 Open Access
 Total Downloads : 1125
 Authors : I.Muthulakshmi, Dr. G. Gnanadurai
 Paper ID : IJERTV1IS3092
 Volume & Issue : Volume 01, Issue 03 (May 2012)
 Published (First Online): 30052012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Efficient MBand Wavelet Based Inpainting Technique to Detect and Impound the Distorted Digital Images
1I.Muthulakshmi and 2Dr. D. Gnanadurai
1Assistant Professor / HOD, CSE Department,
VV College of Engineering, VV Nagar, Tisaiyanvilai 627 657 Tuticorin District.
muthulakshmiphd@gmail.com
2Principal, J.P College of Engineering, Ayikudy, Tenkaasi – 627 852, Tirunelveli District
Abstract
Image in painting or completion is a technique to restore a damaged image. In this paper M band complex wavelet transform is used for frequency domain conversion of the image subsequently using the iterative shrinkage technique, inpainting process of the cracked image is carried out successfully. In the proposed approach Mband dual tree wavelet transform which decompose the each input wavelets into set of subbands with each sub band wavelets occupying a portion of the original frequency band and hence produced better frequency analysis for image in painting process. Each sub bands and its coefficients preferentially will be captured different directions and hence it will be detected cracks in different direction. The proposed technique shows better performance than the conventional wavelet based methods. The performance of the proposed approach is evaluated and analyzed by the various cracked images.
Keywords: In painting, wavelets, DWT, Haar, Daubechies, CWT, 2D Dual tree Complex Wavelet transform

Introduction
Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting is numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects [7]. Image inpainting [1, 5] provides a means to restore damaged region of an image, such that the image looks complete and natural after the inpainting process. Applications of image inpainting range from restoration of photographs, films and paintings, to removal of occlusions, such as text, subtitles, stamps and publicity from images. In addition, inpainting can also be used to produce special effects [8]. Traditionally, skilled artists
have performed image inpainting manually. But given its range of applications, it would be desirable to have image inpainting as a standard feature of popular image tools such as PhotoShop. Bertalmio et al [6] have introduced a technique for digital inpainting of still images that produces very impressive results [8]. Digital techniques are starting to be a widespread way of performing inpainting, ranging from attempts to fully automatic detection and removal of scratches in film [3, 19], all the way to software tools that allow a sophisticated but mostly manual process [4].
Cracks usually have low brightness and therefore it is considered as local intensity minima [2]. Inpainting is a technique, used for altering an image in an undetectable form. The main intention of inpainting is the restoration of damaged paintings and photographs for the purging of selected objects [9]. Inpainting is the process of recreating lost or damaged portions of images and videos [15]. Inpainting is an image interpolation technique [16]. In the mathematical field of numerical analysis, interpolation is a technique of creating new data points within the range of a discrete set of known data points [17].
For the crack detection and analysis, several techniques such as neural network, wavelet transform, grid cell analysis (GCA), genetic algorithm (GA), artificial life (AL), fuzzy set theory, texture classification and more has been employed [10]. Wavelet is a promising method, very useful for the detection of structural damages [13]. The 2D discrete wavelet transformation is applied to the model of digital image data in order to find the locality and length of the crack [18]. In mathematics, a wavelet series is a depiction of a squareintegrable (real or complexvalued) function by a certain orthonormal series created by a wavelet [11]. The wavelet transform itself gives great design flexibility. Basis selection, spatialfrequency tiling, and different wavelet threshold approaches can be optimized to achieve best adaptation
for processing application, data characteristics and feature of interest [12]. In wavelet packet transform, the data is transformed using a far more comprehensive range of spacefrequency analysis functions, which is expected to mine more information of interest [20].
The structure of the paper is organized as follows: A brief review of the researches related to image inpainting is discussed in Section 2. The proposed wavelet transform based image inpainting is given in Section 3. The experimental results of the proposed approach are presented in Section 4. Finally, the conclusion is given in Section 5.

Related Work
Gunamani Jena [21] has presented an inpainting algorithm, which implements the filling of damaged region with impressive results. Many algorithms usually required several minutes on current personal computers for the inpainting of relatively small areas. Such a time is unacceptable for interactive sessions and motivated us to design a simpler and faster algorithm capable of producing similar results in just a few seconds. The results produced by the algorithm are two to three orders of magnitude faster to the existing.

A. Ismail et al. [22] have proposed an integrated technique for the recognition and purging of cracks on digitized images. Using steepest descent algorithm (SDA), initially the cracks have been identified. Then, the identified cracks have been purged using either a gradient Function (GRF) and processed data or a semi
defected area. Then, a fast searching algorithm which uses feature extraction parameters has been proposed to find the defected pixels and to robustly segment it. Their proposed method was appropriate for both texture and non texture images. Consequently, the algorithm has successfully detected the damage in the digital texture image using non texture methods.

Rupil et al. [25] have introduced a digital image correlation technique for recognizing and calculating automatically the micro cracks on the surface of a specimen during a fatigue test. The technique has allowed a quick scanning of the entire surface with all possible (pixelwise) locations of micro crack centers and the detection of cracks containing a subpixel opening. An experimental test case has been presented as a design of the method and a comparison has been conducted with a replica technique
YANG Jianbin et al [14] used dualtree complex wavelet transform tool in signal and image processing. This paper proposed a dualtree complex wavelet transform (CWT) based algorithm for image inpainting problem. The approach is based on Cai, Chan, Shen and Shens frameletbased algorithm. The complex wavelet transform outperforms the standard real wavelet transform in the sense of shiftinvariance, directionality and antialiasing. Numerical results illustrate the good performance of algorithm.


Wavelets Based Image Inpainting
Let a be an image in the domain D
automatic procedure based on region growing. Lastly, crack filling has been performed using the steepest
a { aij ;1 i P,1 i Q }
(1)
descent method. The proposed technique has been
And the
a' be known, observed region and D is
implemented using Matlab, Surfer and Visual Fortran
programming. Experimental results have shown that their technique has performed effectively on digitized
the inpainting domain. The intensity alue
(ai ) 0 (i) (i)
(2)
images suffering from cracks.
Dayal R. Parhi and Sasanka Choudhury [23] have conducted a comprehensive review of several techniques in the field of crack detection in BeamLike Structure. Sensibility analysis of experimentally measured frequencies as a decisive factor for crack identification has been employed widely in the last few decades because of its straightforwardness. But, the determination of crack parameters such as depth and location is complicated. Several techniques have been discussed on the basis of dynamic analysis of Crack. The techniques mostly used for crack detection were fuzzy logic neural network, fuzzy system, hybrid neuro genetic algorithm, artificial neural network, artificial intelligence.
K.N.Sivabala and D.Gananadurai [24] have utilized Gabor filter and Gaussian filter in order to remove the texture elements in the digital image by separating the
in the domain D where is the noise term. The
proposed system finds an image b that matches 0 in D and have meaningful content in the domain D since the value of (ai ) is arbitrary when iD . The proposed system consists of the following steps (a) Initial value assignment, (b) Converting to frequency domain (c) coefficients thresholding , (d) Reconstruction , (e) Iterative image inpainting.

Initial Value Assignment Using Nearest Neighbor Algorithm
Initially the closest entries of a' are identified and replaced using nearest neighbor algorithm. The selection of closest entries can be realized in two methods, first, as is, on the set of entities, and, second by considering only entities with non missing entries in
the attribute corresponding to that of targets missing entry. The proposed system uses the second approach for initial assignment of the damaged portion. The following procedure represents the nearest neighbor algorithm.
m (w)  i sign(w)(w) ,
{m{1,….M 1}
The sign is the signum function and d
designates
Procedure 1: Nearest Neighbor Algorithm
the fourier transform of a function d. The Hilbert condition (4) yields
{m{1,….M 1} H (w)  
(w) 
(5)
m m
The scaling equation leads to
m m
{m{1,….M 1} G (w) eim(w) H (w)
(6)
Where
m is
2 periodic. The frequency phase
functions should also be odd (for real filters) and thus
only need to determined over
[0, ], In the 2 bandcase (under weak assumptions) m
is a linear function

Conversion of Image to Frequency Domain By Means of Wavelet
The proposed system uses the Mband Complex 2 D
on [ , ]. In the M band the constraint is slightly restricted on a smaller interval by imposing
{w [0,2 / M ],0 (w) w where R . It can be
deduced that, Para unitary M band filter bank conditions are obtained by choosing the phase functions defined by
Dual tree wavelet t transform
which posses the unique geometrical features for
{p{0,….[ M ] 1}, w[ pM 2 , ( p 1) 2 ], (d 1 )(M 1)w p , 2 M M 0 2
1
(7)
frequency domain conversion. This decomposition
{m {1….., M 1},
(w) 2 (d 2)w if w [0,2 ],
provides local, multi scale directional analysis. The wavelet transform is self possessed of cascading M band filter banks. The Mband trees are obtained by performing two Mband multi resolution analyses in
Where
m
0
if w 0
(8)
parallel in the real case , or four in the complex case. The dual tree decompositions are shift variant , with each trend keeping the same characteristics when the data is delayed. Different sub bands and two sets of
coefficients preferentially capture different directions.
Where d Z denotes the upper integer part of real
u. The scaling function associated to the dual wavelet composition is such that
1(d 1 )w
{k N , w [2k , 2(k 1) ,H (w) (1) k e 2 . (w)
The Mband biorthogonal wavelet decomposition of L2 (R) is based on the joint use of two sets of basic functions 0 m M , m m M which satisfy the following scaling equations expressed in the frequency domain.
0 0 (9)
Find that except in the 2 band case 0 exhibits discontinuities on 0, due to the p term.
The two dimensional separable Mband wavelets bases can be derived from the 1 D dual tree decomposition. Thus we obtain two bases of L2 (R2 ) . The
m
M1/ 2
m
M1/ 2)
(M ) 0
(M ) 0
( )0 ( )
( )0 ( )
(3)
(4)
first one corresponds to the classical 2d separable
wavelet basis but the second one results from the tensor product of the dual wavelet basis function. A discrete implementation of these wavelet decompositions starts
Here 0 is the father wavelet and m are mother
from level j=1 to go up to the coarsest resolution level
jN * . The decomposition on to the former 2D wavelet
wavelets. m {1,……M 1} which defined a dual
[k, l]
Mband multi resolution analysis. Specifically the
basis function yields coefficients
j,m,m' ,
mother wavelets will be obtained by Hilbert transform. In the Fourier domain the desired property reads,
whereas the decomposition on to the dual basis
generates coefficients
H
j,m,m'
[k, l]m
m
The wavelet transform is a continuousspace formalism which is applied to the discrete image. The analog scene corresponds to the 2D field
k (t)
m (t) 21/ 2
i H(t)
(21)
f ( p, q) f (g,l) x( p g, q l) , (10)
k (t) m (t) i H(t)
g ,l
m 21/ 2 m
(22)
Here the x is the interpolation functions and
The tensor product of the two analytic wavelets
f (g, l) ( g ,l )
is the image sample sequence. The
k
and H
image is project on to the approximation space
m
m '
And the real part of the tensor product is
V span{ ( p g)
(q l)(k, l) Z 2 }
a k H
0 0 0
. (11)
m,m' (x, y) Re{ m ( p) m' (q)
(23)
The projection of f reads
EV0 ( f ( p, q) 0,0,0 [k,l] ( p q) (q l)
(12)
For
m, m' {1,……M 1}2
the Fourier transform
k ,l
of this function is equal to
Where the approximation coefficients are
a
p ( p)m' ( y ) if sign( x ) sign( y),
( , )
(24)
0,0,0 [g,l] f ( y, z)x , 0,0 (k y,l z)
(13)
m,m'
p q
0
if sign( x ) sign( y),
([ p, q)
( p) (q) and ,
(14)
The above function allows us to extract the
Where
0,0
0 0 x
0,0
directions that falling in the first third quadrant of the
Is the crosscorrelation function defined as
x , 0,0 x(u, v) 0,0 (u p)(v q)dudv
(15)
frequency plane. Like wise the real part of the tensor product of an analytic wavelet and anti analytic one is
a
denoted by m,m' . This function is used to select the
Similarly the analog image is projected on to the
dual approximation space
V H span{ H ( p g, q l), (k, l) Z 2 }
frequency components which are localized in the second /fourth quadrant of the frequency plane. This
corresponds to opposite directions to those obtained
o 0,0
H ([ p, q) H ( p) H (q)
(16)
a
'
0
Where
0 , 0
0 (17)
with
m,m
Then the dual approximation coefficients are given
by
m, m' with m 0 and m ' 0 , the directional analysis
H [g,l]
f ( y, z) ,
(k y, l z)
is achieved by computing the coefficients
0,0,0
x 0,0
(18)
1 x y
M
a
Obviously Eq.(13) and (18) can be interpreted as the use of two of pre filters on the discrete image
Cr,m,m' [k,l]( f ( p, q),
M
r m,m' (
r k,
M r l))
(25)
f (g, l)
( g ,l )
before the dual tree decomposition and
Cr,m,m' [k,l] ( f ( p, q),
1 a (
M r m,m '

k,
M r

l))
M r
(26)
the frequency responses of these filters are According to equation (21), (22) and (23) for all
* m, m' {1,…..M 1}2
H1( x, y ) s(wx 2 y , q 2z )0 ( q 2Z )
u v
(19)
1 H
H 2 (
p , q
) ei(d 1/ 2) p , q H1(
p , q )
(20)
Cr,m,m' [k,l] r,m,m' [k,l]
2
r ,m, m'
[k,l](27)
Different kinds of interpolation function may be considered, for instance the separable functions of the
x( p, q) ( p)(q).
C H
r ,m, m'
[k, l] r,m,m' [k, l]1
2
H
r , m, m'
[k, l](28)
form the two pre filters are then
separable with the impulse responses 3.2.2. Coefficients Thresholding
, ( p) ,
(q)
, H (y) , H (z)
Initially the diagonal matrix D
is obtained as
0 , 0
and
o , o
respectively.
3.2.1. Direction Extraction in the Different Sub
follows.
1 if
D ij
aij
'
(29)
Bands
Some linear combinations of the primal and dual sub bands are used to extract the local directions present in the image. The defined analytic wavelets for direction sub bands are
0 if aij
Subsequently the initial guess of the original image is done. by using the For n=1,2,.
n
l
f * Shrink (
, )
. By using the shrinkage
band wavelet transforms and
F1' , F 2'
and
procedures as in [14] are carried out for all the M
( (F1' F1 F 2' F 2)1 correspond to filtering with
bands of 2DCWT coefficients. As follows
frequency responses.
 (F1* ( ) 2 ,  (F 2* (
) 2
1 p q 1 p q
0 if l 
and ( F1 (
) 2  F1 ( ) 2 )1 respectively.
shrink(u,)  l  .l
if l '
(30)
1 p q
2 p q
[l]
Where l is the given intensity. And then the iterative algorithm


Experimental Results
The proposed image inpainting system is
ln1
Dl (I D) fl
(31)
implemented in MATALB platform (version 7.10) and
it is evaluated using the various images. Also the
is repeated until the n convergence. Using [25] ,if
l * (i) 0 for every values is the output of (35) then
I of (1), then it will be the solution of the
interpolation problem. Otherwise the solution
performance of the proposed wavelet based inpainting system is tested and analyzed by increasing the crack level. The (a),(b),(c) of Figure 1,Figure 3, Figure 5,Figure 7 and Figure 9 represents the three levels of cracked images and (d),(e),(f) of those images
l*
l * * Shrink (
,)
will be the denoising and
represents the inpainted images using the proposed
technique. The performance of the proposed technique
interpolation problem.
3.2.3 Reconstruction
f
is analyzed quantitatively by using the metrics Peak
Signal to Noise Ratio(PSNR) and standard deviation to mean ratio(S/M).
The performance of the proposed technique is also
Let
be the vector of image samples,
the
evaluated by comparing it with the inpainting
vector of coefficients produced by the primal M band
H
techniques using the wavelets DWT, Haar, Daubechies,
and CWT based technique. The table1,2 and 3
decomposition and
be the vector of coefficients
represents the psnr values of the inpainted images and
produced by dual one. The global decomposition operator is
1
C D f
D : f
evaluation values. The Figure.11 and Figure. 12
represents the PSNR mean ratio comparison graph of the proposed technique with the other inpainting techniques using the comparison wavelets. Like wise
C H D f
(32)
the Figure.13 represents the S/M comparison graph.
2
Where
D1 U1 F1
and
D2 U 2 F2
F1 and
F 2 being the pre filtering operations and U1 and
g ,lZ
U 2 be the orthogonal m band decomposition then the following can be proved . Assume that x( p g, q l) 2 is an orthonormal family of
L2 (R2 ).
Provided that there exist
I J I (R* )3 for almost all [ , ]2 ,
e e 0 x y
 x( x , y )  < Ie , ( x )  A 0
(33)
 x( p
( p,q)(0,0)
2 y , q
2z ) 2 J
I I
2 4
x 0
(34)
Figure 1: The cracked and inpainted image1(Proposed Approach)
x
The D is the frame operator. The dual frame
reconstruction operator is given by
I (F1' F1 F 2' F 2)1 (F1'U11 F 2'U 21 H )
(35)
Where
F1'
designates the adjoint of an operator
F1 . The formula (32) minimizes the impact of possible errors in the computation of the wavelet
coefficients.
U11
and
U 21
are the inverse of M
Figure 2: In painted output images using various comparison wavelets for image1.
Figure 3: The cracked and inpainted image2(Proposed Approach)
Figure 4: In painted output images using various comparison wavelets for image2.
Figure 5: The cracked and inpainted image3(Proposed Approach)
Figure 6: In painted output images using various comparison wavelets for image3.
Figure 7: The cracked and inpainted image4(Proposed Approach)
Figure 8: In painted output images using various comparison wavelets for image4.
Figure 9: The cracked and inpainted image5(Proposed Approach)
Figure 10: In painted output images using various comparison wavelets for image5.
Table 1: Performance comparison table_1(PSNR)
Image1
Image2
Image3
Image4
Image5
Total
Average
Standard deviation
S/M
DWT
15.682659
17.38496
17.204759
16.78396
19.2851
86.34143
17.26829
1.307092
0.075693
Haar
14.702661
16.87209
17.003936
16.52693
18.72821
83.83382
16.76676
1.43463
0.085564
Daubechies
14.74743
17.06256
17.133125
16.59035
18.96635
84.49982
16.89996
1.506654
0.089151
CWT
15.528204
17.34496
17.211023
16.71531
19.22632
86.02581
17.20516
1.337609
0.077745
Proposed
15.668087
17.365
17.218906
16.77732
19.23909
86.26859
17.25372
1.29388
0.074991
Table 2: Performance comparison table_2(PSNR)
Image1
Image2
Image3
Image4
Image5
Total
Average
Standard deviation
S/M
DWT
12.31413
13.62292
15.3487
14.82145
15.72731
71.83451
14.3669
1.395405
0.097126
Haar
11.84051
13.33268
15.2288
14.66204
15.5623
70.62632
14.12526
1.534543
0.108638
Daubechies
11.86531
13.46868
15.29738
14.73502
15.62703
70.99343
14.19869
1.542118
0.10861
CWT
12.18224
13.58814
15.35559
14.80449
15.70939
71.63985
14.32797
1.444204
0.100796
Proposed
12.29484
13.61145
15.37255
14.82146
15.75517
71.85546
14.37109
1.415034
0.098464
Table 3: Performance comparison table_3(PSNR)
Image1
Image2
Image3
Image4
Image5
Total
Average
Standard deviation
S/M
DWT
11.38259
13.32602
14.08648
13.54998
12.84451
65.18958
13.03792
1.027402
0.078801
Haar
11.18111
13.05052
13.99956
13.38804
12.70727
64.32649
12.8653
1.055409
0.082035
Daubechies
11.16304
13.16882
14.04156
13.44157
12.81528
64.63029
12.92606
1.082766
0.083766
CWT
11.33501
13.28759
14.09376
13.48232
12.84611
65.04478
13.00896
1.037682
0.079767
Proposed
11.45881
13.30298
14.11342
13.52093
12.89258
65.28872
13.05774
0.99662
0.076324
The standard deviation values and S/M values shows the better result of the proposed approach.
Figure 11: Performance Comparison graph_1
Figure 12: Performance Compariosn Graph_2 From the table1, it is clear that, the proposed
approach has achieved( 0.01457,0.486954,0.353755,0.048556 PSNR values
than the DWT, Haar, Daubechies and CWT based inpainting techniques for crack level1. Like wise from table 2 and table3 illustrates that the proposed approach achieved (0.00419, 0.245828, 0.172406 and 0.043122)
and (0.019828, 0.192445, 0.131687 and 0.048789) for
crack level2 and crack level3 respectively. Also the Figure11 and Figure12 represents the higher performance of the proposed inpainting technique. Though the psnr value of proposed approach is little deviated than the dwt based approach for crack level1,

Conclusion
In this paper, 2D CWT_M band based iterative image inpainting approach was proposed. The approach was implemented and experimented with different images with various crack level also the proposed approach was compared with the various inpainting techniques with different wavelets. The analytical results confirmed that the proposed approach has shown a better performance than the other comparative wavelets based approaches. Overall, the proposed approach has achieved 0.032192
%,2.00223%,1.419495%, 0.318375% more PSNR
values than the traditional DWT , Haar, Daubechies and CWT based inpainting techniques (i.e) In the circumstance of achieving 100% performance by proposed approach, the other comparative wavelets based inpainting approaches are able to achieve only 99.97%, 98%, 98.59% 99.68% for DWT, Haar,
Daubechies and CWT) respectively. Such performance has been achieved because of the M band nature of 2d dual tree complex wavelet transform and its improved directional analysis as well as frequential analysis feature.
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