5/3 Lifting Scheme Approach for Image Interpolation

IOSR Journal of Electronics and Communication Engineering (IOSR-JECE)
e-ISSN: 2278-2834,p- ISSN: 2278-8735.Volume 12, Issue 1, Ver. I (Jan.-Feb. 2017), PP 08-13
www.iosrjournals.org
DOI: 10.9790/2834-1201010813 www.iosrjournals.org 8 | Page
5/3 Lifting Scheme Approach for Image Interpolation
Palla Aakreeth Reddy1
, Pulgam Lokanadh Reddy 2
, Thota Vinay Kumar 3
Ace Engineering College, Hyderabad,T.S.
Abstract: In this paper we proposed fast and accurate interpolation and resizing of images using lifting scheme
approach. 5/3 lifting scheme is an accurate and computationally inexpensive interpolation technique for image
resizing. We compared the bilinear interpolation, Haar lifting scheme and 5/3 lifting scheme in this paper. The
lifting scheme algorithm is applied for image interpolation to resize the image. In case of reduction in size, the
image components are reduced and the reconstruction will be carried out to the original image. The
reconstruction results are better by using Mean Squared Error (MSE) and Peak Signal to Noise Ratio (PSNR)
with other techniques like bilinear interpolation and Haar lifting schemes. The interpolation and reconstruction
is executed in much less time with better MSE and PSNR as compared to Bilinear and Haar lifting schemes.
Keywords: Lifting scheme, 5/3 Lifting transform , DWT , IDWT , Image Resizing , Reconstruction.
I. Introduction
Wavelets provide accurate and excellent enough tool to approximate the functions, datasets and signals.
Mother wavelet acts as basic building block to approximate the complex signals and functions at reasonable
computational overhead. In other words, wavelets can approximate complex signals by using a small set of
coefficients without much computational complexity. This is possible because, most data sets have correlation
both frequency domain and in time (or space) domain. Among the family of xxxlet (wavelet, curvelet, ridgelet)
transforms, wavelet is the first one to preserve the correlation in both frequency domain and spatial domain.
Transformations in most of the frequency domain transforms do not preserve the original form of a signal like
image in spatial domain. One cannot make out the visible form of an image from the results of cosine, sine or
Discrete Fourier Transforms. However, the LPLP component of DWT of an image seems to be similar to the
original image and other components are the edges in different directions. Thus DWT may serve as an excellent
tool to simulate human vision phenomenon like continuous interpolation, because of the time-frequency
localization property of wavelets.
Lifting scheme was introduced by Sweldens (1996a; 1996b; 1996c). It is a new method to construct bi-
orthogonal wavelets (Sweldens, 1996b). The basic properties of the image are improved by adding new basis
functions (Daubechies and Sweldens, 1998). The lifting scheme consists of three main steps: SPLIT, which sub
samples the original data into odd and even sets; PREDICT, which finds the wavelet coefficients and UPDATE,
which updates the even set using the wavelet coefficients. Image resizing algorithms using DWT are used to
resize the image to any desired scale (Asamwar et al., 2010; 2009). In the proposed work, the Lifting scheme
divides the complete data set into two equal parts that is even and odd based on the spatial sequence. Hence, the
resizing process based on Lifting scheme is a simple task when the total pixels in the image are 2n
× 2n
in
number. Image with dimensions not equal to 2n
× 2n
is made equal to 2n
× 2n
size by padding zeros at required
row and column positions and then allowed to undergo Lifting Scheme procedure. The resized results can be
compared with traditional methods like bilinear and Haar lifting using MSE and PSNR values.
II. Discrete Wavelet Transform (Dwt)
When digital images are to be viewed or processed at multiple resolutions, the most popular
mathematical tool is discrete wavelet transform [1]. Wavelet transform is the first transform which has been
explored a lot in image and signal processing for its unique property of maintaining the spatial domain and
frequency domain contents. Wavelet series expansion maps a function of continuous variables into a sequence
of coefficients. Discrete wavelet transform decomposes the input image (2-D signal) into four different wavelet
coefficients at the first level. The process of obtaining these four coefficients (Vetterli and Kovacevic, 1995) is
presented in Fig.1. After dividing the data set into two parts that is even and odd, average and difference is
computed. Out of these two coefficients, one is called the average and the other is the difference. These two
coefficients are again processed to compute average-average, average-difference, difference-average and
difference-difference terms. Thus after completion of the whole process we have four coefficients. The
coefficient average-average is also known as LPLP coefficient (this is exactly visually similar to the original
image but 50% in size), average-difference as LPHP coefficient, difference-average as HPLP and

Fig. 1: QMF implementation of DWT
Fig. 2: Lifting scheme forward transform
Difference-difference as HPHP.These four coefficients are exactly 50% of the original image contain the
different information of the original image as given below [2].
1). LPLP: This coefficient contains the low frequency image content and is the reduced resolution
LPHP: It contains Horizontal information of the image
HPLP: It contains vertical information of the image
HPHP: It contains diagonal information of the Image
III. Lifting Scheme
It is a technique to design wavelets and perform the discrete wavelet transform. The forward lifting scheme
wavelet transform divides the data set being processed into an even half and an odd half [3].
3.1. Lifting scheme forward transform
Lifting scheme forward transform consists of three steps:
Split
Predict
Update
A. Split: In split step the data is divided into ODD and EVEN elements.
B. Predict step: The difference between odd and even data forms the odd elements of the next step wavelet
transformation. The predict step, where the odd value of next iteration is “predicted” from the even value of
present step is described by (1). Index ‘j’ represents iteration and ‘i’ represents element:
Oddj+1,j= oddj,i+ P(evenj,i) (1)
Fig. 3: Lifting scheme inverse transform

C. Update step: The update step replaces the even elements of the next step with average of the earlier step.
These results in a smoother input (even element) for the next step wavelet transform. The update step follows
the predict phase. So in calculating average the update phase must operate on the differences that are stored in
the odd elements:
Evenj+1,i=eveni,j+U(oddj+1,i) (2)
A simple lifting scheme forward transform is as shown in Fig. 2.
After dividing the complete data set into two parts that is even and odd, the processing is done as follows: For
the forward transform iteration j and element i, the new odd element j+1, i would be:
Oddj+1,i=oddj,j-evenj,i (3)
Even element of the next step is calculated as: Evenj+1,i=(evenj,i+oddj,i)/2 (4)
The original value of the odd j, i element has been replaced by the difference between this element and its even
predecessor. Simple algebra lets us recover the original value from Eq. 3:
Oddj,i=evenj,i+oddj+1,I (5)
Substituting this into the average that is Eq. 4
Evenj+1,i=(evenj,i+evenj,i+oddj+1,i)/2 (6)
Evenj+1, i=evenj,i+(oddj+1,i)/2 (7)
3.2.Lifting scheme inverse transform
As the name suggests the mirror process of forward lifting scheme transform. We recover the original data
sequence by going upwards as shown in Fig. 3. Additions are substituted for subtractions and subtractions for
additions. The merge step replaces the split step.
IV. Materials And Methods
4.1. Image resizing using lifting scheme
Image is a data set made of rows and columns. Monochrome images are two dimensional with one
intensity value associated with every pixel [4]. Being two dimensional, they are to be processed first row wise
and then column wise assuming that the image signal is orthogonal in both the directions. Here, we present
computation of one dimensional wavelet transform using lifting scheme and then extend it to the 2D images to
compute the Haar Wavelet transform. Consider the following pixel row intensity data array:
S = {44, 20, 30, 48, 49, 39, 40, 35}
Step 1 (split): Initially this data set is divided into two parts. First, third, fifth, seventh pixel are accommodated
in the first part known as odd part. Second, forth, sixth, eighth pixel are accommodated in the second part
known as even part that is:
Odd = 44, 30, 49, 40 and Even = 20, 48, 39, 35
Now this even and odd data parts are processed in the following stepwise manner to obtain average and
difference terms.
Step 2 (predict): The first difference is obtained by subtracting the first even element from first odd element.
Similar procedure is adopted for remaining all elements of odd and even vector sequentially, to result in even
j+1,i and here DIFF1 vector sequence as given below:
evenj+1,i = oddj+1,i – evenj,i and thus
DIFF1= {44,-18,10,5}.
Step 3 (update): The first average term is obtained by adding the first odd and first even element and by
dividing this sum by two. This process is continued for the complete data set of odd and even elements
sequentially and the resultant vector oddj+1,i and AVG1 is obtained as shown below:
Evenj+1,i = (evenj,i + oddj,i)/2 thus
AVG1 = {32, 39, 44, and 37.5}.
Thus initial eight elements present in a data array are reduced to four average and four difference terms.
Same procedure from Step 1-3 is repeated further, considering AVG1 vector as input data array for the next
level forward lifting scheme transformation. Thus the successive transformations can be taken till each
component reaches single coefficient. For image processing applications, this procedure of Step 1-3 is applied to
all the rows present in the image. The resulting matrix is subjected to the same procedure of 1-3 Steps on all
columns in the image. This results in Low Pass Low Pass (LPLP) approximation of the image and is the 50%
reduced version of the original image. Thus at each level, the LPLP (AVG component after row-wise and
column-wise computation of lifting scheme DWT) is considered as the interpolated version of the original
image. The difference components obtained at every stage are used for reconstruction of the original image. For
the next level lifting scheme forward transformation average component of earlier level acts as an input. At
every lifting scheme forward transformation, image size gets reduced to 1/2 of the input image size. Thus finally
individual the component (LPLP, LPHP, HPLP and HPHP) of an image gets reduced to single DWT coefficient
after successive transformations [5].

4.2. Reconstruction of the image using Inverse discrete wavelet transforms
The original data can be reconstructed with the help of inverse lifting scheme. Four pixels obtained in
forward lifting scheme transform now acts as an input to reconstruct average vector in, inverse lifting scheme.
Proceeding further in this manner as explained in materials and methods till all average vectors (obtained in
various levels of forward lifting scheme transformation) gets processed, yields all the recovered elements in the
last step. These reconstructed elements correspond to the respective elements in the original array. Similar
procedure is adopted in row wise and column wise manner for the image. Image resizing with the algorithm as
presented in this paper is not possible if the image is not available in the size 2n
× 2n
. The size of the original
image with rows ‘r’ and columns ‘c’ is converted into the next (2n
× 2n
) size by zero padding at the ends of all
rows and columns i.e., adding (2n
-r) zeros in all the columns and (2n
-c) zeros in all the rows at the ends.
V. 5/3 Lifting Transform
A wavelet 5/3 lifting transform has 3 and 5 taps in the high and low pass analysis filters respectively.
5/3 lifting transform is also known as Le Gall 5/3 transform. Le Gall 5/3 wavelet is the shortest symmetrical bi-
orthogonal wavelet with two vanishing moments. It is the simplest way to decompose the image into one high
frequency component and one low frequency component. The shortest bi orthogonal scaling and wavelet
function with two regularity factors at synthesis and analysis denoted (2,2) is attained with Le Gall 5/3 synthesis
function. Interpolation of data means inserting additional data points into the sequence to increase the sampling
rate.
Let assume the pixels x1, x2, x3, x4, x5, x6 ... and in general we will predict even pixels and update the odd pixels.
The formulas used to predict and update even and odd pixels using 5/3 lifting are
Pn=x4-((x3+x5)/2) (8)
Where Pn is the prediction of even ‘n’ pixels
Un=x3+((p2+p4)/4) (9)
Where Un is updation of odd pixels
5.1.5/3 Lifting Algorithm
1) Read the original image
2) For each scaling factor read the reference image and perform 5/3 lifting to predict and update the image along
horizontal and vertical directions.
3) For each down scaling level, down sample the update coefficients by a factor of two and perform 5/3 lifting
as above step.
4) For each up scaling perform nearest neighbor interpolation on pred samples and apply inverse 5/3 lifting.
VI. Results And Discussion
From the results we can conclude that 5/3 lifting scheme will give more efficient results compared to
Bilinear and Haar lifting schemes. Fig.5 and Fig.7 are the comparisons of Mean Square Error (MSE) and Peak
Signal to Noise Ratio (PSNR). Fig.4 and Fig.6 are the resizing images using 5/3 lifting with different resize
factors for Lena and Zelda. The proposed algorithm is mainly focused on MSE and PSNR values. If we compare
the three algorithm the 5/3 lifting scheme will give the better results compared to bilinear and Haar lifting
schemes.From the table 1 we compared the MSE and PSNR values for Lena image with different resizing
factors. From the table 2 we compared the MSE and PSNR values for Zelda image. From the two tables we can
observe that the 5/3 lifting scheme will gives the better PSNR values and low MSE values than the Haar lifting
scheme and bilinear interpolation. If we increase the resize factor for an image the PSNR value increases and the
MSE value decreases. If we consider large resize factor the time taken by the image to resize increases and the
efficiency of the algorithm may decreases. The lifting scheme allows a faster implementation of the wavelet
transform. Traditionally, the fast wavelet transform is calculated with a two-band sub band transform scheme, in
Fig. 1. In each step the signal is split into a high pass and low pass band and then sub sampled. Recursion occurs
on the low pass band. The lifting scheme makes optimal use of similarities between the high and low pass filters
to speed up the calculation.
VII. Conclusion
In this particular paper we used 2n
× 2n
images for resizing. The proposed algorithm will resize the
image if the image is of the size 2n
× 2n
and zero padding enables application of the scheme on all sizes of
images but with additional computational overhead. If we use the large images for resizing it will take more
time to resize . So we can concentrate on these large images as a future work to resize the images and to increase
the more efficiency.

Table1: Comparision of MSE and PSNR values for Lena Image
Bilinear
Interpolation
Haar Lifting 5/3 Lifting
Image Size Resize
Factor
MSE PSNR MSE PSNR MSE PSNR
Lena 128× 128 0.25 866.8743 18.7512 492.0217 21.2110 332.8713 22.9080
Lena 256×256 0.5 520.6248 20.9656 164.7891 25.9615 87.7668 28.6975
Lena 1024×1024 2 188.9562 25.3672 82.6570 28.9580 38.8632 32.2354
Lena 2048×2048 4 159.0445 26.1156 124.2400 27.1882 77.5673 29.2340
Table2: Comparision of MSE and PSNR values for Zelda Image
Bilinear
Interpolation
Haar Lifting 5/3 Lifting
Image Size Resize
Factor
MSE PSNR MSE PSNR MSE PSNR
Zelda 128× 128 0.25 409.8525 22.0045 176.6397 25.6599 96.9163 28.2668
Zelda 256×256 0.5 210.2132 24.9042 46.1388 31.4901 21.8254 34.7412
Zelda 1024×1024 2 87.2603 28.7226 26.9042 33.8326 12.7555 37.0738
Zelda 2048×2048 4 74.9172 29.3850 41.3637 31.9646 25.5541 34.0562
5.(a)
Fig.4: 5/3 Lifting based Image Resizing with resizing factors of (a) : 0.25 , (b) :0.5,(c) : 2, (d) : 4, For Lena.
5. (b)
Fig. 5: (a) Comparison of MSE and (b) Comparison of PSNR for Lena Image.

Fig.6: 5/3 Lifting based Image Resizing with resizing factors of (a) : 0.25 , (b) :0.5,(c) : 2, (d) : 4 for Zelda
7. (a)
7. (b)
Fig. 7: (a) Comparison of MSE and (b) Comparison of PSNR for Zelda Image.
References
[1]. Asamwar, R.S., K.M. Bhurchandi and A.S. Gandhi, 2009. Piecewise lifting scheme based DWT to model human vision
interpolation phenomenon. Proceeding of the IAENG International Conference on Signal and Image Engineering, July 1-3, IAENG,
London, UK., pp: 822-827.http://www.iaeng.org/publication/WCE2009/WCE 2009_pp822-827.pdf
[2]. Asamwar, R.S., K.M. Bhurchandi and A.S. Gandhi, 2010. Successive Image Interpolation Using Lifting Scheme Approach. Journal
of Computer Science 6 (9): 961-970
[3]. Asamwar, R.S., K.M. Bhurchandi and A.S. Gandhi, 2010.Interpolation of images using discrete wavelet transform to simulate
image resizing as in human vision. Int. J. Autom. Comput., 7: 9-16. DOI: 10.1007/S116333-010-0009-7
[4]. Daubechies, I. and W. Sweldens, 1998. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appli., 49: 247-269. DOI:
10.1.1.106.8124
[5]. Latha, Y.M., B.C. Jinaga and V.S.K. Reddy, 2007. A precise content based image retrieval: Lifting scheme. Int. J. Comput. Sci.
Network Secur. 7: 38-45.

5/3 Lifting Scheme Approach for Image Interpolation

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