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Lifting scheme ：ウィキペディア英語版

Lifting scheme

The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform.
Actually it is worthwhile to merge these steps and design the wavelet filters ''while'' performing the wavelet transform.
This is then called the second generation wavelet transform.
The technique was introduced by Wim Sweldens.
The discrete wavelet transform applies several filters separately to the same signal.
In contrast to that, for the lifting scheme the signal is divided like a zipper.
Then a series of convolution-accumulate operations across the divided signals is applied.
==Basics==

The basic idea of lifting is the following:
If a pair of filters

(h,g)

is ''complementary'',
that is it allows for ''perfect reconstruction'',
then for every filter

s

the pair

(h',g)

with

h'(z) = h(z) + s(z^2)\cdot g(z)

allows for perfect reconstruction, too.
Of course, this is also true for every pair

(h,g')

of the form

g'(z) = g(z) + t(z^2)\cdot h(z)

.
The converse is also true:
If the filterbanks

(h,g)

and

(h',g)

allow for perfect reconstruction,
then there is a unique filter

s

with

h'(z) = h(z) + s(z^2)\cdot g(z)

.
Each such transform of the filterbank (or the respective operation in a wavelet transform) is called a lifting step.
A sequence of lifting steps consists of alternating lifts,
that is, once the lowpass is fixed and the highpass is changed and in the next step the highpass is fixed and the lowpass is changed.
Successive steps of the same direction can be merged.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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