Block transform

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  has properties that vary in time, it is then  more appropriate to decompose  in a block basis that segments the time axis in intervals with sizes that are adapted to the signal structures.

Block Bases 
Block orthonormal bases are obtained by dividing the time axis in consecutive intervals  with

 and .

The size  of each interval is arbitrary. Let . An interval is covered by the dilated rectangular window

Theorem 1. constructs a block orthogonal basis of  from a single orthonormal basis of .

Theorem 1. 
if  is an orthonormal basis of , then

is a block orthonormal basis of

Proof 
One can verify that the dilated and translated family

is an orthonormal basis of . If , then  since their supports do not overlap. Thus, the family  is orthonormal. To expand a signal  in this family, it is decomposed as a sum of separate blocks

and each block  is decomposed in the basis

Block Fourier Basis 
A block basis is constructed with the Fourier basis of :

The time support of each block Fourier vector  is  of size . The Fourier transform of  is

and

It is centered at  and has a slow asymptotic decay proportional to  Because of this poor frequency localization, even though a signal  is smooth, its decomposition in a block Fourier basis may include large high-frequency coefficients. This can also be interpreted as an effect of periodization.

Discrete Block Bases 
For all , we suppose that . Discrete block bases are built with discrete rectangular windows having supports on intervals :

.

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  for any . The proof is straightforward.

Theorem 2. 
Suppose that  is an orthogonal basis of  for any . The family

is a block orthonormal basis of .

A discrete block basis is constructed with discrete Fourier bases

The resulting block Fourier vectors  have sharp transitions at the window border, and thus are not well localized in frequency. As in the continuous case, the decomposition of smooth signals  may produce large-amplitude, high-frequency coefficients because of border effects.

Block Bases of Images 
General block bases of images are constructed by partitioning the plane  into rectangles  of arbitrary length  and width . Let  be an orthonormal basis of  and . We denote

.

The family  is an orthonormal basis of .

For discrete images,we define discrete windows that cover each rectangle

.

If  is an orthogonal basis of  for any , then

is a block basis of

References 
 St´ephane Mallat, A Wavelet Tour of Signal Processing, 3rd

Category:Signal processing