International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
Analysis of Orthogonal and Biorthogonal Mother
Wavelet Using Gaussian noise for Image Denoising
Reena Thakur1
Guru Nanak Institute Of Engineering and Technology,
Nagpur University, Maharashtra, India
Abstract
This paper analyzes the performance of orthogonal and Biorthogonal mother wavelets for image denoising using Gaussian
noise on various images. These images to be tested are of different size and resolution. The performance of denoised image is
measured, subjectively visual quality of image and objectively peak signal to noise ratio and it is found that Biorthogonal wavelets
outperform the orthogonal ones in both the criteria.
Keywords- Denoising , Wavelets, Peak Signal to noise ratio, Orthogonal, Biorthogonal, Mean Squarred error, Mother
Wavelets.
1. INTRODUCTION
Computer becoming more powerful day by day. Image denoising is still a challenging problem for researchers as image
denoising causes blurring and introduces artifacts .It has become a very critical exercise of inverse problems in image
processing. Wavelet denoising using its families is a more successful kind of application of wavelet transforming. The
blemish of signal acquisition devices is added with noises which can be reduced by estimator using prior information on
signal properties. Noise is unwanted signal that hinders with the original signal and disgraces the visual quality of
original image. The key sources of noise in images are imperfect instruments, problem with data gaining process,
intrusion natural phenomena, compression and transmission [1]. Image denoising forms the preprocessing step in the
field of image processing, medical fields, research, and technology, where somehow image has been degraded and needs
to be restored before further processing.
Different types of images inherit different types of noise and different noise models are used to present different noise
types. Denoising is a way to get rid of the effect of noise and to improve the signal-noise ratio and we have used
Gaussian noise.
In [1] the author uses wavelet transform in connection with threshold functions for removing noise and also Universal,
Visu Shrink, Sure Shrink and Bayes Shrink, normal shrink are compared with their threshold function, which improves
the SNR efficiently but depends on the nature of image. In [2] the author proposed a method to remove the noise of
QuickBird images based on the wavelet packet transform after analyzing the characteristics of wavelet packet transform.
The analytical results show that wavelet packet transform performs effectively in removing the noise of QuickBird
images compared with other methods.
Wavelet transform is one of the promising methods of image denoising. The basic measure of the performance of a
denoising algorithm is the quality of image and peak signal to noise ratio, which is defined by the ratio between original
image and denoised image. In the present work, we analyze various wavelet families for image denoising on variety of
test images and then compare the performance of wavelets. According to this analysis, we show the selection of the
optimal wavelet for image denoising taking into account Peak Signal to Noise Ratio (PSNR) as objective and visual
quality of image as subjective quality measure.
As we go down, section II describes the methodology used that is Wavelet Families for denoising , section III and IV
describes quality measures including type of noise used, section V describes experimental work result and discussion of
results and finaly section VI describes conclusion and references.
Organized By: GNI Nagpur, India
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
2. MOTHER WAVELETS
Wavelet families can be divided into two main categories, orthogonal and Biorthogonal wavelets, which have different
properties of basis functions. Orthogonality decorrelates the transform coefficients there by minimizing redundancy.
Symmetry provides linear phase and minimize border arti-facts Other Important properties of wavelet functions in image
denoising applications are compact support, symmetry, regularity and degree of smoothness [3] [4]. Figure 1 illustrates some
of the commonly used wavelet functions in our experiments.
Fig. 1 Some of the wavelet mothers used in our experiments
3. QUALITY MEASURES
The performances of image denoising techniques are mainly analyzed on the basis of : Noise varience and Peak Signal to noise
ratio (PSNR) which is the ratio between noisy image and denoised image.
PSNR provides a measurement of the amount of distortion in a signal [5], with a higher value indicating less distortion. For nbits per pixel image, PSNR is defined as:
PSNR
2n 1
20log10
RMSE
Where, RMSE is the root mean square difference between two images. The Mean Square Error (MSE) is defined as follows [6]:
MSE
1
MN
M 1N 1
y (m, n) x(m, n)
2
m 0n 0
where x(m,n),y(m,n) are respectively the original and recovered pixel values at the mth row and nth column for MxN size
image. PSNR is normally quoted in decibels (dB), which measure the ratio of the peak signal and the difference between two
images (error image). Logically, a higher value of PSNR is good because it means that the ratio of Signal to Noise is higher. So,
if we find a denoised scheme having a high PSNR, we can recognize that it is a better one.
4. GAUSSIAN NOISE
Gaussian noise is evenly distributed over the signal [7]. This means that each pixel in the noisy image is the sum of the true
pixel value and a random Gaussian distributed noise value. This noise has a Gaussian distribution as the name indicates, which
has a bell shaped probability distribution function given by,
Organized By: GNI Nagpur, India
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
1
2
F (g)
e g m /2 2
2
2
where g represents the gray level, m is the mean or average of the function, and σ is the standard deviation of the noise.
5. EXPERIMENTAL RESULTS , ANALYSIS AND COMPARISON
We analyze orthogonal and Biorthogonal wavelet families for image denoising and compare their results. We used four types of
test
images
with
different
frequency
content,
different
resolution
and
different
size:
Rose(128X85),Globe(260X173),Car(259X194),Horse(225X226).
TABLE 1
Wavelet family daubachies
PSNR (in DB)
Horse
db6
db10
db20
0.01
28.68
28.81
28.81
28.77
28.75
0.02
28.36
28.47
28.44
28.46
28.41
0.03
28.26
28.31
28.30
28.30
28.23
0.04
28.21
28.25
28.15
28.21
28.23
28.5
0.05
28.12
28.15
28.15
28.16
28.18
28
0.01
29.15
29.16
29.21
29.16
29.13
0.02
27.64
27.63
27.70
27.64
27.65
27.5
db4
0.03
27.06
27.06
27.05
27.08
27.07
27
0.04
26.85
26.85
26.84
26.84
26.87
db6
0.05
26.73
26.71
26.72
26.72
26.73
0.01
28.58
28.57
28.59
28.58
0.02
28.27
28.30
28.26
28.29
0.03
28.13
28.16
28.16
28.13
28.16
0.04
28.08
28.08
28.10
28.08
28.05
0.05
28.02
28.02
28.04
28.01
28.01
0.01
28.52
28.49
28.56
28.53
28.48
0.02
28.15
28.12
28.18
28.15
28.13
0.03
28.01
28.00
28.02
28.01
27.98
0.04
27.96
27.93
27.93
27.92
27.89
0.05
27.90
27.88
27.91
27.89
27.85
29.5
29
db2
26.5
db10
28.58
26
db20
28.27
25.5
Wavelet family biorthogonal
PSNR (in DB)
Rose
Var
0.01
bior
1.3
28.71
bior
1.5
28.73
bior
2.2
28.76
bior
2.4
28.75
bior
2.6
28.83
0.02
28.32
28.37
28.36
28.45
28.48
0.03
28.28
28.22
28.29
28.30
28.29
0.04
28.21
28.17
28.22
28.20
28.23
0.05
28.13
28.11
28.11
28.14
28.20
0.04
Fig. 2 Graphical Performance of Daubachies family
TABLE 2
Image
0.01
0.03
25
0.05
Car
db4
0.02
Globe
db2
0.04
Rose
Var
0.01
Image
Organized By: GNI Nagpur, India
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
Globe
Car
Horse
0.01
29.18
29.21
29.27
29.25
29.24
0.02
27.69
27.67
27.66
27.64
27.69
0.03
27.07
27.11
27.09
27.06
27.01
0.04
26.85
26.87
26.82
26.84
26.85
0.05
26.72
26.74
26.74
26.72
26.72
0.01
28.54
28.57
28.60
28.58
28.62
0.02
28.25
28.27
28.27
28.26
28.28
0.03
28.10
28.15
28.14
28.12
28.15
0.04
28.07
28.06
28.07
28.08
28.08
0.05
28.02
28.01
28.02
28.03
28.04
0.01
28.56
28.50
28.52
28.59
28.55
27
bior2.2
0.02
28.14
28.14
28.12
28.18
28.18
26.5
bior2.4
0.03
28.00
27.99
27.97
28.00
28.01
0.04
27.93
27.91
27.92
27.92
27.92
26
bior2.6
0.05
27.85
27.85
27.86
27.88
27.88
29.5
29
28.5
28
bior1.3
27.5
bior1.5
25.5
0.02
0.03
0.04
0.05
0.01
25
Fig 3. Graphical performance of Symlet family
TABLE 3
Wavelet family Symlet
PSNR (in DB)
Image
Rose
Variance
Sym2
Sym3
Sym5
Sym10
Sym12
0.01
28.73
28.73
28.79
28.70
28.72
0.02
28.36
28.37
28.44
28.41
28.42
0.03
28.22
28.27
28.28
28.26
28.24
0.04
28.23
28.19
28.23
28.18
28.13
0.05
28.12
28.16
28.18
28.13
28.19
29.5
29
28.5
28
sym2
0.01
29.19
29.21
29.18
29.14
29.22
27.5
0.02
27.60
27.63
27.60
27.67
27.70
27
0.03
27.09
27.06
27.07
27.07
27.06
0.04
26.81
26.82
26.84
26.88
26.86
26.5
sym10
0.05
26.72
26.73
26.71
26.73
26.71
28.58
28.61
28.62
28.57
28.59
26
sym20
0.01
0.02
28.28
28.30
28.28
28.29
28.28
0.03
28.12
28.16
28.17
28.15
28.15
0.04
28.06
28.10
28.08
28.07
28.07
0.05
28.00
28.06
28.01
28.03
28.02
0.01
28.52
28.51
28.50
28.56
27.87
0.02
28.17
28.13
28.51
28.15
28.49
0.03
27.99
27.96
28.00
28.00
28.15
0.04
27.95
27.91
27.91
27.92
27.99
0.05
PSNR (in DB)
27.89
27.86
27.85
27.85
27.87
Car
Horse
sym5
25.5
25
0.01
0.04
0.02
0.05
0.03
0.01
0.04
Globe
sym3
Fig. 4. Graphical performance of Coiflet family
TABLE 4
Waveletfamily Coiflet
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
Rose
Globe
Car
Horse
Variance
coif1
coif2
coif3
coif4
coif5
0.01
28.69
28.83
28.69
28.71
28.74
0.02
28.40
28.43
28.38
28.39
28.39
0.03
28.30
28.30
28.26
28.27
28.25
0.04
28.20
28.19
28.19
28.20
28.23
0.05
28.10
28.20
28.13
28.17
28.19
0.01
29.20
29.16
29.14
29.20
29.19
0.02
27.65
27.66
27.63
27.71
27.70
0.03
27.04
27.06
27.08
27.08
27.06
0.04
26.86
26.85
26.86
26.81
26.85
0.05
26.74
26.72
26.72
26.71
26.73
0.01
28.58
28.59
28.61
28.56
28.58
0.02
28.30
28.26
28.26
28.26
28.29
0.03
28.13
28.15
28.15
28.14
28.13
0.04
28.10
28.08
28.06
28.10
28.03
0.05
28.02
28.03
28.02
28.03
28.02
0.01
28.50
28.53
28.49
28.55
28.53
0.02
28.11
28.13
28.16
28.16
28.13
0.03
27.97
28.02
27.97
28.01
27.97
0.04
27.90
28.01
27.92
27.93
27.91
0.05
27.87
27.87
27.84
27.87
27.86
29.5
29
28.5
28
27.5
27
26.5
26
25.5
25
coif1
coif2
coif3
coif4
coif5
0.01
0.04
0.02
0.05
0.03
0.01
0.04
Image
Fig. 5 Graphical performance of Coiflet family
The visual quality results are shown in figure 2. The images shown here are denoised at the noise varience 0.01 to 0.05 each at
decomposition level of 5, which is optimum level of denoising. The results show that wavelet function BIOR2.2 provides
better results in terms of PSNR for the test image Globe. Also, it is found that the wavelet function BIOR 2.2 gives better visual
quality when the test images are in .png format. Secondly DB4 and DB6 provides better results in terms of PSNR for the test
image Rose.jpg. While COIF2 and BIOR2.6 shows the Competitive PSNR performance for the large noise varience for the
test images Rose.jpg .
The analysis and comparison of the results show that the not only in the BIOR family, the wavelet function BIOR 2.2 gives the
better denoising results ( in terms of PSNR) in all the wavelet families considered in our experiment.
For the denoising performance in terms of visual image quality, the wavelet BIOR 2.2 provides the better results for the test
image. While, the wavelet BIOR 2.6 for the images Bird and Bridge gives the better compression performance in terms of
visual image quality .This shows that objective as well as subjective quality of the compressed image is better for wavelet
family Biortogonal. Motivation following this performance is that Biorthogonal wavelets can use filters with similar or
dissimilar order for decomposition (Nd) and reconstruction (Nr). Therefore Biorthogonal wavelet is parameterized by two
numbers and filter length is {max (2Nd, 2Nr) +2} [8]. Also these are Symmetric and Symmetry provides linear phase and
minimize border arti-facts. In study if decomposition level is increased the compression performance improves but the quality
of image deteriorates. Further, it is also observed that the BIOR wavelet families take much more computational time in
comparison to other wavelet families considered in our experiment. Also it is found that as the filter order increases in a given
wavelet family, the compression performance increases, but the visual quality of compressed image becomes not as good as.
The higher order of filters involves the longer filters, which involves more blurring in the compressed image
Organized By: GNI Nagpur, India
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
A)ORIGINAL IMAGE
b) DENOISED IMAGE BY DB6(VAR 0.01)
& C)DB6 VAR 0.05 D) COIF5, VAR=0.05
F)Denoised by bior2.2 var=0.01G) SYM1,VAR=0.01 H) BIOR 2.6, VAR=0.01
Fig. 6 Visual quality and performance
6. CONCLUSION
This analysis focuses on the performance of orthogonal and Biorthogonal mother wavelets for image denoising using
Gaussian noise on variety of test images. This paper measures the performance of the images in terms of peak signal to
noise ratio and visual quality of the image also. it is found that Biorthogonal wavelets outperform the orthogonal ones in
both the criteria.
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Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
ISSN 2319 - 4847
Special Issue for National Conference On Recent Advances in Technology and Management for
Integrated Growth 2013 (RATMIG 2013)
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Mrs. Reena Thakur has received a B.E.(CSE) degree from Amravati University and Master degree
in
Computer Science and Engineering from Uttar Pradesh Technical University Lucknow(U.P.). She
is having 17 yrs of teaching experience. She has written three books. She has more than 10
publications to her credit in international journals, conferences as well as in IEEE Explore. Her fields
of interest are Image Processing, Data Mining, Computer Graphics. She is presently working in Guru
Nanak Institute of Engineering and Technology, Nagpur.
Organized By: GNI Nagpur, India