Al-Khwarizmi 

Engineering   

Journal 
Al-Khwarizmi Engineering Journal,Vol. 12, No. 2, P.P. 124- 133 (2016) 

 

  

Digital Image Watermarking Using Arnold Scrambling and Berkeley 

Wavelet Transform 
 

Iman M.G. Alwan*             Enas Muzaffer Jamel** 
*,** Department of Computer Science /College of Education for Women / University of Baghdad  

*Email:ainms_66@yahoo.com 

**Email: enasm.altai@gmail.com 

 
 (Received 26 April 2015; accepted 26 November 2015)  

 

 

Abstract 

 
Embedding an identifying data into digital media such as video, audio or image is known as digital watermarking. In 

this paper, a non-blind watermarking algorithm based on Berkeley Wavelet Transform is proposed. Firstly, the 

embedded image is scrambled by using Arnold transform for higher security, and then the embedding process is applied 
in transform domain of the host image. The experimental results show that this algorithm is invisible and has good 

robustness for some common image processing operations. 

 

Keywords: Berkeley Wavelet Transform, image watermarking, Arnold Transform, watermark attack. 

 

 

1. Introduction 
 

      Due to rapid distribution of digital files such 

as text, audio, still images or video over fast 
communication systems, the plans for verification 

of digital contents originality and copyright 

ownership are usually demanded. Digital 
watermarking is a new solution for content 

protection. It draws significant interests and 

becomes an active research field [1]. Generally 

digital watermark is the technology of adding 
verification message or hidden copyright 

information called a watermark or label or 

signature to the digital media so that it can be 
extracted or detected later to make a confirmation 

about the object [2]. 

Several eligible characteristics should be 
available in watermarking system. They are 

Imperceptibility, which means perceptual parity 

between the original and the watermarked images, 

Robustness, which means the immovability of the 
watermark data against changes and alteration to 

the watermarked file, and finally capacity, which 

indicates the amount of information that can be 
saved in the cover media . To reveal the 

watermark information, two techniques are 

followed. They are blind and non-blind 

techniques. In the first one the cover media is not 

required while in the second one, the original 
media is required for extraction.  

Mainly, two alternative watermarking methods 

exist; spatial and frequency (transform) domains. 
The spatial domain techniques are not powerful 

even to the insignificant of altering or removal 

attempts (attacks), hence these techniques have 

become less common. On the other hand, the 
transform domain techniques have taken the most 

interests by the researchers and almost all of the 

known transforms have been used [3]. The 
transforms used are Discrete Cosine Transform 

(DCT), Discrete Fourier Transform (DFT), and 

Discrete Wavelet Transform (DWT) [4,5]. In [3] 
slantlet transform is adapted in watermarking 

process, while in [6] the contourlet transform is 

used as a frequency domain watermarking system. 

The curvelet transform based logo watermarking 
is proposed in [7]. 

In [8] a two-dimensional triadic wavelet 

transform is proposed, it’s called Berkeley 
Wavelet transform. It participates many 

characteristics like oriented, spatial localization, 

and frequency bandpass. It is formed by              
4-complete, orthogonal bases which the transform 

and inverse transform computations are fast. Its 



Iman M.G. Alwan                          Al-Khwarizmi Engineering Journal, Vol. 12, No. 2, P.P. 133- 124(2016) 

 

125 

 

good computation and fine primary simulation 
visually makes it a robust tool in watermarking 

process. In [9], the authors proposed a 

watermarking scheme based on Berkeley Wavelet 
Transform. The embedding process is carried out 

in the DC band of the transformed host image 

only by sub-dividing it into non-overlapping 

blocks, and replacing the binary watermarked 
coefficient with only one designated coefficient of 

middle frequency diagonally, and then an inverse 

Berkeley Wavelet Transform isperformed to get 
the watermarked image. This algorithm embeds a 

limited amount of information since it is 

embedded in one band which is (1/9) of the total 

host image size. They used a host image of size 
(243x243) and the binary watermark image of size 

(27x27).    

In this paper, Berkeley wavelet transform is 
used for grey scale image   watermarking process, 

the grey scale watermark image is first scrambled 

by using Arnold transform to boost the security 
side, and then it is embedded in the whole 

frequency bands of the transformed grey scale 

host image. So the amount of embedded 

information is greater. From simulation results, 
the proposed method shows good performance in 

terms of Peak Signal to Noise Ratio (PSNR) of 

the watermarked image and normalized 
correlation (NC) between the watermark image 

and the extracted one. The performance of the 

system is also tested under various attacks 
including noise, JPEG compression, and cropping. 

The results exhibit a good response of the 

proposed system. Rest of this paper is organized 

as follows: The concepts of Berkeley Wavelet 
Transfrom are introduced in Section 2.  In Section 

3, the proposed embedding algorithm will be 

presented. Finally, the Simulation Results and 
Discussion will be given in Section 4, followed by 

the conclusions in Section 5. 

 

 

2. Berkeley Wavelet Transform 
 
     The Berkeley wavelet transform is a wavelet 

basis, triadic, 2- dimensional transform. It 

presents an effective image representation. Its 

computational prosperities are useful; it is 
complete, orthonormal, and a sparse code for 

images of natural type. BWT includes 4- pairs of 

mother wavelets at 4- orientations. 
At every pair, one of the wavelet is odd 

symmetric, and the other is even symmetric. 

Orthonormal, complete basis in 2- dimensions are 
resulted from the translation and scaling of the 

total set plus a single constant term of wavelet. 

The mother wavelets are piecewise constant 

functions, so each wavelet, 𝛽𝜃 ,𝜙 , can be totally 

represented  as a  matrix 𝑏𝜃,𝜙  of 3x3 dimensions 

[8]:  

 

𝑏|,0 =
1

 6
 
 −1 0 1
 −1 0 1
 −1 0 1

                                    …(1) 

𝑏|,𝑒 =
1

 18
 
 −1  2  −1
−1  2  −1
−1  2  −1

                            …(2) 

𝑏/,0 =
1

 6
 
−1     1     0
   1     0  −1
   0  −1     1

                            …(3) 

𝑏/,𝑒 =
1

 18
 
−1  −1     2
−1     2  −1
   2  −1  −1

                          … (4) 

𝑏−,0 =
1

 6
 
−1  −1  −1
   0     0     0
   1     1     1

                            …(5) 

 

𝑏−,𝑒 =
1

 18
 
−1  −1  −1
   2     2     2
−1  −1  −1

                           …(6) 

𝑏\,0 =
1

 6
 

0  −1  1
1     0 −1

−1     1  0
                              …(7) 

𝑏\,𝑒 =
1

 18
 
   2 −1  −1
−1    2  −1
−1 −1     2

                           …(8) 

 

Daughter wavelets βθ,ϕ
m ,n ,s

 can be produced at 

various positions (i, j) in the x-y plane, and 
various scales, (s) through scaling and translation 

of the mother wavelets  βθ,ϕ  by using the dilation 

equation:  

βθ,ϕ
m ,n ,s

=
1

s2
βθ,ϕ (3

s  x − i , 3s  y − j )            …(9) 

The BWT uses triadic scaling, i.e. the sizes of 
the daughter wavelets are scaled by powers of 3. 

To form an orthogonal set, the possible 

translations in the x-y plane are locked to integer 
multiples of the wavelet size, 3s. All of the BWT 

wavelets have zero mean, and so a single DC term 

is required to represent the mean value of an 
image: 

β0 =  u(x 3 , y 3)  / 9                              …(10) 
Note that this DC term is not subject to the 

dilation equation; instead, it is applied only once, 

at the largest spatial scale. Since the BWT is a 
complete, orthonormal set, it is self-inverting.   

An image can be reconstructed from its BWT 

coefficients using [8]: 



Iman M.G. Alwan                          Al-Khwarizmi Engineering Journal, Vol. 12, No. 2, P.P. 133- 124(2016) 

 

126 

 

𝐼 𝑥, 𝑦 =   𝑤𝑚 ,𝑛 ,𝑠𝛽𝜃 ,∅(3
𝑠 𝑥 −∞𝑚 ,𝑛 ,𝑠=0𝜃∈Θ,𝜙∈Φ

𝑚 , 3𝑠 (𝑦 − 𝑛)                                               …(11) 

where   wm ,n ,s
θ,ϕ

 are the BWT coefficients 

representing the image. 

     Figure (1) shows the output of the Berkeley 

Wavelet Transform of the image Boat as it is 

obtained by the Berkeley Wavelet Transformation 

program. 

 

                      (a)                                      (b) 

Fig. 1.  The original image (a), the transformed 

image (b). 

 

 

3. The Proposed System 
 

The proposed system is based on embedding 
grey scale visual watermark image of size (1/9) of 

the host image into the host grey scale image by 

using Berkeley Wavelet Transform (BWT).      
For security confirmation and robustness 

improvement, the watermark image is scrambled 

before embedding into the host image. Arnold 

transform algorithm is adopted as a scrambling 
scheme to the original watermark image. Arnold 

transform has periodicity process, so the image 

can be reclaimed after the permutation concept. It 
is only appropriate for N×N digital images. It is 

defined as 

 
𝑥′

𝑦′
 =  

1 1
1 2

  
𝑥
𝑦
  mod N                            …(12) 

where (x, y) represent the pixel coordinates of 

the original image, while (x', y') represent the 
pixel coordinates of  the encrypted image.  Let C 

indicate the matrix at the left of the right hand 

side of equation (12), Im(x, y) indicate the original 

image pixels, and I𝑚(𝑥′ , 𝑦′ )𝑛   indicate pixels of 
the encrypted image acquired by n-times Arnold 

transform application. Thence, Arnold transform’s 

image encryption is [10]: 

𝐼𝑚 𝑥′ , 𝑦′  = 𝐶 𝐼𝑚(𝑥, 𝑦) 𝑟−1 (𝑚𝑜𝑑 𝑁)       …(13) 
Where r = 1, 2, ….. n, and  𝐼𝑚(𝑥′ , 𝑦′ )0 =
𝐼𝑚(𝑥, 𝑦). 
 To obtain 𝐼𝑚(𝑥, 𝑦)(𝑟−1), the inverse matrix of C 
can be multiplied at each side of equation (13). 
This means, the encryption of the encrypted 

image can be made by computing the equation 
(14) n-times iteratively. 

𝐷(𝑥, 𝑦) 𝑟 = 𝐶−1  𝐷(𝑥′ , 𝑦′ )𝑟−1  𝑚𝑜𝑑 𝑁      …(14) 

where 𝐷(𝑥′ , 𝑦′ )(0) is the encrypted image’s pixel, 
and 𝐷(𝑥, 𝑦)(𝑟 ) is a decrypted pixel by executing r 
iterations. Figure (2) shows an example of Arnold 

image encryption with n=5 for pepper test image 

of size (256x256). 
 

 
                        

(a)                            (b) 
Fig. 2.   The original image (a),  Arnold transformed 

image (b). 

 
 

For embedding process, the host image should 

be a power of  3𝑛 𝑥3𝑛  where n is an integer 
number, and the bit stream of the encrypted 
watermark image is transformed into a sequence 

wi(1)….wi(L), where L is  the bit stream length, 

and 𝑤𝑖 𝑚 ∈  −1,1 ,  𝑚 = 1, … . , 𝐿 . The 
sequence is used as the watermark.        
 

 

A:  Watermark Embedding Process 

   

Input: Host image, Secret image, 𝛼 the scaling 
factor. 

Output: Watermarked image. 

1. Read the host image I(N,N), the watermark                 
image w(m,m). 

2. Apply Berkeley wavelet transform to I(N,N).                 
Triadic pattern image decomposition is 
applied. 

3. Apply Arnold transform to the grey- scale 
watermark image. 

4. Change the 2-D 8-bit scrambled watermark 
pattern of size m x m into a binary sequence 

 w =  w k  k = 0, … . , m2 }, where w(k) ∈
 0,1 . Then map the watermark information w 
into a bipolar vector 

 

𝑤𝑖𝑏 =  𝑤𝑖𝑏  𝑘  𝑘 = 0, … . . , 𝑚2  and 𝑤𝑖𝑏 (𝑘) is 
denoted as 

𝑤𝑖𝑏  𝑘 = (−1)𝑤𝑖 (𝑘)                                    …(15) 
where   k=0,1,…..,𝑚2 



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5. Embed the binary sequence in the all 
transformed host image’s subbands according 

to the formula 

𝑊𝑡𝐼𝑚𝑔 = 𝐻 + 𝛼 × 𝑤𝑖𝑏                              …(16) 
 
where,  WtImg is the watermarked image, H is the 

host Berkeley Wavelet Transformed image, 𝛼 is 
the scaling factor, lower value of it gives better 

quality for the watermarked image,  and 𝑤𝑖𝑏  is the 
watermark information stream. 

6. Apply the inverse of the Berkeley Wavelet 
transform to get the watermarked image. 

7. End 
 

 

B: Watermark Extracting Process 
 

 It is assumed that the original image, value 

of  𝛼, and number of iteration for Arnold 
scrambling technique n, are known for extraction 

procedure. 

 Input: Watermarked Image. 

 Output: Watermark secret image. 
1. Apply Berkeley Wavelet transform for the 

watermarked image, to get the transformed 

subbands. 
2. Extract the embedded sequence from the all 

subbands of the transformed image according 

to the equation: 

3. 𝑤𝑖𝑏 = (𝑊𝑡𝐼𝑚𝑔 − 𝐻)/𝛼                             … 
(17) 

4. Convert the extracted sequence 𝑤𝑖𝑏  into 0, 1 
sequence, reconstruct the scrambled mxm 

watermark image. 
5. Apply the inverse Arnold transform to get the 

watermark image.  

6. End 

 
 

4. The Simulation Results and Discussion 
 

In this part, the simulation parameters settings 

of the proposed algorithm, the simulation results, 
and discussion of the results are exhibited. 

 

 
 

 

 

 
 

 

 

 

4.1   The Simulation Parameters Settings 
      

      The proposed algorithm was implemented and 

tested on several standard images using 
MATLAB Version 7.0.4.365, and was 

implemented with an Intel core i7, 2.7 GHz 

processor. The host images are of size 243x243, 

and the watermark images are of size 80x80. To 
evaluate the qualityof the watermarked image, 

PSNR criteria were adopted. Greater values of 

PSNR indicate preferable quality of the 
watermarked image. Values of  PSNR more than 

30 dBs denote acceptable image quality, where no 

significant change is made by watermarking 

process [10]. The algorithm was also tested under 
different kinds of signal attack. Figure (3a) and 

(3b) displays the test host, and watermark images. 

Equation (18) represents the peak signal to noise 
ratio [10]. 

𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔10 (
255 2

𝑀𝑆𝐸
)                               …(18) 

𝑀𝑆𝐸 = (
1

𝑁
)2   (𝑝𝑖𝑗 − 𝑝𝑖𝑗

′ )2                       …(19) 

where 𝑝𝑖𝑗 ’s stand for the values of the original 

pixel, 𝑝𝑖𝑗
′

’s stand for the values of the  modified 

pixel, and N is the image dimension. The 
efficiency of the watermark extraction result is 

evaluated by using normalized correlation 

coefficient (NC), for the extracted watermark W' 

and the original watermark W as: 

𝑁𝐶 𝑊, 𝑊′  =
 𝑤  𝑖 𝑤 ′ (𝑖)𝑛𝑖=1

  𝑤 (𝑖)2𝑛𝑖 =1   𝑤
′ (𝑖)2𝑛𝑖 =1

           …(20) 

where  (n×n) are the watermark dimensions. The 

unity value given exact matching between the 
extracted watermark and the original watermark 

images, NC of about 0.7 or above is counted 

passable [11]. 

 
 

 

 
 

 

 

 
 

 

 
 

 

 
 

 

 



Iman M.G. Alwan                          Al-Khwarizmi Engineering Journal, Vol. 12, No. 2, P.P. 133- 124(2016) 

 

128 

 

    
 

a. Test Host Images. 

 

    
 

b. Test watermark Images. 

 

Fig. 3.  Test host and watermark images. 

 

 

4.2. The Simulation Results 
 

    The simulation results of the proposed 

algorithm are shown in Figure (4). It shows the 
watermarked images and the reconstructed 

watermark images. PSNR values of the 
watermarked images are also included. The 

Normalized Correlation (NC) is listed to indicate 

the similarities between the original watermarks 

and the extracted ones. 

 

    
PSNR= 40.88 dB PSNR=40.66 dB PSNR=40.87 dB PSNR=40.7 dB 

    
NC=1 NC=1 NC=1 NC=1 

 
Fig.  4.  The watermarked images and the reconstructed watermark images. 

 

   

 

 

 



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129 

 

Various attacks are used to test the robustness 
of the watermark algorithm, such as noise 

(Gaussian, Salt & Pepper), geometrical distortion  

(cropping), and JPEG compression. Figure (5) 
shows results of applying various attacks to the 

watermarked test image. 
 

    
PSNR= 25.22 dB 

Salt & Pepper noise 

Gaussian Noise mean=0, 

variance=0.00005,         

PSNR= 32.37 dB 

Crop 1/9 of image 

PSNR=15.12 dB 

JPEG compression 

30% 

PSNR= 32.94 dB 

    
NC= 0.8404 NC=0.8071 NC=0.6241 NC=0.8043 

    
PSNR= 25.01 dB 

Salt & Pepper noise 

 Gaussian Noise mean=0, 

variance=0.0005 

PSNR=  34.84 dB 

Crop 1/9 of image 

PSNR= 16.48 dB 

JPEG compression 

30% 

PSNR= 30.99 dB 

    
NC=0.8950    NC=0.8071 NC= 0.8293 NC= 0.7491 

    
PSNR = 24.90 dB 

Salt & Pepper noise(0.01) 

 Gaussian Noise mean=0, 

variance=0.00005 

PSNR= 35.58 dB 

Crop 2/9 of image 

PSNR= 12.93 dB   

JPEG compression 

30% 

PSNR = 36.44 dB 

 



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130 

 

 

Fig. 5. Results of applying various attacks to the watermarked images. 

 

 

The results of the proposed algorithm were 
compared with [10] in which non blind digital 

image watermarking based on discrete wavelet 

transform has been proposed. The watermark 
image was encrypted by Arnold transform and 

Cross chaotic sequence, the encrypted watermark 

image is transformed into wavelet transform 

domain, and the medium frequency coefficients 
are extracted and embedded into the wavelet 

transformed host image. Figure (6), shows the 

PSNR and NC of both algorithm in [10] and the 
proposed one respectively. 

 

 

Fig. 6. PSNR and NC of both algorithms in [10] and 

the proposed. 

 

 
 

 

    
NC=  0.7631 NC=0.6527 NC= 0.5391   NC=  0.7635 

    
PSNR=  27.68 dB 

Salt & Pepper noise(0.004) 

 Gaussian Noise mean=0, 

variance=0.0005 

PSNR=  31.85 dB 

Crop 3/9 of image 

PSNR= 8.14 dB  

JPEG compression 

30% 

PSNR= 30.99 dB 

    
NC=0.8950    NC=0.6642 NC=  0.4417 NC=  0.7930 

Results of algorithm in  

[10] 

Results of the proposed 

algorithm 

 
 

PSNR=33.243dB. PSNR=40.66 dB 

  



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131 

 

4.3.  The Discussion 
 

The simulation results show that the proposed 

algorithm offers good response in terms of 
quantitative parameters (Peak-Signal-to-Noise- 

Ratio and normalized correlation) and in terms of 

visual quality. From Figure (4), the values of 

PSNR of the watermarked images are between 
(40.66- 40.88) dB for the test images which is 

perfectly convenient for the human eyes since 

values of PSNR greater than 30 dBs indicate 
convenient image quality. Also the extracted 

watermark test images have normalized 

correlation values of one which means exact 

similarity between the original and the 
watermarked images. The inviolability of the 

watermark versus remove or degradation attempts 

by attacks of different digital signal processing 
operations like noise (Gaussian, Salt & Pepper), 

different rates of geometrical distortion 

(cropping), and JPEG compression is measured by 
NC criteria. From the results of Figure (5), it is 

seen that watermark can be still extracted despite 

the degradation of image quality due to different 

types of attacks. Also from Figure (6), it can be 
noticed that the proposed algorithm offers better 

imperceptible results. The improvement in the 

PSNR value for Boat image is 7.417 dB. 

 

 

5. Conclusion 
 

In this paper, a non-blind digital image 

watermarking scheme is proposed. It depends on 

using Berkeley Wavelet Transform (BWT), which 
is a complete orthonormal basis functions. The 

watermark image is firstly scrambled by Arnold 

transform to achieve higher embedding security. 

The embedding process is done in the whole 
frequency subbands of the transformed host 

image. The evaluation of the proposed method in 

terms of PSNR and normalized correlation shows 
very good performance as far as invisibility. Also 

the proposed method offers good response under 

various image attacks. 

 
 

Notation 

 
BWT Berkeley Wavelet Transform 
H host image in BWT domain 
Im(x,y) original image pixel 

Im(x′ , y′)n  pixels of encrypted image acquired 
by n- times Arnold transform. 

MSE Mean Square Error 

NC Normalized Correlation 
PSNR Peak Signal to Noise Ratio 
WtImg watermarked image 
w(m,m) watermark image 

 

 
 

Greek letters 

 
𝛼 scaling factor 

β0 DC term of  BWT 
βθ,ϕ

m ,n ,s
 daughter wavelet of BWT 

wm ,n ,s
θ,ϕ

 BWT coefficients 

 

 

6. References 
 
[1] Ramani K., Prasad E.V., Naidu V.L., and 

Ganesh D.,”Color Image Watermarking Bi-
Orthogonal Wavelet Transform”, 

International Journal of Computer 

Application, vol.11, no.9, pp. 25-29, 
December, 2010. 

[2] Ram B., “Digital Image Watermarking 
Technique Using Discrete Wavelet 

Transform and Discrete Cosine Transform”, 
International Journal of Advancements in 

Research & Technology, Vol. 2, No.4, pp.19-

27,April-2013. 
[3] Lafta M.M. and Alwan I.M.,” Watermarking 

in Image Using Slantlet Transform”, Iraqi 

Journal of Science, Vol.52, No.2, pp. 225-
230, 2011. 

[4] Bailey K., Curran K.,” An Evaluation of 
Image Based Steganography Methods”, 

Multimedia Tools & Applications, vol. 30, 
no. 1,pp. 55-88,July 2006. 

[5] Gutub A., Fattani M.,” A Novel Arabic Text 
Steganography Method Using Letter Points 
and Extensions”, WASET International 

Conference on Computer, Information and 

System Science and Engineering, Vienna 
Austria, May 25-27, 2005. 

[6] Taiyue W., Hongwei LI.,” A Novel 
Scrambling Digital Image Watermarking 

Algorithm Based on Counterlet Transform”, 
Journal of Natural Science, vol. 19, no. 4, 

pp.315-322, 2014. 

[7] Hien T.D., Miyara K., Nagata Y.,Nakao Z., 
and Chen Y.W.,” Curvelet Transform Based 

Logo Watermarking”, Innovative Algorithms 

and Techniques in Automation, Industrial 

Electronics and Telecommunications, pp 
305-309, Springer 2007. 



Iman M.G. Alwan                          Al-Khwarizmi Engineering Journal, Vol. 12, No. 2, P.P. 133- 124(2016) 

 

132 

 

[8] Willmore B., Prenger R.J., Wu M. C-K, and 
Gallant J.L.,” The Berkeley Wavelet 

Transform: A biologically-inspired 

orthogonal wavelet transform”, MIT Press 
Journals, Neural Computation, vol. 20, no.6, 

pp. 1537-1564, June 2008.  

[9] Ravindran R. P., Soman K.P.,” Berkeley 
Wavelet Transform Based Image 
Watermarking”, International Conference on 

Advances in Recent Technologies in 

Communication and Computing, India, pp. 
357-359, 2009. 

[10] Pradhan C., RathS. and Kumar Bisoi A. ,       
” Non Blind Digital Watermarking 

Technique Using DWT and Cross Chaos”,  

2nd International conference on 
communication, computing, and security, 

India, Procedia Technology, Elsevier Ltd., 

no.6, pp. 897-904, 2012. 

[11] Pan J.S., Huang H.C., and Jain L.C.,  
”Intelligent Watermarking Techniques”, 

World Scientific Publishing Co. Pte. Ltd., 

2004. 

 



 (2016) 133- 124، صفحة 2، العذد12دجلة الخىارزمي الهنذسية المجلم                                           ايمان محمذ جعفر                

133 

 

 

 
 Berkeley Waveletو تحىيلة  Arnoldالعالمة المائية الرقمية بإستعمال تحىيلة 

 

 **ايناس مظفر جميل*             جعفر علىان ايمان محمذ
 جاهعت بغذاد / كليت التزبيت للبٌاث / قسن الحاسباث** ,*

  ainms_66@yahoo.com :البزيذ األلكتزوًي*

 enasm.altai@gmail.com:البزيذاأللكتزوًي**

 

 

 الخالصة

 
يقذم هذا البحث  خىارسهيت و تعزف عوليت تضويي البياًاث في وسط رقوي كولف الفذيى, أو الولف الصىتي , أو الصىرة  بالعالهت الوائيت الزقويت

 Arnoldفي هذٍ الخىارسهيت الوقتزحت يتن اوال تشفيز الصىرة السزيت باستعوال تحىيلت .  Berkeley Waveletالعالهت الوائيت  التي تعتوذ علً تحىيلت 

أظهزث الٌتائج العوليت جىدة هذٍ الخىارسهيت الوقتزحت وهتاًتها . هي ثن تٌفذ عوليت االخفاء في هجال التحىيلت للصىرة الوضيفتو, لتحقيق هستىي أهاى أعلً

. عالجت الصىريتلبعض عولياث الن