APPLICATIONS OF LIFTING SCHEME
General Concept of Lifting
(even j-1;odd j-1)=split(s j-1)
d j-1 =odd j-1 -P(even j-1)
even j-1 =even j-1 +U(odd j-1)
The lifting scheme allows for an
in-place transformation on a
signal: prediction replaces the
odd values and update replaces
using the prediction results.
Three Steps in lifting scheme
x[n] is split to two subsets, one even (s0[n]) and other odd (d0[n]). This splitting is called
lazy wavelet transform, and is used for the ease of reconstruction.
Now for a sparser approximation of one of x[n] subsets the next step with linear
combination of elements in one subsequence is used to predict the values of the other
subsequence. Here it is assumed that two subsequences produced in the splitting step are
correlated. If correlation present in x[n] is high, the predicted values will be closer to the
actual ones. Thus d[n] is predicted using samples in s[n] and replaces the samples in d[n]
using prediction error i.e. given by d[n] ← d[n] - P(s[n]). The predict step results in the
loss of some basic properties of the signal like mean value, which needs to be preserved.
The update step lifts the even sequence values using the liner combination of the
predicted odd sequence so that the basic properties of the original sequence is preserved.
Therefore the data in s[n] is updated using data in d[n] i.e. given by s[n] ← s[n] + U
Lossless compressive image used in
Heavily edited images.
A common characteristic of most images is that the neighboring pixels are
correlated, and therefore contain redundant information. The foremost task
then is to find less correlated representation of the image.
Spatial Redundancy or correlation between neighboring pixel values.
Spectral Redundancy or correlation between different color planes or spectral
Temporal Redundancy or correlation between adjacent frames in a sequence of
images (in video applications).
Image compression research aims at reducing the number of bits needed to
represent an image by removing the spatial and spectral redundancies as much
INTEGER FAST FOURIER TRANSFORM
Basic butterfly structure for N-point FFT using split-radix structure.
Lifting scheme that allows perfect reconstruction
Converting complex numbers into real lifting steps
Consider a complex number that is quantized to have a magnitude 1 then
its represented as
a=c+js were c and s are real numbers and c2 +s 2 = 1 and let aq
be quantised version of a such that
aq = cq + jsq where cq and sq are finite wordlength approximations of c and s.
hence the reciprocal of aq is
1 = cq + s q
aq [aq ]2 [aq ]2
In general, [aq ] is not one, although [ a]=1 . Instead, [aq ] -1 may not even
be a finite word length complex number, even though [aq ] is. This is the
reason why the conventional fixed-point arithmetic does not preserve the
y =a*x=(cx r-sx i) +j(cx i +sxr)
y=[1 j] c
where R =
The PR property can be preserved via the lifting scheme. R can be decomposed
into three lifting steps
-s = 1 (c-1)/s 1 0
Butterfly structure for
implementing a complex
Lifting structure for implementing a
complex multiplication and its
ADVANTAGES OF LIFTING SYSTEM in IFFT
number of real multiplications is reduced from four to three
it allows for quantization of the lifting coefficients and the quantization of the
result of each multiplication without destroying the PR property. To be
specific,instead of quantizing a directly, the lifting coefficients c and (c-1)/s are
quantized, and therefore, its inverse also consists of three lifting steps with the
same lifting coefficients but with opposite signs.
nonlinear operators can also be applied to the product at each lifting step such
as rounding or flooring operation. As an example Eight-Point Split-Radix IntFFT
is shown.W18 and W38 are implemented using proposed conversion. There are
24 butterflies with coefficients 1, -1, j and –j that do not require any
multiplication or addition. These 24 butterflies can be implemented
usingcomplex adders or 96 real adders. The rest of the computation is based on
the binary representation of the lifting coefficients + 1/(2) ½ and + ((2) ½ -1).
Lattice structure of eight-point IntFFT using split-radix
The lifting scheme is a simple method for designing
customized biorthogonal wavelets and offers several
Allows a faster implementation of the wavelet transform,
Saves storage by providing an in- place calculation of the
Simplifies determining the inverse wavelet transform,
Provides a natural way to introduce and think about
Swanirbhar Majumder, N. Loyalakpa Meitei, A. Dinamani Singh, Madhusudhan
Mishra,” Image Compression Using Lifting Wavelet Transform”
Wade Spires,”Lossless Image Compression Via the Lifting Scheme”
Soontorn Oraintara,Ying-Jui Chen,Truong Q. Nguyen,”Integer Fast Fourier
A.Alice Blessie, J. Nalini, S.C.Ramesh,”Image Compression Using Wavelet
Transform Based on the Lifting Scheme and its Implementation”
The method used here is very simple. There are 5 steps mainly. Firstly
the image is under gone lifting wavelet based DWT for the desired
number of levels as per the size of the image. This is followed by
undergoing zig-zag scan, to convert it to one dimensional format. Then
it is uniformly quantized and encoded using the run length encoding
(RLE). Finally the RLE encoded data is again encoded using Huffman
coding such that the Huffman dictionary has a length equal to the one
more than the number of quantization levels used. The decompression
method is just the reverse of the compression method .Here
compression is mainly achieved by removing spectral redundancy in the
DWT domain, and achieving some amount of loss of data due to
quantization. But as both the encoding methods used are lossless so no
data loss is undergone due to the encoding and decoding steps.
Moreover it has been seen that more amount of compression is achieved
if Huffman encoding is undergone after RLE. Thus they have been used
in this order.