Wavelet packet bases are designed by dividing the frequency axis in intervals of varying sizes. These bases are particularly well adapted to decomposing signals that have different behavior in different frequency intervals. If
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Block Bases
Block orthonormal bases are obtained by dividing the time axis in consecutive intervals
The size
Theorem 1. constructs a block orthogonal basis of
Theorem 1.
if
is a block orthonormal basis of
Proof
One can verify that the dilated and translated family
is an orthonormal basis of
and each block
Block Fourier Basis
A block basis is constructed with the Fourier basis of
The time support of each block Fourier vector
and
It is centered at
Discrete Block Bases
For all
Since dilations are not defined in a discrete framework,we generally cannot derive bases of intervals of varying sizes from a single basis. Thus,Theorem 2. supposes that we can construct an orthonormal basis of
Theorem 2.
Suppose that
is a block orthonormal basis of
A discrete block basis is constructed with discrete Fourier bases
The resulting block Fourier vectors
Block Bases of Images
General block bases of images are constructed by partitioning the plane
The family
For discrete images,we define discrete windows that cover each rectangle
If
is a block basis of