14, the fusion frame operator formula_80 can be bounded by
formula_81,
where formula_82 is the identity operator. Therefore, the fusion frame operator formula_80 is positive and invertible.
Representation of fusion frame operator.
Given a fusion frame system formula_84 for formula_1, where formula_86, and formula_87, which is a dual frame for formula_88, the fusion frame operator formula_80 can be expressed as
formula_90,
where formula_91, formula_92 are analysis operators for formula_88 and formula_94 respectively, and formula_95, formula_96 are synthesis operators for formula_88 and formula_94 respectively.
For finite frames (i.e., formula_99 and formula_100), the fusion frame operator can be constructed with a matrix. Let formula_6 be a fusion frame for formula_102, and let formula_103 be a frame for the subspace formula_12 and formula_105 an index set for each formula_106. With
formula_107
and
formula_108
where formula_109 is the canonical dual frame of formula_110, the fusion frame operator formula_111 is given by
formula_112.
The fusion frame operator formula_113 is then given by an formula_114 matrix.