J. Software Engi neeri n g & Applications, 2010, 3, 939-943
doi:10.4236/jsea.2010.310111 Published Online October 2010 (http://www.SciRP.org/journal/jsea)
939
Digital Image Watermarking Algorithm Based on
Fast Curvelet Transform
Jindong Xu, Huimin Pang, Jianping Zhao
College of Physical Engineering, Qufu Normal University, Shandong, China.
Received July 15th, 2010; revised August 18th, 2010; accepted August 21st, 2010.
ABSTRACT
A digital image watermarking algorithm based on fast curvelet transform is proposed. Firstly, the carrier image is de-
composed by fast curvelet transform, and, the watermarking image is scrambled by Arnold transform. Secondly, the
binary watermarking image is embedded into the medium frequency coefficients according to the human visual charac-
teristics and curvelet coefficients. Experiment results show that the proposed algorithm has good performance in both
invisibility and security and also has good robustness against the noise, cropping, filtering, JPEG compression and
other attacks.
Keywords: Digital Image Watermarking, Fast Curvelet Transform, Human Visual Characters, Robustness, Invisibility
1. Introduction
In the past two decades, more and more researchers have
devoted to the transform domain which shows good per-
formance and robustness. Recently, Candès and Donoho
[1,2] have developed a new multiscale transform which
is called the Curvelet transform. It is anisotropic with
strong direction, and provides optimally sparse represen-
tations of objects along a general curve with bounded
curvature. Hence, it has been widely used in image proc-
essing field. At the same time, many research papers
proposed to embed watermark based on the first curvelet
transform [3-8].
Shi et al. [3] proposed a semi-fragile watermarking
algorithm by embedding the wartermark in the maximum
module of curvelet coefficient. This algorithm keeps
good tolerance against JPEG compression. The method
of Thai et al. [4] embedded a watermark in curvelet coef-
ficients which are selected by a threshold. This method
has good invisibility and robustness. Thai et al. [5] em-
bedded the watermark in the curvelet transform which
contain as much edge information as possible. It has
good invisibility but poor robustness. The implementa-
tion of the first curvelet transform includes subband de-
composition, smooth partitioning, renormalize and ridgelet
analysis. So these algorithms have many problems such as
implementation complexity and large amount of redun-
dancy.
This paper proposes a digital image watermarking al-
gorithm based on fast curvelet transform. Firstly, we take
a binary-valued image which contains copyright infor-
mation as watermark, and adapt Arnold transform to it to
improve its security. Secondly, according to the human
visual characteristics, we embed the watermark in
curvelet coefficients of the original image. This algo-
rithm is simple and easy to implement. Experiment re-
sults show that our algorithm has good invisibility and
security, and is robustness to the noise, cropping, filter-
ing, JPEG compression and other attacks.
2. Curvelet Transform
Similar to the wavelet transform and the ridgelet trans-
form, the curvelet transform theory is based on sparsity
theory [8]. The idea of curvelet is to calculate the inner
relationship between the signal and the curvelet function
to realize the sparse representation of the signal.
2.1. Continuous—Time Curvelet Transform
The curvelet transform can be expressed as


,,
,, :,
j
lk
cjlk f   (1)
here, j = 0, 1, 2, …, is a scale parameter; l = 0, 1, 2, …, is
an orientation parameter; and k = (k1, k2) ∈ Z2 is a
translation parameter. The mother curvelet is φj (x), its
Fourier transform is φj (ω) = Uj (ω), where Uj is fre-

Digital Image Watermarking Algorithm Based on Fast Curvelet Transform 940
quency window defined in the polar coordinate system
such as:


3/4 2/2
,2 22
jj
j
j
UrW rV









l

(2)
W and V are radial and angular windows respectively and
will always obey certain admissibility conditions.
Curvelet at scale 2-j, orientation θand position


,1
12
2, 2
l
jl
j
j
k
xRk k


 can be expressed as:


,
,, l
jl
jlk jk
xRxx








(3)
So, for f ∈ L2 (R2), curvelet transform is expressed
as:
 

 
,,
2
2
1ˆˆ
,,:( )d
2
1ˆexp i,d
2l
jlk
j,l
jθk
cilkf
fURωx
 








(4)
2.2. Digital Curvelet Transform
Digital curvelet transform is linear and takes as input
Cartesian arrays of the form f [t1, t2], 0 ≤ t1, t2 < n, which
allows the output as a collection of coefficients:
  
12
12,.12
0,
,, :,,
D
jlk
tt n
cjlkftt tt 

D (5)
In order to improve the curvelet transform—in the
sense that they are conceptually simpler, faster and far
less redundant. Paper [2] proposed the Fast Discrete
Curvelet Transform (FDCT). There are two digital im-
plementation of FDCT. The first is based on un-
equally-spaced fast Fourier transform (USFFT) while the
second is based on the wrapping of specially selected
Fourier samples. The FDCT-Wrapping uses simpler
choice of spatial grid to translate curvelets at each scale
and angle. It needs less two-dimensional FFTs than
FDCT-USFFT, so it is quickly.
The architecture of FDCT via Wrapping is then
roughly as follows:
1) Apply the 2D FFT and obtain Fourier sample


12
,
f
nn

, -n/2 ≤ n1, n2 ≤ n/2 (n is the size of the picture).
2) For each scale/angle pair (j, l), form the product




,12 12
ˆ
,,
jl
Unnfnn
.
3) Wrap this product around the origin and obtain



,12, 12
ˆ
,
jl jl

,
f
nnWU f nn

n n0
, where the range for
1and 2 is now 11,
j
nL and 22, 0
j
nL (for

in the range (/4,/4

 ).
4) Apply the inverse 2D FFT to each,
j
l
f
, hence col-
lecting the discrete coefficients .

lk,,
D
cj
3. Watermarking Algorithm Based on FDCT
3.1. Watermark
In this algorithm, the watermark is a binary-valued image
with 32 × 32 pixels (Figure 1).
Arnold transform is used to increase the data security,
and its function is defined as (6).
'
'
11 mod
1
xx
N
kky
y



 


(6)
Adapt n (here, n = 8) times Arnold transform to the
original watermark W, then we obtain scrambling wa-
termark (Figure 2).
3.2. Watermark Embedding
According to the Human Visual Characteristic, we
choose to embed the watermark into medium-frequency.
In selected scale, the coefficients of each orientation are
sorted and find criterion Ti by amplitude factor λ (λ ≥ 0).
Then choose the minimum Ti as embedding parameter T.
Finally, the watermark is embedded to coefficients which
are chosen by T .
The embedding procedure (Figure 3) is described as
follows.
1) Apply FDCT to the original image, get curvelet co-
efficient C.
2) Select the curvelet coefficient to embed a bit wa-
termark according to the following conditions.



For 00,,2or,,32
For 12,,
CjlkTT CjlkT
TCjlkT
 

(7)
Then record the positions of these coefficients and the
positions of 0 and 1 in the watermark.
3) The embedded coefficients are modified by the fol-
lowing equations.




 

 
'
'
Embed 0,,,,
mod, ,,4
Embed 1,,,,
mod, ,,34
CjlkC jlk
CjlkTT
CjlkC jlk
CjlkTT




(8)
4) Do inverse FDCT to C’, obtain the watermarked
image.
3.3. Watermark Extraction
The extraction procedure is composed of 4 steps and
each step is described as follows.
Figure 1. Watermark.
Figure 2. Scrambled watermark.

Digital Image Watermarking Algorithm Based on Fast Curvelet Transform 941
IFDCT
1j

2
j

3
j

4j

5
j


3
j

1
l
2
l
3
l
n
l
 2/,,0 TkljC 
 2/3,, TkljCT 







 4/,,,mod,,,,
'
TTkljCkljCkljC 
 TkljCT,,2/







 4/3,,,mod,,,,
'
TTkljCkljCkljC 
Figure 3. Embedding procedure .
4.1. Invisibility Tests
1) Apply curvelet transform to watermarked image,
obtain curvelet coefficient C’. The watermarked image is shown in Figure 5(a), the
detected watermark from watermarked image is shown in
Figure 5(b).
2) Locate the watermark positions from the original
image by using embedding procedure step 2. Then ex-
tract the coefficients C’ from the watermarked image
using those watermark position.
3) Extract the watermark W’ from C’ with the follow-
ing rule




'
'
1mod,,,
0mod,,,
i
ifCljkTT
ifCljkTT






2
2
(9)
4) Apply T’-n (T’ is Arnold transform period) times
Arnold transform to W’, obtain the binary watermark
image. Figure 4. Original image.
4. Experimental Results
In this program, the image is transformed through FDCT
via Wrapping. We use standard 512 × 512 pixel image
‘Lena’ (Figure 4) for evaluation of our proposed method,
and conduct experiments binary-valued image water-
marking with the noted above parameter scale = 3, λ =
0.75.
We have investigated the invisibility and the robust-
ness of our watermarking system, analyzed the algorithm
performance by objective and subjective standards.
(a) (b)
Figure 5. Invisibility Tests. (a) Watermarked image [PSNR
60.8028dB]; (b) Detected watermark [NC = 1.00]. =

Digital Image Watermarking Algorithm Based on Fast Curvelet Transform 942
Table 1. JPEG compression attack.
Q 90 80 70 60 50 40 30 20 15
PSNR 25.2093 25.1357 25.0710 25.0186 24.9718 24.8890 24.8226 24.6714 24.4813
NC 0.9988 0.9975 0.9901 0.9877 0.9914 0.9741 0.9532 0.9470 0.9113
Detected
watermark
Table 2. Gaussian low pass filtering attack.
Standard Deviation
(Window) 0.5(3) 1.5(3) 0.5(5) 1.5(5) 3(5)
PSNR 40.8289 32.4277 40.8027 29.9117 28.7562
NC 0.9914 0.9704 0.9914 0.9113 0.8658
Detected
watermark
Table 3. Other attacks.
Attacks Gaussian noise
(0.001)
Salt & Pepper noise
(0.01)
Random cropping
(1/16)
Random cropping
(1/32) Change contrast
PSNR 19.4926 25.0252 19.6377 21.7762 25.2690
NC 0.8116 0.9200 0.8645 0.9372 0.8756
Detected
watermark
Subjectively, the watermarked image has good invisi-
bility. Compare Figure 5(b) with Figure 2, they are fully
consistent. The original watermark is accurately recov-
ered.
4.2. Robustness Tests
To evaluate the robustness of algorithm, all the attacks
are tested by the software of Stirmark [10,11]. The Peak
Signal to Noise Ratio (PSNR) is employed to evaluate
the quality of watermarked image after attack, and the
Normalized Correlation Coefficient (NC) is used to
evaluate the quality of extracted watermark for some
attacks such as JPEG compression (Quality factor Q),
Gaussian low pass filtering, adding noise, cropping and
so on. The simulation results are shown as following
Tables.
It can be seen from Table 1 and Table 2, the proposed
method show very good robustness to JPEG compression
and Gaussian low pass filtering. From Table 3, the
method also has good robustness against noise, cropping
and so on.
5. Conclusions
This paper proposes a method by embedding a water-
mark into the original image based on FDCT. At the
same time, Arnold transform is applied to improve the
security of the system. The experimental results show
that the watermarked image has good invisibility and
robustness against JPEG compression and Gaussian low
pass filtering.
There are many unstable coefficients are discovered
from the experiments. For instance, a small change of the
image will arouse big changes of these coefficients.
These unstable factors can influence the extracting of
watermark. Therefore, the proposed algorithm can not
give a good performance of rotation and scaling, etc. In
the future, we will unceasing devote ourselves to the
study of the robustness watermark system against geo-
metric attacks based on curvelet transform.
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