Journal of Signal and Information Processing, 2011, 2, 308-315
doi:10.4236/jsip.2011.24044 Published Online November 2011 (http://www.SciRP.org/journal/jsip)
Copyright © 2011 SciRes. JSIP
1
A Dyadic Wavelet Filtering Method for 2-D Image
Denoising
Yonggui Zhu*, Xiaolan Yang
School of Science, Communication University of China, Beijing, China.
Email: *ygzhu@cuc.edu.cn
Received January 11th, 2011; revised July 16th, 2011; accepted July 24th, 2011.
ABSTRACT
We improve spatially selective noise ltration technique proposed by Xu et al. and wavelet transform scale ltering
approach developed by Zheng et al. A novel dyadic wavelet transform ltering method for image denoising is proposed.
This denoising approach can reduce noise to a high degree while preserving most of the edge features of images. Dif-
ferent types of images are employed to test in the numerical experiments. The experimental results show that our
ltering method can reduce more noise contents while maintaining more edges than hard-threshold, soft-threshold
lters, Xus method and Zhengs method.
Keywords: Dyadic Wavelet Transform, Image Edges, Denoisin
1. Introduction
Wavelet transform is a multi-resolution representation of
a signal or image. It is a powerful tool in several areas of
applications like signal processing, image processing,
pattern recognition, data compression, commutation, etc.
Singularities and irregular structures often carry essential
information in signals and images. For example, the dis-
continuities of the intensity of an image indicate the lo-
cations of edges.
The local regularity is characterized by the decay of
the wavelet transform amplitude across scales. Signal
singularities and image edges can be detected by the dy-
adic wavelet transform modulus maxima across scales
[1,2]. In mathematics, singularities are generally measu-
red with Lipschitz exponents. The wavelet theory proves
that these Lipschitz exponents can be calculated from the
propagating amplitude values of the different modulus
maxima across scales.
The original signal or image has singularities whose
Lipschitz exponents are greater than or equal to zero, and
the noise has singularities whose Lipschitz exponents are
less than zero. Thus, the amplitudes of signal or image
modulus maxima increase when the scale increases, wh-
ile the amplitude of noise modulus maxima decrease str-
ongly when the scale increases. By using these properties,
the noises can be eliminated from the noised signals or
images. The approaches for separating signal and noise
in wavelet scale space are proposed by many researchers.
For example, the original signal can be extracted from
the noisy version by estimating the signal modulus max-
imum at small scales [1,2]. Adaptive Wiener ltering
were used to remove noise in signals and images [3-5].
The selective noise ltration technique and adaptive thr-
esholding function in image denoising were developed
[6-8]. The scale space ltering algorithms applied to im-
age denoising were also proposed [9,10]. In addition,
many other novel approaches for image denoising have
been presented by some researchers [11-13] recently. In
this work, we develop an image denoising approach by
improving spatially selective noise ltration technique
proposed by Xu et al. [6] and wavelet transform scale
ltering approach given by Zheng et al. [9]. Hard-thr-
eshold and soft-threshold lters that were proposed by D.
L. Donoho [14,15] are widely used in image denoising
processing. We will compare our ltering approach with
hard-threshold ltering, soft-threshold ltering, Xu’s me-
thod and Zheng’s method in the numerical experime- nts.
Peak-Signal-Noise-Rate (PSNR) and Root-Mean-Square-
Error (RMSE) are employed to estimate the quality of
restored images.
2. 2-D Dyadic Wavelet Transform
Let

12
,
k
x
x
(k = 1, 2) be wavelet functions.
We denote that






112
12
2
212
12
2
1
2
1
,,
222
1
,,
222
j
j
jjj
jjj
xx
xx
xx
xx










(1)
A Dyadic Wavelet Filtering Method for 2-D Image Denoising309
The wavelet transform of


22
12
,
f
xxL R at the
scale 2
j
Is








11
12 12
22
22
12 12
22
,,
,,
jj
jj
Wfxx fxx
Wfxx fxx


(2)
The set of functions




12
12 12
22
,, ,
jj
j
WfW fxxWfxx
is called 2-D dyadic wavelet transform of
12
,
f
xx . The
Fourier transforms of

1
12
,
x
x
and

2
12
,
x
x
are


1
12
ˆ,

and

2
ˆ

12
,


1
We suppose that12

,
x
x
,

2
12
,
x
x
are recon-
structed wavelet functions. If their Fourier transforms
satisfy








11
12 12
11
12 12
ˆˆ
2,2 2,2
1
ˆˆ
2,2 2,2
jj jj
jj jj
j






(3)
Then
12
,
f
xx can be reconstructed from their dyadic
wavelet transform i.e.

 

 

11
12
22
12 11
12
22
,
,
,
jj
jj
j
Wf xx
fxx
Wf xx





(4)
Because of the limitation of image’s resolution, we int-
roduce a smoothing function

12
,
x
x
Whose Fourier transform satises







11
212 12
12 22
112 12
ˆˆ
2,2 *2,2
ˆ,
ˆˆ
2,2 2,2
jj jj
jj jj
j

  



(5)
We define the smoothing operator 2
j
S by
 

12 12
22
12
12
2
,,
1
,,
222
jj
jjjj
Sfxx fxx
xx
xx

 



(6)
The wavelet transform between the scales 1 and





12
12 12
22
1
2,,,
jj
J
j
J
W fxxWfxx
provides the details that are in but that have
lost in .
112
,Sfx x

12
2,
j
Sfxx
Mallat [1] has given the fast algorithm for the discrete
dyadic wavelet transform. The fast dyadic wavelet trans-
form can also be calculated with a lter bank algorithm
called the algorithm trous proposed by Holschneider,
Kronland-Martinet, Morlet and Tchamitchian [16]. In
this paper, we use trous algorithm to reconstruct the
image.
a
a
3. Dyadic Wavelet Transform Filtering
Algorithm
In recent years, some denoising techniques based on the
wavelet transform have been studied by many authors
[2,6,12,17]. The edge modulus maxima can be distin-
guished from noise modulus maxima by analyzing the
singularity properties of wavelet transform domain ma-
xima of a signal or image across scales [2]. Y. Xu [6]
developed wavelet transform domain lters based on the
direct spatial correlation of the wavelet transform at sev-
eral adjacent scales. Y. Zheng [9] proposed a wavelet
transform scale filtering algorithm by using the proper-
ties of signal and noise modulus maxima across large
scales. Our approach relies on the variations of the dy-
adic wavelet transform data across all scales to remove
noises rather than extracting edges directly.
For a 2-D image, the discrete sampling of


12
21
,
j
k
j
J
Wfxx
is given by





2
12 12
12 12
22,,
,, 1
d
jj
kk
xx kk
WfkkWfxxjJ

(7)
The discrete coarse smoothed image is denoted by
 

2
12 12
12 12
22,,
,,
JJ
d
xx kk
Sfkk Sfxx
(8)
In the scale space, the modulus maxima of 2
across scales produced by image edges have positive
correlation. When the scale increases, the amplitudes
of modulus maxima coeffcients will increase or retain
constant. On the contrary, the modulus maxima produced
by noises have negative correlation and the amplitudes of
their coeffcients decrease as increases.

k
d
j
Wf
j
j
Dene 2-scale correlation as







12 12
2, ,1,
1, 2,,1,,,1,,
kkk
CrmnWc mnWc mn
mJnnnnn


,
N
.
(9)
where J re-
presents the maximum scale of the decomposition.





12
22
,,
d
mm
kk
kd
Wcm nWfnWfnn
The 2-scale direct correlation sharpens and enhances
major edges while suppressing noise and small features.
So comparing the values of and
can separate important edges from noise in
images. Before comparison, needs to be
rescaled to Xu’s rescaling scheme is

2,
k
Crmn

2
k
Crmn


,
k
Wcm n

,

,
k
Wcm n

 


2, 2,2
kkk
Crm nCrmnPWmPCrm
k
,
(10)
where



22
k
k
n
PCrmCrm n
and




2
,
k
k
n
P
Wm Wcmn.
Copyright © 2011 SciRes. JSIP
A Dyadic Wavelet Filtering Method for 2-D Image Denoising
310
Zheng et al. use the modulus maxima rescaling method
at large scales, and apply the above mentioned rescaling
method at small scales. Let S be the upper limit of small
scales, assume






2
12
,
max2, ,1
kk
nnn
M
crmCrm nmS

 (11)
and




,,1
kk
M
wcmWcmm S
 (12)
The modulus maxima rescaling formula as follows:








2, 2,,1
k
kk
k
Mwc m
Crm nCrm nmS
Mcr m




(13)
At small scales, noise in the noised image is dominat-
ing except some sharp image edges. According to Xu and
Zheng’s ideas, if compare

2,
k
Crm n
with


,
k
Wcm n directly, then too much noise will be ext-
racted as edges. To avoid this drawback, we apply the
modulus maxima rescaling at all scales and renew the
formula (13) as








2, 2,
1, 2,,1,
k
kk
k
mk
Mwc m
CrmnCrmn Mcr m
mJ





,
(14)
where

k
m
is a weight parameter with respect to the
scale m.
After rescaling to for all
m and n, the important edges can be identified in
by comparing the absolute values of

2,
k
Crmn

,
k
Wcm n

,
k
Wcm n

2k
Cr
and . If

k
Wc m
,n




2,Crmn
,
kk
Wc mn
e
at
the point , we retain the value of
at the point
. We use a new
matrix name
,n to represent the retained value,
,.n
Making comparison at th
1
,nn



,
km n
d ne


k
new
Wmn

2
n

k
w
W
,Wc
Wc
i.e.
12
,nnn

k
m
m

scale m for all points
 
12 12
,1,nnn nnN

,mn

2,
k
Crmn


, we
identify which represents the most important
features of the image edges. We set the values of
and to 0’s at the positions
identied and thus obtain a new set of and

,
k
new
Wmn

,m n


2
k
Cr

k

k
Wc
,Wcm n labelled as 2,Crmn
k and


,
k
Wc .
Next we go to rescale and compare
m n

2,
k
Crm n
with


,
k
Wcm n and extract the wavelet transform coeffcients
that correspond to the next most important features of the
image edges. We repeat this process until all major image
Figure 1 shows
edges are acquired.
the effect of this wavelet transform
ltering method at the scale m = 3 and

327
k
mm
.
In this gure, (a) is the original Lena ima
image containing Gaussian white noise with the standard
deviation 35
ge, (b) is noised
.


13,Wcn and


23,Wcn are
given in Fic)ectivel

gures 1( and (d), respy.



11
3, 4,WcnWcn and


are shown in Figures 1 (f). Figures 1(g) and


2
4, n
2
3,Wcn Wc
(e) and (h)
present


13,
new
Wn and


we acquire the fil-
te
1
23,
new
Wn.
By thentioneh, above med approac
red dyadic wavelet transform data


k
1
new mJ
W

12
,n, we can rec
. Let




on-


k
kk d



2
,,
new J
WJnWJnWfn
struct the ltered image from the set

 
12
,, ,,
J
newnew 21
j
J
Wj  nWjnSnwhere
Figure 1. The effect of the new wavelet filtering at the scale
m = 3 and

327.

k
mm (a) The original Lena image; (b)
the noised

image; (c) W
(
; (h)

23
new
Wn

13,cn; (d)


23,Wcn; (e)




11
3, 4,WcnWcn ;


3, nW; (g)

,.
f)

Wc

2
4,c n
2

13,
new
Wn
Copyright © 2011 SciRes. JSIP
A Dyadic Wavelet Filtering Method for 2-D Image Denoising
Copyright © 2011 SciRes. JSIP
311
thugh the inverse dyadic wav-
elet transform. The fil
Transform Filtering Algorithm for
the discrete wavelet transform of
no




12
2
11
,,
j
k
kd
j
Jj
Wjn Wfnn

12
22
,
JJ
SnS nn
dfro
J

and
12
22
,
JJ
d
SnSfnn
tering algorithm is summarized as
follows.
Wavelet Step 2. Initialize:






,,
kkk
Wbm nWcmnWmn,,
Image Denoising
Step 1. Compute


,
k
N
N
Filterm nO
 
kk
ised image

12
,
f
xx and its the lower-frequency sm-
oothed image:
 


2, ,1,,1
k
CrmnWc mnWcmnmJ1

for iteration
{
Loop for the scale m
Step 3. Loop

11mJ

{






2
12
,
max2 ,
kk
nnn N
M
crmCrm n









,,2,
kkk k
M
wc mWc mwhereCrmMcr m


Loop for each pixel point

12
,nnn
{



 

2
kk
Cr b, 2,kkk
m
m nCrm nMwcMcrm

end loop n }
Loop for each pixel point
nn

12
,n
{
If




2, ,
kk
CrbmnWcm n
0
e m
for the pixel


2,
k
Crmn


,0
k
Wcm n


k
Filterm n,1
end if
}end loop n
}end loop m
}end iteration
Scale space ltering:
Loop for the scal
{
Loop

12
,nnn
{

k


,
velet coefficient at the maxi-
m


,,
kk
new
WmnFilterm n

Wbmn
}
end loop n
}
end loop m
Step 4. Compute the wa
um scale
J
:




,,
kk
new
WJnWJn
Step 5. Reconstruct the image fromed wavelet
da
filter
ta



k
1
,
new
j
J
Wjn
 and

Sn
2J
The reco the set nstruction from

 

k
21
,,J
new
j
J
WjnSn
 through the inverse dyadic
wavelet transform will yi
inverse dyadic wavelet transform that we implemented in
our technique uses a trous algorithm and the quadratic
spline scaling functions and wavelets given in [18].
Now we give some comments on the choice of the
number of iterations and weight parameter

k
m
. We can
design wavelet ltering iteration times and parameter

k
m
according to the user’s request. For thumber of
iterations, when it is too small, we can not obtain a
oth estimate. If the number of iterations is too large,
most of the edge information of reconstructed image will
e n
smo
eld the final filtered image. The
A Dyadic Wavelet Filtering Method for 2-D Image Denoising
312
be eliminated. Thus we should choose a tradeoff between
the number of iterations and the estimation of filtered
image. It is well known that the Lipschitz exponents of
image and noise are different. At the finer scales such as
1
2 and 2
2, the modulus maxima mainly produced by
noise, while at coarser scale, most modulus maxima
duced image. So if we set the different value of

k
m
pro by
at the different scale m, noise will be eliminated
more effectively. In general, let parameter

k
m
be lager
e larger scale.
4. Experimental Results
at th
We use Peak-Signal-Noise-Ratio
Square-Error (RMSE) to evalua
(PSNR) and Root-Mean-
te restored results. PSNR
and RMSE is defined by the following:


2
255
,10log10
1
PSNRu w2
,,
,
ijij
ij
uw
mn
(14)


2
,,
,
1
,ijij
ij
RMSEuwuw
mn

, (15)
where and denotes the pixel valu
processed and the original images respectively. Hard-
ldin sof
entioned Lena im-
ag
presses more noise while preserves
m
sian noise with the standard
de
,ij
w,ij
ues of the
threshog and t-thresholding are widely used for
denoising in image processing by many researchers.
Therefore, in the following tests the hard-thresholding
method, soft-thresholding method, Xu’s method and
Zheng’s method will be used to compare with the dyadic
wavelet transform ltering algorithm.
Example 1: When we use our filtering method to do
denoising experiment for the above m
e corrupted by additive noise, the restored result is
Figure 2(a). If apply softthreshold, hard-threshold, Xu’s
method and Zheng’s method to filter the noised image,
the result is in Figures 2(b)-(e).
Table 1 presents the values of PSNR and RMSE for
each of the schemes.
From all the five restored images, it is clear that our
proposed method sup
ore fine details and small structures in the image. In
addition, from the values of PSNR and RMSE for res-
tored image, our method increases the PSNR by 1 - 3 dB
and reduce the RMSE 5 - 6.
Example 2: A texture image is used in the second test.
It is added by the white Gaus
viation 30
. Use our ltering method, soft-thr-
esholding method, hard-thresholding method, Xu’s met-
hod and Zmethod to process noised image, we
can find that our method is also better than other four
methods. Figure 3(a) is the original image, Figure 3(b)
is the noised image. Figures 4(a)-(e) show the results
50
100
200
150
250
25020015010050
aOur method.
50
100
200
150
250
25020015010050
bSoft-threshold
50
100
200
150
250
250
200
150
100
50
cHard-threshold.
50
100
200
150
250
250
200
150
10050
dXu’s Method.
50
100
200
150
250
250
200
150
100
50
eZheng’s Method.
heng’s
processed by the five methods.
Figure 2. Filtered results. (a) Our method; (b) Soft-threshold;
(c) Hard-threshold; (d) Xu’s method; (e) Zheng’s method.
Table 2 is the comparison of the values of PSNR and
RMSE for restored images.
Copyright © 2011 SciRes. JSIP
A Dyadic Wavelet Filtering Method for 2-D Image Denoising313
Table 1. PSNR and RMSE for each of the schemes.
threshold method method
Method Our
method
Soft-
threshold
Hard- Xu’s Zheng’s
PSNR 25.2035 dB 23.1125 dB 22.4597 dB 24.0583 dB23.1079 dB
RMSE 14.6 17.19.15.0078203 2111 9818 17.8297
50
100
200
150
250
250
200
150
100
50
aOriginal image.
50
100
200
150
250
250200150100
50
bNoised image.
Figure 3. Texture image. (a) Original image; (b) Noised
image.
50
100
200
150
250
25200150100
50
aOur method.
0
50
100
200
150
250
250200150100
50
bSoft-threshold.
50
100
200
150
250
250200150100
50
cHard-threshold.
50
100
200
150
250
250
200150
100
50
dXu’s Method.
50
100
200
150
250
250
200150
100
50
eZheng’s Method.
Figure 4. Results for the texture image. (a) Our method; (b)
Soft-threshold; (c) Hard-threshold; (d) Xu’s method; (e)
Zheng’s method.
od
Soft-thresh
old
Hard-thresh
old
Xu’s
method
Zheng’s
method
Table 2. PSNR and RMSE for restored images.
Method Our meth
PSN74 dBR 24.1536 dB 22.5607 dB 22.5533 dB 22.3974 dB22.77
From ts ot
visual
ti r
pr
he results above, it is obviouthat nonly for
quality of images, but
u
also for quantitative evalua-
rdon of
ocessi
esimtored ages, o metho tuin texre ime ag
ng is still better than other four methods.
Example 3: We use a man image containing both a
human face and some textures to do the third test. The
challenge with this image is to keep both texture details
and smooth transitions in the human face in the process-
ing. We add the original image (Figure 5(a)) with the
white Gaussian noise with σ = 30, and get a noised image
(Figure 5(b)). Figures 6(a)-(e) are the results obtained
by five methods
50
50
100
200
150
250 250
200150
100
50
aOriginal image.
100
200
150
250
250
200150
100
50
bNoisy image.
Figure 5. Man image. (a) Original image; (b) Noisy image.
50
100
200
150
250
25
200150
100
50
aOur method.
0
50
100
200
150
250
250
200150
100
50
bSoft-threshold
50
100
200
150
250
250
200150
100
50
cHard-thres hold.
50
100
200
150
250
250
200150
100
50
dXu’s Method.
50
100
200
150
250
250
200150
100
50
eZheng’s Method.
Figure 6. Results for man image. (a) Our method; (b) Soft-
threshold; (c) Hard-theshold; (d) Xu’s method; (e) Zheng’s
method.
RMSE 15.8075 18.9892 18.9966 19.3488 18.5212
Copyright © 2011 SciRes. JSIP
A Dyadic Wavelet Filtering Method for 2-D Image Denoising
314
Table 3 is the quantitative comparison among the ve
methods.
The results above reveal that our method not only
maintain more texture details and smooth transitions in
the face but also suppress more noise than other methods
after processing. Additionally, our method can increase
more PSNR and decrease RESE than other methods.
At last, we give some other types of images. And we
only present results recovered by our mthod. A MR
w
aintain all important information and lter out
m
wavelet transform ltering
method method
e
image (see Figure 7(a)) has been corrupted with white
Gaussian noise (σ = 20) and become a noised image, see
Figure 7(b). After the noised image has been processed
ith our method, we can see that our restoration scheme
is able to m
uch noise, see Figure 7(c).
Figures 8(a), (b) and (c) are an original building im-
age, the noised image, and the processed result by our
method. We can see that the recovered image can pre-
serve more image edge details.
A ngerprint image is used in the last test. Figures
9(a)-(c) are the original image, the noised image with σ =
15, and the recovered result with our scheme. From the
visual quality, it is obvious that restored image is as good
as the original one.
5. Conclusions
We have introduced the dyadic
Table 3. Comparison of PSNR and RMSE for res tored ima ges.
Method Our
method Soft-threshold Hard-threshold Xu’s Zheng’s
PSNR 25.7523
dB 23.8359 dB 23.5585 dB 24.4842
dB
23.7759
dB
RMSE 13.1500 16.3963 16.9284 15.217116.5099
50
100
200
150
250
50 250200150
ginal image.
100
aOri
50
100
200
150
250
10050
bNoised
250200
= 20.
150
image with σ
50
100
200
150
250
25020015010050
cThe result recovered with our scheme.
50
100
200
150
250
25020015010050
aOriginal image.
50
100
200
150
250
25020015010050
bNoised image with σ = 30.
50
100
200
150
250
250
200150
100
50
im-
age with σ = 30; (c) The result recovered with our scheme.
cThe result recovered with our scheme.
Figure 8. Building image. (a) Original image; (b) Noisy
50
100
200
150
250
25020015010050
aOriginal image.
50
100
200
150
250
25020015010050
bNoised image with σ = 15.
50
100
200
150
250
250
200150
100
50
cThe result recovered with our scheme.
Figure 9. Fingerprint image. (a) Original image; (b) Noised
image with σ = 15; (c) The result recovered with our
scheme.
technique for denoising in image processing. Our ltering
algorithm is superior to soft-thresholding method, hard-
thresholding method, Xu’s method and Zheng’s method
because important edge features in the wavelet transform
domain are preserved while much noise is suppressed.
The other filtering methods perform very poorly in image
denoising because they tends to remove the high-fre-
quency component exclusively, which yields smooth im-
ages and blurs the image edge features.
Figure 7. MR image. (a) Original image; (b) Noised image
with σ = 20; (c) The result recovered with our scheme.
Copyright © 2011 SciRes. JSIP
A Dyadic Wavelet Filtering Method for 2-D Image Denoising
Copyright © 2011 SciRes. JSIP
315
6. Acknowledgements
This work was supported by the Key Project of Ch
Ministry of education (No. 109030), 382 Training Pr
amme of CUC(G08382316), and Science Research Pro-
ject of Communication University of China(XNL1003).
The authors would like to thank the reviewers for ve
useful comments, which improved the manuscript sig-
nificantly.
tiscale Edges,” IEEE Transactions on Pattern Analy-
inese
ogr-
ry
REFERENCES
[1] S. Mallat and S. Zhong, “Characterization of Signal from
Mul
sis and Machine Intelligence, Vol. 14, No. 7, 1992, pp.
710-732. doi:10.1109/34.142909
[2] S. Mallat and W. L. Hwang, “Singularity Detection and
Processing with Wavelets,” IEEE Transactions on Infor-
mation Theory, Vol. 38, No. 2, 1992, pp. 617-643.
doi:10.1109/18.119727
[3] J. L. Starck and A. Bijaoui, Filtering and Deconvolution
by the Wavelet Transform,” Signal Processing, Vol. 35,
No. 3, 1994, pp. 195-211.
doi:10.1016/0165-1684(94)90211-9
[4] A. Bijaoui, “Wavelets, Ga
Signal Processing, Vol. 82
ussian and Wiener Filter
, No. 4, 2002, pp. 709-712.
ing,”
doi:10.1016/S0165-1684(02)00137-8
[5] P. L. Shui, “Image Denoising Algorithm via Best Wavelet
Packet Base Using Wiener Cost Function,” Institution of
Engineering and Technology Image Processing, Vol. 1,
No. 3, 2007, pp. 311-318.
[6] Y. Xu, J. B. Weaver, et al., “Wavelet Transform Domain
Filters: A Spatially Selective Noise Filtration Technique,”
IEEE Transactions on Image Processing, Vol. 3, No. 6,
1994, pp. 747-758.
[7] C. Okechukwctive Noise Filtra-
Image Denois
eucom.2008.04.016
u and C. Ugweje, “Sele
tion of Image Signals Using Wavelet Transform,” Imag-
ing Measurement Systems, Vol. 36, No. 3-4, 2004, pp.
279-287.
[8] M. Nasri and H. Nezamabadi-pour, “ing in
the Wavelet Domain Using a New Adaptive Thresholding
Function,” Neurocomputing, Vol. 72, No. 4-6, 2009, pp.
1012-1025. doi:10.1016/j.n
8, 2000, pp. 1535-1549.
[9] Y. Zheng, D. B. H. Tay, et al., “Signal Extraction and
Power Spectrum Estimation Using Wavelet Transform
Scale Space Filtering and Bayes Shrinkage,” Signal Proc-
essing, Vol. 80, No.
doi:10.1016/S0165-1684(00)00054-2
[10] Y. Leung, J. S. Zhang and Z. B. Xu, “Clustering by
Scale-Space Filtering,” IEEE Transaction on Pattern An-
alysis and Machine Intelligence, Vol. 22, No. 12, 2000,
, pp. 446-
pp. 1396-1410.
[11] J. Kalif, S. Mallat and B. Rouge, “Deonvolution by
Thresholding in Mirror Wavelet Bases,” IEEE Transac-
tion on Image Processing, Vol. 12, No. 4, 2003
457. doi:10.1109/TIP.2003.810592
[12] Y. F. Zheng and R. L. Ewing, “Feature-Based Wavelet
Shrinkage Algorithm for Image Denoising,” IEEE Trans-
action on Image Processing, Vol. 14, No. 1
2024-2039.
2, 2005, pp.
y and Machine, Vol. 37, No. 4, 2007, pp.
[13] Z. Y. Chen, X. P. Guo, X. L. Zhang, W. J. Cram and Z.
W. Li, “A Novel Method for Analysis of Single Ion
Channel Signal Based on Wavelet Transform,” Com-
puters in Biolog
559-562. doi:10.1016/j.compbiomed.2006.08.006
[14] D. L. Donoho and I. M. Johnstone, “Ideal Spatial Adap-
tion via Wavelet Shrinkage,” Biometrika, Vol. 81, No. 3,
1994, pp. 425-435. doi:10.1093/biomet/81.3.425
[15] D. L. Donoho, “De-Noising by Soft-Thresholding,” IEEE
Transaction on Information Theory, Vol. 41, No. 3, 1995,
pp. 613-627. doi:10.1109/18.382009
[16] M. Holschneider, R. Kronland-Martinet, J. Morlet and P.
of 8th
Tchamitchian, “Wavelets, Time-Frequency Methods and
Phase Space, Chapter A Real-Time Algorithm for Signal
Analysis with the Help of the Wavelet Transform,”
Springer-Verlag, Berlin, 1989, pp. 289-297.
[17] A. Witkin, “Scale Space Filtering,” Proceedings
International Joint Conference on Artificial Intelligence,
Karlsruhe, 1983, pp. 1019-1022.
[18] S. Mallat, “A Wavelet Tour of Signal Processing,” 2nd
Edition, Academic Press, New York, 1999.