De-Noising with Spline Wavelets and SWT

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 10 | Oct -2017 www.irjet.net p-ISSN: 2395-0072
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De-Noising with Spline Wavelets and SWT
1Asst. Prof. Ravina S. Patil, 2Asst. Prof. G. D. Bonde
1Asst. Prof, Dept. of Electronics and telecommunication Engg of G. M. Vedak Institute Tala. Dist. Raigad
2Asst. Prof. Dept. of Electronics and telecommunication Engg J. T. Mahajan college of Engg. Faizpur.
Dist. jalgaoan
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Abstract - This paper explores the difference in
performance of spline wavelets of the bi-orthogonal type
in denoising images corrupted by Additive White Gaussian
Noise. The dependence of the peak signal-to-noise ratio
and the mean squared error on the filter characteristics of
the wavelets, when stationary wavelet transform is used in
the de-noising process, is investigated. It is found that the
de-noising action augments with use of wavelet of lower
effective length for its high pass reconstruction filter. For
wavelets with equal effective lengths for their high pass
reconstruction filters, a relation similar to the above exists
for the high pass decomposition filters. ‘Bior1.1’(bi-
orthogonal spline wavelet 1.1) is found to be the most
suitable wavelet in the family, for de-noising. ‘Bior 3.1’ is
found to be an odd member in the family and is not at all
suitable for de-noising, the reason for which is traced to
the lack of smoothness of its decomposition scaling
function
Keywords - Spline wavelet, stationary wavelet
transform, bi-orthogonal wavelets, DWT,
thresholding.
1. INTRODUCTION
1.1 Image Denoising
The digital image processing is concerned with image
de-noising. Noise is defined as the transmission medium
and error occur during measurement and quantization
process of the data for digital storage. This including
algorithm and routine goal oriented image Processing.
The reduction of degraded images that are Incurred
image is being obtained by the image restoration.
Degradation comes from blurring as Well as noise due to
various sources. Blurring is a form of bandwidth
reduction in the image caused by the imperfect image
formation process like relative motion between the
camera & the object or by an optical system which is out
of the focus. For remote sensing purposes when aerial
photographs taken atmospheric turbulence introduces
blurs, optical system aberration and relative motion
between camera and the ground with these blurring
effects. The noise can also corrupted by recorded image.
Each element in the imaging chain such as film, lenses,
digitizer, etc. contribute to the degradation. The field of
photography or publishing is used for the purpose of
image denoising. In that an image is somehow degraded
but it required improving before it can be printed. Such
kind of application we need to know about the
degradation process in order to design a model for it.
Figure – 1: shows the images with noises
2. LITERATURE SURVEY
The bilateral filtering is applied to the approximation
sub-bands. Unlike The difference in performance of
spline wavelet of the bi-orthogonal type in denoising
images corrupted by Additive White Gaussian noise. The
dependence of the peak single to noise ratio and mean
squared error of the filter of the wavelet. This stationary
wavelet transform is used in the denoising process are
investigated. For wavelet with equal effective length for
their high pass reconstruction filter similar to the high
pass decomposition filter ‘Bior1.1’ (bi-orthogonal spline
wavelet 1.1) is the most suitable wavelet in the family,
for de-noising. ‘Bior 3.1’ is an odd member in the family
and is not at all suitable for de-noising, the reason for
which is traced to the lack of smoothness of its
decomposition scaling function. Image noise reduction
or denoising is an active area of research although many
of the techniques. The literature mainly target additive
white noise. The standard single-level bilateral filtering,
this multi resolution bilateral filtering has the potential
of eliminating low-frequency noise components. (This
will become evident in our experiments with real data.)
The approximation sub-bands in addition works in
Bilateral filtering it is possible to apply wavelet
thresholding to the detail sub-bands, where some noise
components can be introduced and removed effectively.
The new image denoising framework combines bilateral
filtering and wavelet thresholding. Chang and Vetterli [8]
proposed an adaptive, data-driven threshold for image

denoising using the wavelet soft-thresholding. The
application of image processing used in the threshold is
derived in a Bayesian framework, and the prior used on
the wavelet coefficients is the generalized Gaussian
distribution (GGD). The proposed threshold is closed-
form and adaptive to each sub-band. This method, so
called Bayes Shrink [8], outperforms Donoho and
Johnstone‟s Sure Shrink [7] most of the time. Since
wavelet coefficients of real images have significant
dependencies, Sendur et al. [9] considered the
dependencies between the coefficients and their parents
in the detail coefficients part. The purpose of the non-
Gaussian bivariate distributions is proposed, and
corresponding nonlinear threshold functions are derived
from the models using Bayesian estimation theory. The
new shrinkage functions do not assume the
independence of wavelet coefficients. However, the
performance of this method is not very well.
3. MATERIALS AND METHODS
3.1. Bi-orthogonal wavelets
The meaning of ‘bi-orthogonal’ is two functions or
‘bases’ which are mutually orthogonal to each other , but
each of these two functions need not form an orthogonal
set. Two different scaling functions and two different
wavelet functions are used for bi-orthogonal wavelets.
The decomposition step scaling and wavelet functions
(Φ and Ψ) are used and in the reconstruction step the
other set (Ψ and Ψ ) is used. This provides interesting
features are not possible by using one and same filters
for decomposition and reconstruction in the orthogonal
case. Also, filter banks comprising bi- orthogonal filters
are more flexible and can be designed easily. The linear
phase which is good for reconstruction of images has Bi-
orthogonal wavelets. The bi-orthogonal spline wavelets
listed as: 'bior 1.1', 'bior 1.3', 'bior 1.5', 'bior 2.2', 'bior
2.4', 'bior 2.6', 'bior 2.8', 'bior 3.1', 'bior 3.3', 'bior 3.5',
'bior 3.7', 'bior 3.9', 'bior 4.4', 'bior 5.5' and 'bior 6.8'
3.2. Stationary Wavelet Transform
The Stationary wavelet transform (SWT) is similar to the
DWT. Signal is never sub-sampled and instead the filters
are up sampled at each level of decomposition. The
stationary wavelet transform (SWT) is a wavelet
transform algorithm designed to overcome the lack of
translation-invariance of the discrete wavelet
transform (DWT). Translation-invariance is achieved by
removing the down samplers and up samplers in the
DWT and The up sampling the filter coefficients by a
factor of in the level of algorithm. The SWT is an
inherently redundant scheme as the output of each level
of SWT contains the same number of samples as the
input – so for a decomposition of N levels there is a
redundancy of N in the wavelet coefficients.
Figure- 2: Haar Stationary Wavelet Transform of Lena
3.3 Thresholding :
Thresholding is a simple non-linear technique, these
operates on one wavelet coefficient at a time. In its most
basic form of each coefficient which is smaller than
threshold set to zero. The small co-efficient are
dominated by noise, while coefficient with large absolute
value carry more signal information than noise.
Replacing noise co-efficient (small coefficients below a
certain threshold value) by zero and an inverse wavelet
transform. This thresholding idea is based on the
following:
1) The de-correlating property of wavelet transform
creates a sparse signal. Most untouched coefficient is
zero or close to zero.
2) Noise is spread out equally along all co-efficient.
3) The noise level is not too high so that one can dis-
tinguish the signal wavelet coefficients from binary ones.
This method is an effective and thresholding is simple
and efficient method for noise reduction
4. RESULTS AND DISCUSSION
The noisy image is shown in Fig.1, Fig. 2 and Fig.3 The
de-noised images with the obtained maximum and
minimum values of PSNR, respectively. The MSE and
PSNR corresponding to de-noising with the different bi-
orthogonal wavelets are shown in Table1.The
decomposition process using SWT involves convolution
of the image matrix with a low pass filter and a High pass
filter. These filters are LoD and HiD and the values of
their effective lengths are in Table1. Similarly the
reconstruction process involves convolution of the
image matrix with another set of filters containing a low
pass filter and a high pass filter indicated as LoR and HiR.
The values of the effective lengths of these filters are

given in Table 1. Usually the high frequency components
in an image comprise the noise in the image.
Figure-3: Noisy image Figure-4: Image de-
noised with ‘bior 1.1’
Figure -5: Image de-noised with ‘bior 3.1’
Therefore it is reasonable for us to examine the features
of HiD and HiR to relate the same to the variations in the
denoising performance of the different wavelets used for
the study. The output of low pass filter contains
approximation of the image. It is observed that the
estimated values of the PSNR (and MSE) vary with the
different wavelets used in the SWT for the de-noising
Table-1: Variation on the PSNR value
Process. From Table 1 it can be seen that the variations
in the PSNR values have some amount of relationship
with the effective lengths of HiR and HiD. A detailed
inspection of the corresponding values leads to the
following observations:
1. The PSNR decreases with increase in the effective
Length of HiR. This fact is observed to be true in all the
denoising cases under consideration, except in the cases
Of denoising with ‘bior 3.1’, ‘bior 3.3’ and ‘bior 3.5’. ‘Bior
3.1’ gives the lowest PSNR (34.5294) even though this
wavelet has a low value 4 for effective length of HiR. In
fact, ‘bior 3.1’ has the second lowest effective length of
HiR when we consider the corresponding values of all
the other members in the bi-orthogonal spline wavelet
family. Thus ‘bior 3.1’ is found to have an odd behavior,
the reasons for which shall be explored later.
Hence the following discussion skips ‘bior 3.1’, for the
time being. As we move from ‘bior 2.8’ to ‘bior 3.3’, the
PSNR decreases even though the effective length of HiR
has decreased. The reason for this is an increase in the
PSNR, MSE and effective filter lengths of the wavelets
Wavelet MSE PSNR dB Effective length of filters
LoD HiD LoR HiR
bior 1.1 12.4334 37.1849 2 2 2 2
bior 1.3 13.0208 36.9844 6 2 6 2
bior 1.5 13.2655 36.9036 10 2 2 10
bior 2.2 13.0943 36.9600 5 2 2 5
bior 2.4 13.2460 36.9100 9 3 3 9
bior 2.6 13.3893 36.8632 17 3 3 17
bior 2.8 13.5105 36.8241 4 4 4 4
bior 3.1 22.9163 34.5294 8 4 4 8
bior 3.3 13.6426 36.7818 12 4 4 12
bior 3.5 13.4364 36.8480 16 4 4 16
bior 3.7 13.5213 36.8206 20 4 4 20
bior 4.4 13.4691 36.8374 9 7 7 9
bior 5.5 13.6570 36.7773 9 11 11 9
bior 6.8 13.8360 36.7207 17 11 11 17

actual values of HiR represented by the increased value
of HiRmax (maximum value of HiR) given in Table 2. Due
to this increase in values of HiR, high amplitude
coefficients containing noise are retained. An effect just
opposite to this is observed in the case of ‘bior 3.5’. ‘Bior
1.1’ gives the maximum value of PSNR which is 37.1849.
Also, the effective length of HiR has the lowest value for
‘bior 1.1’. This fact agrees with our above observation
regarding relation between PSNR and effective length of
HiR. Large effective length of HiR means large number of
nonzero filter points in the filter. Since this high pass
Filter with the large number of non-zero coefficients is
convolved with the coefficients resulting from
decomposition of the noisy digital image which have
subsequently been thresholded such a convolution gives
rise to high frequency components spread over a large
extent and carries the noise components that have not
been removed in the thresholding process. This explains
the reduction in PSNR with increase in effective length of
HiR.
2. When the effective lengths of HiR for two different
Wavelets are equal, the PSNR is found to decrease with
Increase in the effective length of HiD. This is evident by
Observing the PSNR values of the set of wavelets
comprising ‘biro 2.4’, ‘biro 4.4’ and ‘biro 5.5’, each of
which has an effective length 9 for HiR. The PSNR values
obtained on denoising with these wavelets decrease
regularly as the effective lengths of HiD increase. This is
shown separately in Table 3 for easy reference. An
identical effect is noticed on observing the de-noising
performance of ‘bior 2.8’ and ‘bior 6.8’. Both of these
wavelets have effective length 17 for HiR. The PSNR is
found to have decreased as the effective length of HiD
has increased
3. The influence of effective length of HiD on de-noising
Performance is considerably less than that of HiR. This
can be established in the following way. We have already
established above that (i) the PSNR decreases with
increase in the effective length of HiR and that (ii) when
the effective lengths of HiR of 2 bi-orthogonal spline
wavelets are equal, the PSNR decreases with increase in
the effective length of HiD. As we move from ‘bior 1.5’ to
‘bior 2.2’ the effective length of HiD increases from 2 to
3, effective length of HiR
Table- 2: Maximum values of HiR for 'bior 2.8’,’bior 3.3’
and ‘bior3.5’
Φ (t) = Σ 𝑘 p (k) Φ (2t − k), kε Z,
Where Φ (t) is the scaling function and p (k) is the
discrete
Sequence of coefficients resulting from the
decomposition.
PSNR and effective lengths of HiD for wavelets
with HiR of effective length 9.
Wavelet Effective length of HiD PSNR dB
bior 2.4 3 36.9100
bior 4.4 7 36.8374
bior 5.5 11 36.7773
Wavelet HiRmax
bior 2.8 0.4626
bior 3.3 0.9944
bior 3.5 0.9667
Table-3: PSNR and effective lengths of HiD for wavelet
Decreases from 10 to 5 and PSNR increases. Also when
we move from ‘bior 3.9’ to ‘bior 4.4’, the effective length
of HiD undergoes an increase from 4 to 7; at the same
time, effective length of HiR decreases from 20 to 9 and
the PSNR value increases. Here the effective length of
HiD has increased by 1 point in the former case and by 3
points in the latter case. On the other hand, the effective
lengths of HiR in these cases have had decrease and that
by considerably larger numbers of points. In both the
instances the PSNRs have only increased; the increase in
PSNR accompanies the decrease in effective length of
HiR. In this context it may be noted that the aforesaid
increases in effective lengths of HiD have had
nonoticeable effect on the PSNR. This establishes that
effective length of HiD has considerably lesser influence
on de-noising performance, compared to effective length
of HiR. The apparent dominance of the dependence of
effective length of HiR on PSNR, compared to that of HiD,
is consequent of the larger value of effective length of
HiR compared to that of HiD, or in other words, due to
the larger numbers of non-zero filter points of HiR when
compared to those of HiD; it can be seen that in most
cases, the effective length of HiR is 2 to 4 times that of
HiD. Now, we may investigate the reason for the odd
behavior of ‘bior 3.1’. The decomposition scaling
function of ‘bior 3.1’ is shown in Figure 4. As what can be
seen from Figure 4, this function is not at all a smooth
one. It is scaling function bases that generate the wavelet
basis functions [2]. Hence the decomposition wavelet
function of ‘bior 3.1’ is also not smooth. The basic two-
scale relation in MRA is:

Figure - 6: Decomposition scaling function
of ‘bior 3.1’
Table -4: Variance of the effective lengths of LoD
This equation indicates that the scaling function at a
particular resolution can be decomposed in to a linear
combination of scaling functions at the next higher
resolution [2]. The discrete sequence p (k) of the
coefficients resulting from the decomposition constitutes
the low pass filter in the wavelet decomposition. Since
the decomposition scaling function is not smooth, its
regularity is poor and the decomposition low pass filter
has a high variance. This is also evident from Table 4
Which shows the variances of LoD (Var (LoD)) of the
different wavelets. It can be seen that ‘bior 3.1’ has the
highest value for “variance” or “dispersion” of LoD. This
explains the reason for the odd behavior and the poor
de-noising performance of ‘bior 3.1’. Also the visual
quality of the denoised images is found to have changes
following the changes in the PSNR values
5. CONCLUSIONS
This paper explores de-noising performance of the
different bi-orthogonal spline wavelets, when SWT is
used as the transform for the de-noising operation. The
denoising action is found to improve with the use of bi-
orthogonal wavelet of lower effective length for its high
pass reconstruction filter. When the effective lengths of
high pass reconstruction filter for any two bi-orthogonal
spline wavelets are equal, the PSNR decreases with
increase in the effective length of high pass
decomposition filter. The influence of effective length of
high pass decomposition filter on denoising performance
is considerably less than that of high pass reconstruction
filter; this is due to the fact that the latter has larger
number of non-zero filter points than the former. The
maximum value of PSNR is obtained by de-noising with
the bi-orthogonal spline wavelet with the minimum
effective reconstruction filter length which is ‘bior 1.1’.
‘Bior 1.1’ is hence the most suitable bi-orthogonal spline
wavelet for de-noising images corrupted by AWGN. ‘Bior
3.1’ is found to be an odd member in the bi-orthogonal
spline wavelet family. This wavelet gives the lowest
PSNR. Therefore ‘bior 3.1’ is not at all suitable for de-
noising. The odd behavior and the worst de-noising
performance of ‘bior3.1’ are traced to be consequent of
the lack of smoothness of its decomposition scaling
function. It is also found that the visual quality of the
images resulting from de-noising using the different bi-
orthogonal spline wavelets follow the changes in the
PSNR values obtained.
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Variances of the effective lengths of LoD
Wavelet Var (LoD)
bior1.1 0
Bior1.3 0.1396
Bior1.5 0.0956
Bior2.2 0.2208
Bior2.4 0.1350
Bior2.6 0.0983
Bior2.8 0.0777
Bior3.1 0.6667
Bior3.3 0.2662
Bior3.5 0.1696
Bior3.7 0.1255
Bior3.9 0.1001
Bior4.4 0.0934
Bior5.5 0.0567
Bior6.8 0.0565

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