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Arti Sharma, Prof. Preety D Swami / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 4, Jul-Aug 2013, pp.2055-2062
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Redundant Wavelet Transform Based Image Super Resolution
Arti Sharma, Prof. Preety D Swami
Department of Electronics &Telecommunication Samrat Ashok Technological Institute Vidisha
Department of Electronics & Instrumentation Samrat Ashok Technological Institute Vidisha
Abstract
The process of Super Resolution (SR)
aims at extracting a high resolution image from
low resolution image. The proposed technique
uses Redundant Wavelet Transform to enhance
the resolution of an image using a single low
resolution image. The proposed method
decomposes the input image into different
subbands. Then all subbands are interpolated.
Combining all the interpolated subbands using
Inverse Redundant Wavelet Transform provides
the proposed super resolution image. The
algorithm is tested with various wavelet types and
their performance is compared. The proposed
technique has been tested on Lena, Elaine,
Pepper, and Baboon images. The proposed
method gives higher quantitative peak signal-to-
noise ratio (PSNR) and visual results in
comparison to other conventional and state-of-art
image Super resolution techniques.
Index Term — Image Super Resolution,
Interpolation, Discrete Wavelet Transform,
Redundant Wavelet Transform.
I. INTRODUCTION
The aim of Super Resolution is to overcome
the limitation of the image acquisition device or ill
posed acquisition condition [1]. A Super Resolution
image is useful for better classification of remote
sensing image or to assist radiologist for making
diagnosis based on a medical imagery [2].
The most direct approach of obtaining
higher-resolution images is to improve the image
acquisition device (e.g., digital camera) by reducing
the pixel size on the sensor (e.g., charge-coupled
device). However, there is a problem in reducing the
sensor’s pixel size. When the sensor’s pixel size
becomes too small the captured image quality will be
degraded [3]. It produces shot noise that degrades the
image quality. Another approach for increasing the
super resolution is to increase the chip size, which in
turn increase the capacitance. Since large capacitance
makes it difficult to speed up charge transfer rate this
approach is not effective [4]. A new approach for
increasing the resolution which overcomes all these
problems is known as Super Resolution. The Super
Resolution image processing has grown very rapidly
after it was first researched by Tsai and Huang [5] in
1984. They applied Discrete Fourier Transform for
Super Resolution. The drawback of this method is
that it is insufficient to handle the real-world
applications. Many researchers have mentioned the
use of wavelet transform for addressing a Super
Resolution problem to recover the detailed
information (usually the high-frequency information)
that is lost or degraded during the image acquisition
process [6,7]. The drawback of this method is that
they applied multilevel wavelet transform. Ur and
Gross introduced Super Resolution through
interpolation [8]. The interpolation based Super
Resolution approach constructs a high resolution
image by projecting all the acquired low-resolution
images to the reference image. The information
available from each image is then fused together
because each low resolution image provides an
amount of additional information about the image
and finally deblurs the image [9]. The drawback of
this method is that the interpolation algorithm cannot
do super resolution of single image since it cannot
produce those high-frequency components that were
lost during the image acquisition process. G.
Anbarjafari and H. Demirel proposed a new
technique for image Super Resolution by combining
both the wavelet transform and interpolation. This
technique reduces all the drawbacks of above
mentioned techniques. However, applying
interpolation in high frequency sub-bands introduces
aliasing effects [10, 11].
The proposed technique also combines the
wavelet transform and interpolation. This method
uses redundant wavelet transform instead of discrete
wavelet transform. The proposed method is tested
with different types of wavelets & interpolation
methods & results are compared.
Rest of the paper is organized as follows:
Section 2 presents general description of discrete
wavelet transform and redundant wavelet transform.
Section 3 describes the interpolation and the various
interpolation methods available. Section 4 introduces
a discrete wavelet transform based DASR technique
for super resolution. Sections 5 elaborates the
proposed technique. Section 6 contains the
experimental results and comparison. At last, Section
7 concludes the paper.
II. WAVELET TRANFORM
Wavelet is a mathematical tool for
hierarchical decomposition of a signal. It allows a
function to be described in terms of coarse overall
shape along with details that range from broad to
narrow [12]. Regardless of whether the function of
interest in an image is a curve, or a surface, wavelet

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offers an elegant technique for representing the levels
of detail in the function.
2.1 Discrete Wavelet Transform
The Discrete wavelet transform (DWT)
analyzes the signal at different frequency bands with
different resolutions by decomposing the signal into
coarse approximation and detail information. DWT
adopts two sets of function called scaling function
and wavelet functions which are associated with low
pass and high pass filtering. In other words,
decomposition of the image into different frequency
bands is simply obtained by successive high pass and
low pass filtering and downsampling [13] of the
image as shown in Fig.1
Fig.1 Discrete Wavelet Transform
2.2 Redundant Wavelet Transform
The DWT remove the redundant coefficients
which are not necessary to perfectly reconstruct the
signal. This makes wavelet compression algorithms
more computationally efficient. However, in many
image processing applications the redundant wavelet
coefficient are useful. This work interpolates missing
pixels of an image in the wavelet domain based on
the values of the surrounding pixels. As much as
possible information is needed to accurately
interpolate the missing pixels. Decimation removes
potentially valuable information. In cases like this it
is beneficial to remove the decimators [14]. This is
known as the Redundant Wavelet Transform (RWT)
as shown in Fig. 2.
Fig. 2 Redundant Wavelet Transform
III. INTERPOLATION
Interpolation is the process of estimating the
values of a continuous function from discrete
samples. Interpolation based image processing
applications include image magnification or
reduction, sub pixel image registration, correction of
spatial distortions, image decompression and many
more. Many image interpolation techniques are
available amongst which nearest neighbor, bilinear
and bicubic are the most common [9].
 Nearest Neighbor Interpolation
The easiest interpolation from a
computational stand point is nearest neighbor. The
nearest neighbor algorithm selects the value of
nearest point and does not consider the values of
neighboring points at all, yielding a piece-wise
constant interpolation. This technique is also known
as pixel replication.
 Bilinear Interpolation
Bilinear interpolation considers the closest
2x2 neighborhood of known pixel values surrounding
the unknown pixel and calculates weighted average
of these 4 pixels to arrive at its final interpolated
value. This results in much smoother images than
nearest neighbor interpolation method.
 Bicubic Interpolation
Bicubic goes one step beyond bilinear
interpolation by considering the closest 4x4
neighborhood of known pixels (i.e., total of 16
pixels). Since these pixels are at various distances
from the unknown pixel and closer pixels are given a
higher weighting in the calculation, bicubic
interpolation produces sharper images than the
previous two methods.
IV. DASR BASED IMAGE SUPER
RESOLUTION (DASR)
The main loss of image after doing super
resolution by applying interpolation is on its high
frequency components, which results in smoothing of
the image Hence, for increasing the quality of
interpolated image, edges should be preserved
essentially. G. Anbarjafari and H. Demirel proposed
a method [10] that preserves the high frequency
components of the image. In this technique, Discrete
Wavelet Transform (DWT) separates the image into
different subband images represented by the LL (low-
low subband), LH (low-high subband), HL (high-low
subband) and HH (high-high subband). High
frequency subband contains the high frequency
components of image. All subbands are interpolated
by interpolation factor of 2. In order to extract the
high frequency component from input low resolution
image, difference between the input image and low
frequency subband image is added to all high
frequency subbands. Input low resolution image and
all estimated high frequency subbands are
interpolated by the interpolation factor 2. The
reconstruction of super resolution image is done by
applying Inverse Discrete Wavelet Transform
(IDWT) to the interpolated subbands. The block
diagram of DASR method is shown in Fig.3
Input
image
i
a
L
L
H
L H
H
Lowpassfilter
Highpassfilter
Lowpassfilter
Lowpassfilter
Highpassfilter
Highpassfilter
LL
LH
HL
HH

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V. PROPOSED SUPER RESOLUTION
METHOD
The proposed method is carried out in 4
steps as shown in the block diagram of Fig.4.
Step 1: In the first step, an input low resolution
image is generated from original high resolution
image through Gaussian down sample function. In
this function firstly the original high resolution image
is passed through a Gaussian low pass filter, which
passes only low frequency component of image. The
image is then down sampled row and column wise by
a factor of 2. This converts the original high
resolution image of size 512×512 into low resolution
image of size 128× 128.
Step 2: In the second step, Redundant Wavelet
Transform (RWT) is applied to the input low
resolution image. The input low resolution image is
decomposed through the RWT in four sub bands
represented by LL (low-low), LH (low-high), HL
(high-low) and HH (high-high).
Step 3: This step is the interpolation step. This
technique uses bicubic interpolation which gives
sharper image as compared to other interpolation
methods. All subbands are interpolated by
interpolation factor of 4 and then we achieve
interpolated subbands of size 512× 512.
Step 4: The final resolution enhanced image is
generated by employing the Inverse Redundant
Wavelet Transform (IRWT) to the interpolated
subbands. In this technique the required interpolation
method is same for all subbands.
I. EXPERIMENTAL RESULTS AND
COMPARISION
The performance of the proposed super resolution
algorithm is tested on four gray scale images show in
Fig. 5, each having a size of 512×512. These test
images are first converted to their low resolution
version through Gaussian down sample function and
then Redundant Wavelet Transform is applied to the
low resolution image. The performance of the
proposed method is compared by applying wavelets
namely Haar, Db8, Db9/7, Sym4 and Sym8. In the
next step, bicubic interpolation by factor 4 is applied
to LL, LH, HL, HH subbands. Table 1 shows the
experimental results in terms of Root Mean Square
Error (RMSE) and Peak Signal to Noise Ratio (PSNR
=20log10255/RMSE).
Low resolution
image (mxn) DWT
LH
HL
LL
HHInterpolated
LL
Bicubic
Interpolation
With factor2
Bicubic
Interpolation
With factor2
Bicubic
Interpolation
With factor2
Difference
image
Bicubic
Interpolation
With factor2
Estimated
LH
Estimated
LH
Estimated
LH
IDWT
HighResolution
Image
(αmXαn)
Bicubic interpolationwithfactor α/2Bicubic interpolationwithfactor α/2
Fig.3. DASR Method of image super resolution

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Table 2 shows the comparison of super
resolution of the proposed method with the
interpolation method and DASR method for
different wavelets in terms of RMSE and
PSNR. It is clear from the comparative table
that the PSNR of super resolution image of
proposed method is better as compared to
interpolation method and DASR method.
Qualitative results of Super resolution of
Lena, Elaine, Baboon and Peppers images by
different methods (Interpolation method,
DASR method and proposed method) are
shown in the Fig. 6, Fig. 7, Fig. 8 and Fig. 9
respectively.
Fig. 5 Test image (a) Lena, (b) Elaine, (c) Baboon, (d) Peppers.
(a) (b)
(c) (d)
Input Low
resolution
image
(128X128)
LL(128X128)
)
LH (128X128)
HL(128X128)
HH(128X128)
RWT
Interpolated
LL (512X512)
Interpolated
LH (512X512)
Interpolated
HL (512X512)
Interpolated
HH (512X512)
Interpolation with
factor 4
Interpolation with
factor 4
Super
Resolution
Image
(512X512)
Interpolation with
factor 4
IRWT
Interpolation with
factor 4
Fig. 4 Proposed Method

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Table 1. Numerical results on the test images with different wavelet types.
Wavelet type Lena Elaine Baboon Peppers
RMSE PSNR RMSE PSNR RMSE PSNR RMSE PSNR
Haar 5.66 33.08 5.94 32.65 8.17 29.33 5.84 32.80
Db8 7.23 30.94 7.84 30.24 9.29 28.77 7.32 30.83
Db9/7 8.29 29.75 9.33 28.73 9.84 28.27 8.49 29.55
Sym4 8.29 29.74 9.34 28.72 9.84 28.27 8.51 29.53
Sym8 9.30 28.76 10.07 28.07 10.18 27.97 9.45 28.62
Table 2. Comparison between the Interpolation super resolution method [9], DASR method [10] and the
proposed method in terms of RMSE and PSNR.
Methods
Images
Interpolation method DASR method Proposed method
Interpolation
type RMSE PSNR
Wavelet
type RMSE PSNR
Wavelet
Type RMSE PSNR
Lena Nearest
neighbor
Bilinear
Bicubic
11.33 27.04
10.87 27.40
10.60 27.62
Haar
Db8
Db9/7
Sym4
Sym8
15.01 24.60
14.69 24.79
15.06 24.57
15.06 24.57
14.99 24.61
Haar
Db8
Db9/7
Sym4
Sym8
5.66 33.08
7.23 30.94
8.29 29.75
8.29 29.74
9.30 28.76
Elaine Nearest
neighbor
Bilinear
Bicubic
10.34 27.84
9.67 28.42
9.51 28.56
Haar
Db8
Db9/7
Sym4
Sym8
15.95 24.10
15.74 24.19
15.92 24.09
15.92 24.09
15.95 24.10
Haar
Db8
Db9/7
Sym4
Sym8
5.94 32.65
7.84 30.24
9.33 28.73
9.34 28.72
10.07 28.07
Baboon Nearest
neighbor
Bilinear
Bicubic
21.06 21.66
20.89 21.73
20.72 21.80
Haar
Db8
Db9/7
Sym4
Sym8
15.59 24.27
15.47 24.34
15.61 24.26
15.61 24.26
15.57 24.28
Haar
Db8
Db9/7
Sym4
Sym8
8.17 29.33
9.29 28.77
9.84 28.27
9.84 28.27
10.18 27.97
Peppers
Nearest
neighbor
Bilinear
Bicubic
12.33 26.33
11.54 26.88
11.37 27.01
Haar
Db8
Db9/7
Sym4
Sym8
15.38 24.39
15.15 24.52
15.40 24.38
15.40 24.38
15.36 24.40
Haar
Db8
Db9/7
Sym4
Sym8
5.84 32.80
7.32 30.83
8.49 29.55
8.51 29.53
9.45 28.62

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(a) (b)
(c) (d
Fig. 6 (a) Lena low resolution image, [b-d] Super resolution methods (b) Interpolation (Bicubic), PSNR=27.62
dB (c) DASR (Db8 + Bicubic), PSNR =24.79 dB, (d) Proposed (Haar + Bicubic), PSNR= 33.08 dB.
(a) (b)
(c) (d)
Fig. 7 (a) Elaine low resolution image, [b-d] Super resolution methods (b) Interpolation (Bicubic)
PSNR= 28.56dB, (c) DASR (Db8 + Bicubic), PSNR = 24.19 dB, (d) Proposed method (Haar + Bicubic),
PSNR= 32.65 dB.

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(a) (b)
(c) (d)
Fig. 8 (a) Baboon low resolution image, [b-d] Super resolution method (b) Interpolation (Bicubic), PSNR=
21.80 dB, (c) DASR (Db8 + Bicubic), PSNR = 24.34dB, (d) Proposed (Haar + Bicubic), PSNR= 29.33dB.
(a) (b)
(c) (d)
Fig. 9 (a) Peppers low resolution image, [b-d] super resolution methods (b) Interpolation (Bicubic), PSNR=
27.01 dB, (c) DASR, (Db8 + Bicubic), PSNR= 24.52 dB, (d) Proposed (Haar + Bicubic), PSNR= 32.80 dB.

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II. CONCLUSION
This paper proposes a new super resolution
technique that uses RWT instead of DWT. RWT
reduces aliasing effect as it does not down sample the
image. This technique uses RWT to decompose an
input image to different subband images. The
subbands are then interpolated by factor 4.
Afterwards, these entire images are combined using
IRWT to generate super resolution image. The
experiment is repeated for various wavelets and
interpolation methods. The best result is obtained by
combining Haar wavelet with bicubic interpolation.
The proposed method has been tested on the well
known benchmark images, where their PSNR and
visual quality show the superiority of the proposed
technique over the conventional and state of art
image resolution enhancement techniques.
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