Block transform


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, suppose that. Discrete block bases are built with discrete rectangular windows having supports on intervals :
Since dilations are not defined in a discrete framework,bases of intervals of varying sizes from a single basis cannot generally be derived. Thus, Theorem 2 supposes an orthonormal basis of for any can be constructed. The proof is:

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. The following can be denoted:
The family is an orthonormal basis of.
For discrete images, discrete windows that cover each rectangle can be defined
If is an orthogonal basis of for any, then
is a block basis of