1. science
2. mathematics
3. statistics

International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
Noise Reduction Based on Multilevel DTCWT
Using Bootstrap Method
T.Eswar Reddy and K.M.Hemambaran

Abstract- The main aim of this project is to reduce a
noise introduced by new method which is known as
bootstrap method. Image enhancement technique is
clearly based on the random spray sampling technique.
The output image of this exhibits noise with some
unknown statistical distribution. Non enhanced image is
consider to be free of noise or effected by certain levels of
noise. By taking an advantage of highest sensitivity of
HVS to change in brightness. The whole analysis is done
on luma channel of both noise free and added noisy
images. For giving an importance to directional content in
human vision perception, the analysis is performed
through the DTCWT. With the replacement of DWT we
are using DTCWT for allowing data directionality. Every
level of transform the standard deviation of noise free
image is computed across the six orientations of the dtcwt.
The obtained results show a map of directional structures
present in the noise free image. According to data
directionality in the dtcwt the shrunk coefficients and also
the coefficients from the non enhanced image are then
mixed. Finally by doing the inverse wavelet transform we
are obtaining the resultant image as a noise reduced
version of the enhanced image.
Keywords: Bootstrap Method, Dual tree complex
wavelet transform (DTWCT), Image enhancement, noise
reduction, random sprays.
I. INTRODUCTION
Image enhancement is a very important application in
digital world, as per consumer status it has never stopped
evolving. A novel multi resolution denoising method is used
to find a specific image quality problem this can be avoided
by using image enhancement algorithm based on random
spray sampling.
Random sprays are implemented in spatial domain to
collect a two dimensional points and used to sample an image
support in place of other techniques, and have been
previously used in works such as Provenzi [2], [3] and Kolas
[4]. It can be implemented when image is Human Visual
System (HVS).
The existing Denoising algorithm is gives good noise
reduction via coefficient shrinkage. And also the noise free
image is introduced in the form of partial reference images.
Having a reference allows the shrinkage process to be
Manuscript received Nov, 2014.
T.Eswar Reddy, ECE, SITAMS/JNTUA, Chittoor, India.
K.M.Hemambaran, ECE, SITAMS/JNTUA, Chittoor, India
ISSN: 2278 – 1323
data-driven and strong advantage of this method is to work
based on Dual-tree Complex Wavelet Transform.
.
Finally, Proposing a Bootstrap method to prevent data
loss in noise images compare to other methods i.e. existing
denoised method and RSR(Random Spray Retinex) and
RACE (combination of RSR and ACE(Automatic Color
Equalization)) and main advantage in image analysis is that
no assumptions have to be previously stated about the
distribution of elements within the population. It is an
empirical nonparametric technique commonly used to obtain
several measures of a given statistic when the underlying
statistical model is not known. Fig 1 shows traditional method
for recovering noise free image from noisy image.
Fig 1: Traditional Denoising Method
The paper is organized as follows. Section II contains
brief overview of Dual-tree Complex Wavelet Transform.
After that, Section III and IV contain the proposed Denoising
method and The Bootstrap Method, experimental results is
implemented in Section V and finally, Section VI gives the
conclusion.
II. DUAL -TREE COMPLEX WAVELET TRANSFORM
Discrete wavelet transform is introduced for application
of digital image processing like image denoising to pattern
recognition, image encoding and more. Being a complete and
invertible transform of 2D data the Discrete Wavelet
Transform gives rise to „checker board‟ pattern phenomenon.
To avoid these two problems we use steerable filters
introduced by Adelson and Freeman [19]. Which can be used
to decompose an image into a Steerable Pyramid, by means of
the Steerable Pyramid Transform (SPT) [8]? While, the SPT
is an over-complete representation of data , data orientation
will be shift invariant And it also having drawbacks that are
perfect reconstruction is not possible and computational
efficiency can be a concern.
After SPT, thus a further development has been
accomplished the Complex Wavelet Transform in order to
compute the energy response by using the Hilbert pair of
filters. To recovering the whole Fourier spectrum similar to
SPT this transform needs to over complete by a factor of 4
means there are 3 complex coefficients for each real one. The
CWT is also efficient and it can be computed through
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International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
separable filters, even though it still lacks the perfect
reconstruction property.
Later the Dual tree complex wavelet transform is
proposed by Kingsbury and referred as work by Selesnick
[21] for a comprehensive coverage on the DTCWT. The 2D
DTCWT is implemented using the scaling and wavelet
coefficients as shown in the below equations,
=
,
=
=
,
=
,
=
,
,
,
,
=
=
=
(1)
,
,
the chroma channel, and the whole operation is done only on
the luma channel by using this wavelet space. Here we are
using only the luminance channel because it does not provide
any visible color.
At final, we have to assume that the input is consider to
be free of noise or affected by certain levels of noise. If we
assume like this the original image contains some amount of
information for reduction of noise successfully. For using the
reference input image we have to reduce the noise in a noise
affected image perfectly by using DTCWT. The algorithm for
the proposed method is
ALOGRITHM: Algorithm for Proposed Noise Reduction
Method
(2)
The low pass and high pass wavelet filters h and g is as
shown below
(
repeat
Here j is the decomposition level.
is iteration dependent
When combining DTCWT coefficients the bases give rise
to two sets of real and 2D oriented wavelets
for j=1
do
for k=1
do
end for
for k=1
do
+
Rank of
if
then
Shrunk coefficients from
else
Coefficients from
end if
end for
end for
Fig. 2. Quasi-Hilbert pairs wavelets using in dual-tree complex wavelet
transform. Even part shows on top and odd part on bottom and also each
Pair is shown in a column
The most important characteristic of wavelets is they are
approximately Hilbert pairs and the coefficients deriving
from one tree is imaginary and the other tree is real and also
we obtain the desired 2D DTCWT.
)
until ssim ( ,
=concat (
ycbcr2rgb (
Inverse DTCWT
,
,
)
III PROPOSED NOISE REDUCTION METHOD
Main idea of this method can be simplified as follows:
user information conveys to the Human Visual System.
According to data directionality, the proposed method
using DTCWT produces wavelet coefficients by using those
coefficients we have to shrink the coefficients from non
enhanced image. The propose algorithm is as shown below.
HVS has been more sensitive to changes occurred in the
luma channel than the chroma channel [35]. And hence the
proposed method first converts the original image into
YCbCr. image and then the luma channel is separated from
ISSN: 2278 – 1323
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International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
a normalized map of directionally sensitive weights for a
given level j can be obtained as
Where the choice of γ depends on j as explained later on.
A shrunk version of the enhanced image‟s coefficients,
according to data directionality, is then computed as
+
+
Since the main interest is retaining directional information,
we obtain a rank for each of the non-enhanced coefficients as
,
(b)
where ord is the function that returns the rank according to
natural ordering. The output coefficients are then computed as
follows
(c)
Fig.3. Proposed method flowchart. (a) Luma channels for both non enhanced
and the enhanced images that are transformed using DTCWT and the
obtained coefficients are enlarged. By using inverse DTCWT the output
coefficients are transformed into the output image‟s luma channel (b) and (c)
Computation indicated by the box in Fig. 3(a) is performed per level of the
decomposition. (b) Directional energy map is first computed as the standard
deviation of sum-of-squares of the coefficients. A weight map is then
obtained by using the Bootstrap Method. The even coefficients of the
enhanced image are also ranked according to their magnitude. (c) Weight
map is used to scale the coefficients of the enhanced image. The resulting
scaled coefficients and the coefficients from the non enhanced image are
mixed according to the ranking. The process in (c) is illustrated for even
coefficients only, but it is repeated identically for odd coefficients.
A. Wavelet Coefficients Shrinkage
Assuming level j of the wavelet pyramid, one can compute
the energy for each direction of the non-enhanced image k
ε{1, 2, . . . , 6} as the sum of squares of the real coefficients
and the complex ones
Coefficients associated with non-directional data will have
similar energy in all directions. On the other hand, directional
data will give rise to high energy in one or two directions,
according to its orientation. The standard deviation of energy
across the six directions k = 1, 2. . . 6 is hence computed as a
measure of directionality.
=
Since the input coefficients are not normalized, it naturally
follows that the standard deviation is also non-normalized.
The Bootstrap Method is thus applied to normalize data range
and also for estimating standard errors. Such function is
sigmoid-like and it has been used to model the cones
responses of many species. The equation is as follows:
BM(x,
=
Where x is the quantity to be compressed, γ a real-valued
exponent and μ the data expected value or its estimate. Hence,
ISSN: 2278 – 1323
where ord is the function that returns the index of a
coefficient in
=1,2,...,6 when the set is sorted in a
descending order. The meaning of the whole sequence can be
roughly expressed as follows: where the enhanced image
shows directional content, shrink the two most significant
coefficients and replace the four less significant ones with
those from the non enhanced image.
The reason why only the two most significant coefficients
are taken from the shrunk ones of the enhanced image is to be
found in the nature of “directional content”. For an content of
an image to be directional, the responses across the six
orientations of the DTCWT need to be highly skewed. In
particular, any data orientation can be represented by a strong
response on two adjacent orientations, while the remaining
coefficients will be near zero. This will make it so that the two
significant coefficients are carried over almost un-shrunk. To
help for understanding the energies of the decompositions is
as shown in equation 9.
B. Parameter Tuning
When dealing with functions with free parameters, a
fundamental problem is finding optimum parameters values.
While this can often be attempted with optimization
techniques, such methods proved unfeasible in the case. To at
least provide a reasonable default value for γj and J (the
parameter of the Bootstrap method and the depth of the
complex wavelet decomposition, respectively) tests were
performed on three images from the USC-SIPI Image
Database [37].We are taking Lenna image, Splash image and
Girl image for showing the experimental results. In different
rounds, Gaussian, Poissonian and Speckle noise was added to
the luma channel of said images and the proposed noise
reduction method was run with a J = 2 and values for γj
varying from − 5 to 10 in unit steps for the two levels of the
decomposition. This allowed us to determine that values of
γ1, γ2 and γ3 equal to 1 represent a reasonable choice,
although non-optimal for all inputs. It should be noticed that
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International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
by doing so the Michaelis-Menten function is reduced to the
Naka–Rushton formulation [38]. Since J proved to be
drastically more dependent on the input image than γj, it was
impossible to determine a single optimum value. Therefore, J
was left as the only user set parameter of the method, with
reasonable bounds of Jmin = 1, Jmax = 2
IV PROPOSED METHOD
The Bootstrap Method:
The advantage of using bootstrap method with DTCWT
for analyzing the image is, there is no assumption on previous
statements about distribution of pixels within the original
image. This bootstrap is commonly used for which is
generally used for obtaining several measurements like mean
and median values. This is used to collect number of data
pixels in an image and then resample those collecting data
pixels. The proposed data set is formed by the pixels of the
original image is resample. Hence the several bootstrap data
sets are generated for a previously number of times and also
data collection is explained. The coefficients that are having
in the bootstrap data sets are generally used to estimate mean
and median values and also standard errors and unknown
values. The noise is to be determined for a typical number of
collected data sets and the bootstrapping value ranges from up
to 300.The bootstrap standard error (SEBn) represents the
standard deviation of replication of the bootstrap method
SEBn=
(18)
Where b represents the measured parameters that is mean
and median and so on. B represents the bootstrap number that
means the number of random data sets and n represents the
sub sampling used in this method and
(19)
The bootstrap coefficient of variation (CVBn) is estimated as
follows
CVBn =
(20)
Generally the bootstrap method is obtained for represents
of this sampling areas and here we assume that the area of the
whole image occupied is consider to be an object for
analyzing purpose and it contains the total pixels of original
image and hence the sampling area is considering as equal
value of sizes. Every bootstrap data set is formed for
measuring the sampling area of original image and the
calculation of unknown statistical parameters that is mean and
median etc. This will results that a large number of
estimations of the unknown statistics parameters i.e. one for
every bootstrap data range. It is used for determining the
coefficient of variation (CVBn) and standard error (SEBn) for
the particular sampling area is as shown in the below Figure 1.
The minimum value for coefficient of variation or the
standard error is defined as wit increasing the size of the
sampling area and it is possible to determine the
representative sampling area size. The advantage of the
bootstrap method is its simplicity. It is straight forward way
for estimating the standard errors of complex coefficients.
And also it is an appropriate way for controlling and checking
stability of the results. The general applications of bootstrap
method are data missing problems and censored data.
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Estimation of Noise Statistics by Bootstrapping:
Bootstrapping provides a practical procedure for estimating
the parameters such as the mean and standard deviation etc.
Depending on resampling process the parameters can be
measured from an approximated distribution. This
distribution is obtained from the available bootstrap data set.
From the general Gaussian distribution we have to assume
that a set of observations and it is implemented by using a
number of resample from the bootstrap data that are all equal
size. Every resample is obtained by sampling one by one witjh
replacement from the original image pixel values. Another
advantage of bootstrapping is it provides easy procedure for
estimating the noise pixels in an enhanced image from
complex transform coefficients of the observed bootstrap
resampling pixel values. This bootstrap method is useful
while satisfying the below two that is (i) the theoretical
distribution of one statistic is unknown (ii) for statistical noise
the size of the sample is not enough.
Here we are using bootstrap method for estimating the
noise intensity level
according to the noise corrupted
pixels. We are taking the noise pixel values while sample the
image at the first time the noise intensity level
is only one
parameter of that image. Therefore the value of
is to be
estimate from the resampling original image. Our main idea of
implementing the bootstrap method is more number of
samples from the same pixels that can be efficiently replaced
by resampling one that is by doing first sampling. By using the
mathematical notations and equations let be the mean based
on one random bootstrapping sample of size
m
. The estimated noise
intensity level
is obtained by selecting the median value
of
based on Z different bootstrapping samples
where each sample is of size m. The following formula can be
used to calculate different
and based on calculated
the
can be obtained.
Where
is the absolute value of
element in
bootstrapping sample pixel. The estimation of
is clearly
obtained by using bootstrapping process. This methodology is
a flexible and robust method for deriving sampling
distributions and also standard errors.
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International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
V EXPERIMENTAL RESULTS
In this proposed Bootstrap Method with DTCWT the peak
signal to noise ratio is increased compared to other denoising
methods. Here we are adding different noise like Gaussian
noise, poission noise and speckle noise for the enhanced
luminance image. We are effectively reducing the all type of
noises and increase PSNR value. The peak signal to noise
ratio (PSNR) is defined as,
PSNR=10
The below figures shows the comparison results of DTCWT
and DTCWT using Bootstrap Method with different noise
affected images.
(c)
(d)
Figure 5: (a) Input image
(b) poission noise image
(c) Denoised luminance image (d) Denoised RGB image
PROPOSED METHOD:
Lenna image:
EXISTING METHOD:
Lenna image:
(a)
(b)
(a) (b)
(c)
(d)
Figure 6: (a) Input image
(b) Gaussian noise image
(c) Denoised luminance image (d) Denoised RGB image
(c)
(d)
Figure 4: (a) Input image,
(b) Gaussian noise image
(c) Denoised luminance image (d) Denoised RGB image
Girl image:
Girl image:
(a)
(a)
ISSN: 2278 – 1323
(b)
(b)
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International Journal of Advanced Research in Computer Engineering & Technology (IJARCET)
Volume 3 Issue 11, November 2014
(c)
(d)
Figure 7: (a) Input image
(b) Poission noise image
(c) Denoised luminance image (d) Denoised RGB image
The below table shows the comparisons of PSNR values of
existing and proposed method of different images at different
noises.
TABLE 1: EXISTING METHOD
Noise type
Image name
Gaussian
Noisy
Denoised
PSNR
PSNR
Lenna
27.68
33.90
Girl
28.24
35.39
Splash
27.27
37.15
VI CONCLUSION&FUTURE WORK
In this paper, we are using wavelet based transforms for image
denoising. Noise is one of the major problems in image
processing that occurs while capturing the image and the
image is transmitting through a channel. Here we are focused
on different types of noises like Gaussian, poission and
speckle noises. In this propose method we are using Bootstrap
Method with DTCWT. The DTCWT using Bootstrap Method
has several advantages over traditional denoising methods
that are it is having shift invariance property and good
directional selectivity.
The above experimental results show that different images
are affected by different type of noises. By using our proposed
algorithm we have to denoised the noise affected images
effectively and also calculate the image quality parameter
which is known as PSNR value. Comparison results for
existing and proposed methods are as shown in the above
tables 1 and 2. The proposed method produces good quality
output and removing noise without changing the directional
structures in the image.
Furthermore, improve the speed of the algorithm by
avoiding iterations and also DTCWT is a tool for other
activities such as image quality measures. Thus in future
applied for audio and video signals denoising.
REFERENCES
Poission
Speckle
Lenna
28.50
33.92
Girl
29.08
34.38
Splash
30.26
37.13
Lenna
27.66
33.89
Girl
28.27
35.47
Splash
27.28
36.98
Noisy
Denoised
PSNR
PSNR
Lenna
27.68
37.94
Girl
28.24
42.23
Splash
27.26
47.10
Lenna
28.50
37.82
Girl
29.08
40.50
Splash
30.26
45.08
Lenna
27.66
40.87
Girl
28.27
44.03
Splash
27.28
41.10
TABLE 2: PROPOSED SYSTEM
Noise type
Image name
Gaussian
Poission
Speckle
ISSN: 2278 – 1323
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ABOUT THE AUTHORS
T.Eswar Reddyreceived B.Tech degree from
JNT University, Anantapur and is pursuing his
M.tech
degree
from
JNT
University,
Anantapur.Hisareas of interest are digital image
processing and networking.
Mr. K.M Hemambaran working as Assistant
Professor in the department of ECE in SITAMS,
chittoor. He received AMIE degree from
institution of engineering (India) and he received
M.tech degree from JNT University, Hyderabad.
His areas of interest are Digital systems and
Digital image processing.
ISSN: 2278 – 1323
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