DIGITAL WATERMARKING TECHNIQUE BASED ON MULTI-RESOLUTION CURVELET TRANSFORM

International Journal in Foundations of Computer Science & Technology (IJFCST) Vol.7, No.2, March 2017
DOI:10.5121/ijfcst.2017.7201 1
DIGITAL WATERMARKING TECHNIQUE BASED ON
MULTI-RESOLUTION CURVELET TRANSFORM
Ranjeeta Kaushik1
, Sanjay Sharma2
and L. R. Raheja3
1
Department of Computer Science and Engineering, Thapar University, Patiala, Punjab,
India
2
Department of Electronics and Communication Engineering, Thapar University, Patiala,
Punjab, India
3
Department of Navel Engineering, IIT, Kharagpur, India.
ABSTRACT
In this paper an efficient & robust non-blind watermarking technique based on multi-resolution geometric
analysis named curvelet transform is proposed. Curvelet transform represent edges along curve much more
efficiently than the wavelet transform and other traditional transforms. The proposed algorithm of
embedding watermark in different scales in curvelet domain is implemented and the results are compared
using proper metric. The visual quality of watermarked image, efficiency of data hiding and the quality of
extracted watermark of curvelet domain embedding techniques with wavelet Domain at different number of
decomposition levels are compared. Experimental results show that embedding in curvelet domain yields
best visual quality in watermarked image, the quality of extracted watermark, robustness of the watermark
and the data hiding efficiency.
1. INTRODUCTION
Due to popularity of internet and its increasingly easy access to digital multimedia, a number of
powerful tools are available for editing of digital media without the loss of quality. So
authentication and the intellectual property right of digital media is an important issue i.e to
protect the intellectual property right (IPR) of the author (copyright protection). There are many
solutions to this problem such as digital signature and digital watermarking. In digital
watermarking we embed the watermark in the original work such that it preserves quality of
watermarked data. The watermark can later be extracted for the purpose of author’s identification
and integrity verification. The main goal of watermarking is to hide a message m in some audio
or video (cover) data d, to obtain new data d', practically indistinguishable from d, by people, in
such a way that an eavesdropper cannot remove or replace m in d'. In the present work the
existing techniques of digital watermarking are briefly described and compared with respect to
invisibility, efficiency and the quality of extracted watermark using appropriate metrics. The aim
of the digital watermarking is to embed some specific information about the author in the form of
text or image in to the original work without making it visible [1]. The purpose is not only to hide
the data but prevent any modification or removal of embedded watermark. The original image is
called as cover image and the output image is called as watermarked image In section II
watermark system evaluation metric are presented. Wavelet transforms and the two algorithms
one is embedding watermark in HL, LH frequencies and second is embedding in LL frequencies
are presented in session III. The curvelet transform and proposed algorithm are described in

2
section IV. The experimental results are given in section V. Section VI presents the comparison
of proposed algorithm with wavelet transform and finally section VII provides the conclusion
2. WATERMARK SYSTEM EVALUATION
A watermarking system has a number of requirements. Obviously, different applications have
different concerns; therefore, there is no set of properties that all watermarking systems have to
satisfy.
2.1 INVISIBILITY
Embedding extra information in the original signal will cause degradation and perceptual
distortion. [4]. The most common evaluation method is to compute the peak signal-to-noise ratio
(PSNR) between the host and watermarked signals. PSNR is defined as follows:
= 10 log (i)
More PSNR means better image quality.
Where MSE (Mean Square Error) is define as
= ∑ ( ( − ( 2
(ii)
Where Im = Original image, Iw= watermarked image
2.2 EFFECTIVENESS
Digital watermarking systems have a dependence on the input signal. Effectiveness refers to
whether it is possible to detect a watermark immediately following the embedding process [6].
Although 100% effectiveness is ideal, it is often not possible to achieve such a high rate.
The quality of extracted watermark can be accessed by using Normalized Correlation between the
original and extracted watermark. Normalized (NC) coefficients are defined as follow.
! =
∑ ∑ ("#$%& "#$%'''''''''() ("*$%&"*$%'''''''''
+∑ ∑ ("#$%& "#$%
''''''''''' ("*$%& "*$%
''''''''''%$
(iii)
Where Wo-. is the original watermark image with m*n size, Wx-. is the extracted watermark
image with m*n size.
Wo-.
'''''''' , Wx-.
''''''''Are the means of the original watermark and extracted watermark images
respectively?
2.3 EFFICIENCY
Efficiency refers to the embedding capacity. For images, it is usually expressed in bits of
information per pixel (bpp). The desired size of the watermark is application dependent. Although
it is possible to embed as much information as one wants, the sacrifice is often the signal quality

3
or the invisibility. A common evaluation method of the watermarking system is to plot PSNR vs.
Efficiency, where one can examine both properties together.
2.4 ROBUSTNESS
Robustness refers to the ability for the detector to detect the watermark after signal distortion,
such as format conversion, introduction of transmission channel noise and distortion due to
channel gains
2.5 SECURITY
One of the major goals of a digital watermarking system is to protect digital content from illegal
use and distribution. However, the protection is diminished if the attackers can estimate, remove,
or insert a watermark [6]. Extraction processes such as Pubic, Semi Build, and Build.
3. WAVELET TRANSFORM
3.1 CONTINUOUS WAVELET TRANSFORMATION (CWT)
It is defined as
CWT
ψ
x
(τ, s = ψ*
6
(τ, s =
7|9|
: x(t ψ∗
=
>&?
9
@ dt (iv)
Where is translation, s is scale parameter, ψ (t) is mother wavelet prototype for generating the
other window functions.
Inverse continuous wavelet transform (ICWT) is given as follow
B(C = DE
: : Ψ*
G
HI
(J, K I
L =
M&H
I
@ NJNK (v)
Where OP = Q2S :
|PT (U |
|U|
V
&V
NWX
/
< 0 (vi)
Where L[(B is the FT of (C equation (vi) implies that the L[(B = 0 which is
] (C NC = 0
3.2 DISCRETE WAVELET TRANSFORM (DWT)
As it is well known wavelet transform provides information about frequency domain as well as
time domain at same instant. Embedding of watermark in wavelet transform is relatively of better
quality and robustness. Discrete wavelet transform coefficients are obtained by standard
procedure in which a dyadic grid is used for scale and translation parameter [7]. Signal sequence
is decomposed in to sub bands of frequencies at different levels for eg. To start with the signal
will be decomposed in to two bands manly low and high (L & H) frequency i.e if 500 Hz is the
highest frequency by passing it though a low pass and high pass filter and convoluted then the
two band will contains the coefficients from 0 to 250 Hz and 250 to 500Hz respectively.

Thereafter each sub band will be further decomposed in to two sub bands LL1, LH1 and HL1,
HH1. This procedure will continue to obtain the higher level of decomposition. As shown in Fig.
1.
3.2.1 ALGORITHM FOR EMBEDDING
FREQUENCIES
Embedding watermark LL, HL & LH frequencies regions allows us to increase the robustness of
our watermark; at little to no additional impact on image quality [9].In the algorithm given below
we have used Haar wavelet filter and a threshold (tr) value of the frequency which is the
values of HL and LH frequencies.
Algorithm
Step 1: Read cover image and watermark image
Step 2: Resize the watermark image in to vector and set threshold i.e. Tr
Step3: Apply 2 levels DWT to the original image. Four frequencies matrices are generated called
as LL, HL, LH, and HH
Step4: if the watermark > threshold (Tr)
HL coefficient value = HL coefficient value+ watermark
Else LH coefficient value = LH coefficient value +watermark
Step 5: Repeat step 4 until the length of watermark! = 0
Step 6: Apply inverse wavelet transform to transform a frequency domain image in to spat
domain.
3.2.2 ALGORITHM FOR EMBEDDING
FREQUENCIES
Embedding in LL frequency band the threshold tr has been taken as zero. This is done because
the LL band of frequency in general contains maximum information of the image.
Algorithm
Step 2: Apply 2 levels DWT to the original image. Four frequencies matrices are generated called
as LL, HL, LH, and HH
Step3: Resize the watermark image in the size of LL matrix.
Step4: LL coefficient value = LL coefficient value+ watermark
sub band will be further decomposed in to two sub bands LL1, LH1 and HL1,
Fig 1.Wavelet Transform
MBEDDING THE WATERMARK IN, LH, HL SUB
at little to no additional impact on image quality [9].In the algorithm given below
we have used Haar wavelet filter and a threshold (tr) value of the frequency which is the
values of HL and LH frequencies.
watermark image
Step 2: Resize the watermark image in to vector and set threshold i.e. Tr
Step4: if the watermark > threshold (Tr)
oefficient value = HL coefficient value+ watermark
Else LH coefficient value = LH coefficient value +watermark
Step 6: Apply inverse wavelet transform to transform a frequency domain image in to spat
MBEDDING THE WATERMARK IN LL SUB
in general contains maximum information of the image.
watermark image in the size of LL matrix.
Step4: LL coefficient value = LL coefficient value+ watermark
4
sub band will be further decomposed in to two sub bands LL1, LH1 and HL1,
UB BANDS OF
at little to no additional impact on image quality [9].In the algorithm given below
we have used Haar wavelet filter and a threshold (tr) value of the frequency which is the average
Step 6: Apply inverse wavelet transform to transform a frequency domain image in to spatial
UB BANDS OF

5
Step 6: Apply inverse wavelet transform to transform a frequency domain image in to spatial
domain.
4. CURVELET TRANSFORM
Wavelets don’t describe well highly anisotropic elements and contains only fixed number of
directional elements, independent of scale. Wavelet is non-geometrical and do not exploit the
regularity of the edge curve.
Curvelet transform has been developed to overcome the limitations of wavelet. Though wavelet
transform has been explored widely in various branches of image processing, it fails to represent
objects containing randomly oriented edges and curves as it is not good at representing line
singularities.[9][10]
4.1 CONTINUOUS CURVELET TRANSFORM
Fig. 1.1 Polar coordinates
Dilated Basic curvelet in polar coordinate.
∅_`, , (a, = 2&
bc
d e(2&`
a fg`(
(vii)
Where W and V are window function
Nj=Number of wedges in the circular ring at scale 2-j
V(w) = h
1 |w| ≤
k
Cos l
m
f(3| | − 1o
k
≤ | |
0 else
≤ 2/3q (viii)
e(a =
r
ss
t
ss
u cos l
m
w
f(5 − 6a o k
≤ a ≤ 5/6
cos l
m
w
f(3a − 4 o 4/3 ≤ a ≤ 5/3
1
{
≤ a ≤
w
k
0 |}K|
~ (ix)

6
Where v(x) is a smooth function satisfying
•(B = Q
0 B ≤ 0
1 B ≥ 1
~
•(B + •(1 − B = 1
•(B , where for the tiling of a circular ring in to N wedges, Where N is arbitrary positive integer.
We need to 2π periodic non negative window with support inside [-2 π/N, a π/N) such that
[15][16]
∑ fg = − 2π
ƒ
„
@ = 1 ∈ {0, 2π g&
‡ (x)
The sum of squared rotation curveletfunction depends only on w (2-j
r) follows that
∑ ˆ2
bc
d ∅_`, , ‰a, −
m‡
gc
Š ˆ = ‹ (2&`
a ∑ fg
gcŒ•
‡ ( −
m‡
gc
‹
gcŒ•
‡ = Ž (2&`
a Ž (xi)
O`,•,‡(• = < •, ∅`,•,‡are called curvelet transform
O`,•,‡(• = : •(‘’
∅`,•,‡(‘ dx = : •“(W ”•`,•,‡ (W NW
= : •“(W ”•`, , – —, `,‡W˜| (™š
c,›
,U
NW (xii)
4.2 DIGITAL CURVELET TRANSFORM
Digital curvelet transform is different from continuous as its take input as Cartesian Array instead
of polar tilling, its make virtual concentric squares instead of circle [2].
Fig. 1.2 Cartesian array on input
Basic curvelet according to new tiling rotation is replaced by shearing.
∅œ, ,
• (W = 2&k`/w
e–2&`
W ˜ f(
Œ
c
U
U•
(xiii)
The adopted basic curvelet∅`,ž,ž determine the frequencies in the trapezoid
Ÿ(W1, W2 : 2`&
≤ W1 ≤ 2`¡
, −2
¢
c
~ £
~
~ .
k
≤
U
U
≤ 2&
c
. ~

k ¥ (xiv)

7
In order to replace rotation of curvelet elements by shearing in the new grid, there is need to
consider the cones of each side separately instead of equi-spaced angles, we define as of equi-
spaced slopes for all cones
C¦§¨`,‡ = }2&©`/~ £~
Where } = −2©`/~ £~
+ 1 … … … . . 2©`/~ £~
− 1
Now the curvelet like function be given by
L`,•,‡(B = L`, , ( «,`,‡
M
–B − ¬•
`,‡
˜ (xv)
With shear matrix « l
1 0
tan ¨ 1
o
¬•
`,‡
= «,`,‡
&M
(¯ 2&`
, ¯ 2&`/
So the digital curvelet transform
L`,•,‡(W = |& (
¬•
`,‡
, W L`, , – «,`,‡
&M
, W˜ = |& (
¬•
`,‡
, W 2&k`/w
e–2&`
W ˜ f(
Œ
c
U
U•
(xvi)
We find the Cartesian counterpart of the coefficients in equation by
Õ`,•,‡(• = < •, ∅_`,•,‡ > = ] •“(W
ℝ
”•`, , (K—,`,‡
&
W | (™š
Œc,›
,U
NW
= : •“( —,`,‡Wℝ
”•`, , (W | (•c,U
NW (xvii)
With¯` = ‰¯ 2&`
, ¯ 2&
c
Š , (¯ , ¯ )
4.3 ALGORITHM
1. Compute the Fourier transfrom of f by means of a 2D FFT. Let
³(B = K(B K(B Cℎ K(B = (1 − |B | µ|& , |(B [2]
•(B = ∑ ∑ • = •
g
, g
@g&g&
•
K̃( B − § , B − § (xviii)
•“(W = ∑ ∑ • = •
g
, g
@g&g&
•
|&
¶((•·•¸( ·
¹
“
=
U
g
@ (xix)
The 2D FFT of length N gives us the samples from [14][15]
2. Computer •“– —,`,‡ W˜by interpolation
3. Compute the product •“– —,`,‡ W˜”•`, , (W
4. Apply the inverse 2D FFT in order to obtain the discrete coefficients that is an
approximation of the coefficients.

8
Fig.2 Curvelet Transform
Fig. 2.1 Coefficients of curvelet
The above fig gives the curvelet coefficients. The low frequency coefficients are stored at the
centre of the display. The Cartesian concentric coronae show the coefficients at different scale.
The outer coronae correspond to higher frequencies. There are four strips associated to each
corona corresponding to the four cardinal points; these are further subdivided in angular panels.
Each panel represents coefficients at a specified scale and along the orientation suggestion by the
position of the panel.
4.4 WATERMARK EMBEDDING ALGORITHM
1. Apply USFFT digital curvelet to the n*m size cover image, get curvelet coefficients
matrix C.
2. Read the watermark and covert in into binary image then resize the watermark in some
small size.
3. Select the scale and oriantation for embedding let in this algo we use scale 2 and
orientation due to the advantages of low frequencies
4. Select the curvelet coefficients to embed a bit of watermark in the curvelet coefficents
with 2 scale as
C’(j,l,k) = C(j,l.k) + W[k];
5. Do Inverse curvelet trasform to obtain the watermarked image i.e C’
5. EXPERIMENT RESULTS
We used a gray level cameraman image for experiment. The cover image is size of 256*256 and
the watermark image is binary copyright image. Fig 3 and Fig 4 shows the cover image and

9
watermark image respectively. Fig 5 to Fig 7 shows the results of above discussed algorithms. Fig
5 & Fig 6 show embedding in HL &LH frequencies, LL frequencies in wavelet transform
respectively. Fig 7 shows the embedding of the same in curvelet transform. Fig 8 to fig 10
presents the extracted watermark of above discussed techniques
Fig. 3 Cover Image Fig. 4 Watermark Image
Fig. 5 Wavelet Transformation embedding in HL & LH Frequencies PSNR =69.014
Fig. 6 wavelet transform in LL frequencies PSNR = 72.0251
Fig. 7 Curvelet Domain in 2 scale PSNR = 77.0099

10
5.1 RESULTS OF EXTRACTED WATERMARK
Fig.8 wavelet in HL LH Freq PSNR =57.7729 Fig.9 wavelet in LL Frequencies PSNR =53.9633
Fig.10 Curvelet Transform PSNR= 64.2457
5.2 INVISIBILITY AND EFFICIENCY TEST
Table 1.1 shows the PSNR values, NC values and the embedding capacity of the watermarked
image at different scales in curvelet domain. From the table it very much clear that the embedding
in lower scales give better invisibility without significantly affecting the PSNR values.NC of
watermarked image is 1 for all the scales confirming the quality of extracted watermark. Further
all scales have the same efficiency with regard to data hiding capacity as because the watermark
of the size of the cover image can embedded with significantly effect on PSNR of watermarked
image. So efficiency of data hiding is very high.
Table 1.1 Comparision Of Perposed Algorithm Embedding In Different Scales
5.3 ROBUSTNESS TEST
Robustness is the most important metric for watermark system evaluation. Robustness is the
ability of watermark detection after some attacks or distortion of image. In table 1.2 the PSNR is
employed to evaluate the quality of watermark image after different types of attacks on different
scales in curvelet domain. Table 1.3 gives the normalize correlation (NC) between the original
watermark and extracted watermark image after different types of attacks. Attacks are like adding
of noise &filtering of image. It can be seen from both table 1.2 and 1.3 that the embedding in

11
scale 6 in curvelet domain gives better robustness against the pepper & salt noise, Gaussian noise,
low pass filtering and median filtering compared to lower scales.
Table 1.2 Psnr Value Of Extracted Watermark After Some Image Processing Operations
Table 1.3 NC Value of Extracted Watermark after Some Image Processing Operations
Table 1.4 contains the NC values of extracted watermark of pepper & salt noise attack at different
density parameter. Table 1.5 contains the NC value of extracted watermark image of Gaussian
noise at different standard variance with same mean. It can be seen from both these tables that
embedding in curvelet domain in 6th
scale gives the better robustness even at higher levels of
noise.
Table 1.4 NC Value of Extracted Watermark after Adding Pepper & Salt Noise

12
Table 1.5 NC Value of Extracted Watermark after Adding Gaussian Noise
5.4 ANALYSIS
In order to analyze different embedding techniques we compared our proposed scheme with
wavelet transform technique at different levels. Using out algorithm we have embedded the
watermark in HL domain and a copy of same in LH domain. In second algorithm embedding
is performed in LL frequencies only. Table 1.6 contains the PSNR of watermarked image,
PSNR of extracted watermark and the embedding capacity of the domain. From the table it
clear that the embedding in LL frequencies give better invisibility of watermarked image but
the efficiency of data hiding reduces. The Maximum PSNR of watermarked image is
achieved in 3rd
level of decomposition i.e. embedding in LL frequencies .By comparing the
table 1.1 and 1.6 it observed that the embedding in curvelet domain at any scale gives the
better PSNR of watermarked image and better information hiding capacity. Further, PSNR of
watermarked image at any scale is more than the any decomposition level of wavelet domain
confirming the superiority of curvelet domain.
Table 1.6 Psnr Value Of Watermarked Image, Extracted Watermark & Embedding Efficiency Of Different
Decomposion Level In Wevelet Transform Domain
6. CONCLUSION
The curvelet transform a technique is used for embedding the watermark has been analyze with
respect to invisibility, efficiency- data hiding capacity, effectiveness and robustness. It has been
concluded that lower scales in curvelet domain give better invisibility without affecting the data
hiding efficiency significantly. With addition of noise as attack it is found that higher scales give
more robustness along with good PSNR value of watermarked image. The results have also been
compared with wavelet transform technique .It shows that curvelet domain at any scale gives the
better PSNR of watermarked image and better information hiding capacity.

13
REFERENCES
[1] R. Wolfgang and E. Delp, ” A watermark for digital images,” Proceeding of IEEE International
Conference on Image Processing, Vol. 2, pp.319-222, 1996.
[2] Jianwei MA, GerlindPlonka “A review of curvelet and recent applications” pp. 1-29
[3] C. T. Hsu and J. L. Wu, ”Hidden digita1 watermarks in images," IEEE Transactions on Images
Processing, Vo1.8, pp.58-68, Jan, 1999.
[4] P. C. Su, C. C. J. Kuo, and H. J. M Wang, "Blind digital watermarking for cartoon and map images,"
in Proceedings of SPIE Security and watermarking of multimedia contents, vol. 3657, pp.296-306,
January1999
[5] R. Singh. “Emerging technologies that will change the world: Digital rights management”. MIT
Technology Review,2001.
[6] I. J. Cox, M. L. Miller, and J. A. Bloom, “ Digital Watermarking”. San Diego, CA, USA: Academic
Press, 2002.
[7] P. Tao, A. M. Eskicioglu, “A Robust Multiple Watermarking Scheme in the Discrete Wavelet
Transform Domain”, Optics East 2004, Internet Multimedia Management Systems V Conference,
Philadelphia, PA, October 25-28, 2004.
[8] Parameswaran, L. “Content Dependent Image Signature for Authentication Using Wavelets”,
Proceedings of NCIS, Karunya Deemed University, Coimbatore, Nov.2005.
[9] Emmanuel J. Candes, and David L. Donoho, ‘‘New Tight Frames of Curvelets and Optimal
Representations of Objects with Piecewise C2 Singularities’’, Communication on Pureand Applied
Mathematics, Vol.57, No.2, 2004, pp.219-266.
[10] E. Cand`es and D. Donoho, ‘‘Curvelets: A Surprisingly Effective Nonadaptive Representation of
Objects with Edges’’, In C. Rabut A. Cohen and L. L. Schumaker, editors,Curves and Surface,
Vanderbilt University Press, Nashville, TN. 2000, pp. 105---120.
[11] H. Y. Leung, L. M. Cheng and L. L. Cheng. “A Robust Watermarking Scheme Using Selective
CurveletCoeffi-cients,” International Conference on IntelligientInforma-tionHiding and Multimedia
Signal Processing, Harbin, August 2008, pp. 465-468
[12] Minh N. Do, and Martin Vetterli, ‘‘The Finite Ridgelet Transform for Image Representation’’, IEEE
Transactions on Image Processing, Vol.12, No.1, 2003, pp. 16-28.
[13] I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, ‘‘Secure spread spectrum watermarking for
multimedia’’ , IEEE Transactions on Image Processing, Vol. 6, No. 12, 1997, pp. 1673-1687.
[14] J. P. Shi and Z. J. Zhai, “Curvelet Transform for Image Authentication,” Rough Set and Knowledge
Technology, LNAI 4062, 2006, pp. 659-664.
[15] Emmanuel J. Candes, and David L. Donoho, ‘‘New Tight Frames of Curvelets and Optimal
Representations of Objectswith Piecewise C2 Singularities’’, Communication on Pureand Applied
Mathematics, Vol.57, No.2, 2004, pp.219-266.
[16] E. Cand`es and D. Donoho, ‘‘Curvelets: A SurprisinglyEffectiveNonadaptive Representation of
Objects withEdges’’, In C. Rabut A. Cohen and L. L. Schumaker, editors,Curves and Surface,
Vanderbilt University Press, Nashville,TN. 2000, pp. 105---120.

14
Authors
Er. Ranjeetapursuing Ph.D.in Computer Science and Engineering department from
Thapar University, Patiala. Working as Associate professor in Chandigarh Engineering
College, Landran, Mohali, Punjab. She is having 12 years of teaching experience. She is
having 13 national, international and conferences publications. Her area of research is
Watermarking in curvelet domain.
Dr. Sanjay Sharma is currently working as Professor and Head of the department in
Electronics and Communication Engineering of Thapar University, India. He has done his
B. Tech in ECE from REC in 1993, Jalandhar, M. E. in ECE from TTTI, Chandigarh in
2001 and Ph.D. from PTU, Jalandhar in 2006. He has completed all his education with
honors. He has published many papers in various journals and conferences of
international repute. He has to his credit the implementation of research projects worth
12000 USD. His main interests are VLSI Signal Processing, Wireless System Design using Reconfigurable
Hardware, Digital communication, Channel Coding etc.
Dr. Lajpat R. Raheja Completed his Ph.D. in Applied Mathematics, from IIT
Kharagpur, West Bengal. He has more than 30 years of teaching and research experience.
He worked in number of research projects in field of computer sciences, mechanical and
naval engineering. He is recipient of 3 internal awards Alexander von Humboldt
Fellowship, West Germany, March1971-June1972, Research Scientist,
InstitutfürSchiffbau, University of Hamburg, Hamburg, West Germany, Jan.1985-Jan. 1987 and Exchange
Program, Department of Naval Architecture, University of Zagreb, Yugoslavia, April-June, 1979.

DIGITAL WATERMARKING TECHNIQUE BASED ON MULTI-RESOLUTION CURVELET TRANSFORM

More Related Content

What's hot (17)

Similar to DIGITAL WATERMARKING TECHNIQUE BASED ON MULTI-RESOLUTION CURVELET TRANSFORM (20)

More from ijfcstjournal (20)

Recently uploaded (20)

DIGITAL WATERMARKING TECHNIQUE BASED ON MULTI-RESOLUTION CURVELET TRANSFORM