A Digital Image Watermarking Method Based on Tight Framelet

Runhai Jiao School of Control and Computer Engineering

North China Electric Power University Beijing, China

runhaijiao@ncepu.edu.cn

Biying Lin School of Control and Computer Engineering

North China Electric Power University Beijing, China

linbiyingw@126.com

Abstract. Compared with wavelet, tight framelet has good time-frequency analysis ability. In this paper, we propose a new digital image watermarking algorithm based on tight framelet. In our method, the host image is firstly decomposed by framelet. Secondly, the watermark image is transformed by Arnold scrambling and then is embedded in the lowest frequency subband of the host image. Experimental results show that the proposed method gains good performance in robustness and imperceptibility with respect to traditional image processing including noise attack, filtering and JPEG compression. Moreover, the watermark imperceptibility of our method is better than that of wavelet.

Keywords-image; watermarking; tight framelet; robustness; imperceptibility

I. INTRODUCTION With the rapid development of internet and

multimedia technology, the transmission and access of digital media production (such as image, video and audio) becomes more convenient. This has created an environment that the digital information is easy to distribute, duplicate and modify, which leads to need for effective copyright protection techniques. Researchers have proposed many kinds of solutions. Watermarking is one kind of popular technical that the owner embeds a digital marking into a host media to verify its ownership. Digital watermark has several characteristics such as imperceptibility, unambiguous, robustness and security[1]. Among these characteristics, imperceptibility and robustness is the basic but the most important. The watermark should be minimally perceptible, and at the same time it should not be removed or modified easily.

In the past few years, researchers have put much attention to this area and proposed many methods. Generally, according to the way of embedding watermark, these methods can be divided into two kinds: pixel domain[2,3] and frequency domain[4-6]. Experimental results show that the watermark embedded in frequency domain is more tolerant to attacks and image processing than that in pixel domain, which leads the research focus to frequency domain. Compared with DCT, wavelet has both time and frequency analysis capability, and is more according with Human Visual System(HVS), which is helpful for watermark embedding. That is, embedding the watermark in the eye insensitive area will improve the watermark imperceptibility. Therefore, wavelet domain watermarking method prevails in the recently years.

Framelet is very similar to wavelets but has some important differences. Framelet has two or more high frequency filter banks, which produces more subbands in decomposition. This can achieve better time-frequency localization[7] ability in signal processing. Moreover, framelet is more robust. There is redundancy between the framelet subbands, which means change on coefficients of one band can be compensated by other subbands coefficients. This can help to improve the imperceptibility of the watermark.

In this paper, we propose a new image watermarking method in framelet domain. The rests of the paper are organized as follows. Section Ⅱ analyzes the characteristic of framelet transform. Section Ⅲ illustrates the embedding and extracting process of our method. Experiments are given in section Ⅳ. Section Ⅴ draws the conclusion.

II. FRAMELET TRANSORM ANALYSIS Framelet has important difference with wavelet.

In[8-9], A.Ron, Z.shen and I. Daubechies gave the definition of MRA-based framelet.

Suppose ( )2dL Rφ ∈ , 0V is the space generated

by ( ){ }, dx k k Zφ − ∈ . We define ( )( ) ( )2 2dDf y f y= ,

0 ,j jV D V j Z= ∈ . φ Generates a MRA, and

1 ,j j jj

V V V+∞

+=−∞

⊂ ⊂… … ∪ is dense in ( )d dL R .

{ }1 2, ,..... nW ψ ψ ψ= is the mother wavelet function and satisfies 1W V⊂ . Suppose

( ) ( )2, 2 2 , , ,

jdj d

i kX W x k W j Z k Zψ ψ ψ⎧ ⎫⎪ ⎪= = − ∈ ∈ ∈⎨ ⎬⎪ ⎪⎩ ⎭

(1)

( )X W is a tight framelet in ( )d dL R . φ is called as scaling function, and functions in W are called as mother wavelet functions.

From above definition, we can see that framelet has at least one mother wavelet function. By comparison, wavelet has one scaling function and one mother wavelet function. Therefore, wavelet can be thought as the special kind of framelet. Compared with wavelet, framelet is more flexible and has higher free degree. Considering the computation complexity, we adopt the framelet with one scaling function ( )tφ and two mother

2010 International Conference on Web Information Systems and Mining

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DOI 10.1109/WISM.2010.86

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wavelet functions 1( )tψ and 2 ( )tψ , which can be defined by the following low pass(scaling) filter 0 ( )h n and high pass(wavelet) filters 1( )h n and 2 ( )h n :

0( ) 2 ( ) (2 )

( ) 2 ( ) (2 ), 1, 2n

i in

t h n t n

t h n t n i

φ φ

ψ φ

= −

= − =

∑

∑ (2)

Under the guide of the decomposition method proposed by Burt and Adelson, Mallat proposes the framelet pyramid decomposition structure for two dimension signal[10].

'

''

1

, , 1

', 1

'', 1

, 1,..., 4; 1,2; 1,2

, 5,6; 1,2

, 7,8; 1,2

j c r jij l c m r j

ij c m r j

ij l c r j

C H H CD G G C i l m

D H G C i m

D G H C i l

−

−

−

−

== = = =

= = =

= = =

(3)

In above, ' ''

, , ,i i ij j j jC D D D are corresponding to the

low frequency part and high frequency part of image 1jC − . One-level framelet decomposition on image generates nine subbands, and the structure is illustrated in Fig. 1.

HH HG1 HG2

G1H G1G1 G1G2

G2H G2G1 G2G2

Figure 1 One-level framelet decomposition

HH is the approximate part and the others are the detail parts. Each subband size is quarter of the original image size because the sample is down-sampled in the decomposition stage[11]. That is, the subband size is (N/2, N/2) if the original image size is (N, N). Performing decomposition on HH again, we can obtain nine coarse subbands and each coarse subbands size is (N/4, N/4). Continuous decomposition on HH will generates the pyramid structure of framelet decomposition.

Framelet transforms the image from pixel domain to frequency domain. Subbands of Multi-level framelet decomposition constitute a multi-resolution analysis of the original image. Each subband is the representation of the original image in specific frequency range. Most of the image energy is compacted into the lowest frequency subband, and the framelet coefficients usually have big magnitude. This means that we can enlarge the embedding strength on these coefficients, which will improves watermark robustness.

Framelet breaks out the restriction of orthogonality, and there are correlations between the framelet filters. After framelet decomposition, the coefficients in one subband have correlation with coefficients in other subband. This means that changes on one coefficient can be compensated by its related coefficients in reconstruction stage, which produces less noise in the

original signal. This characteristic is very helpful for watermarking. In frequency domain watermarking, the host image is first transformed to frequency coefficients, then watermark is embedded into the coefficients, finally these coefficients were reversibly transformed to generate the watermarked image. Thus, embedding watermark into coefficients can be thought as adding noise to host image, which makes the watermarked image quality degrade compared with the original host image. Since framelet is not orthogonal, the noise produced by watermark embedding can be compensated by its related coefficients, which will be help to keep the watermarked image quality.

III. PROPOSED METHOD

A. Watermark embedding After framelet decomposition, most of the image

energy is compacted into to the lowest subband. In our method, watermark is embedded into this subband. The framework of watermark embedding is in Fig. 2.

Figure 2 Watermark embedding procedure

In order to improve the security and robustness of the watermark, this paper use Arnold transform to scrambling the watermark. The transform can be shown as follows

'

'

1 1(mod )

1 2XX

NYY

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

(4)

Because Arnold transform is cycle, if the watermark image is transformed n(n<N) times we can get the scrambled image. The scrambled image is turned N-n times we can get the original watermark image.

The host image used in this paper is 256×256, and the watermark image is 32×32. We apply three-level framelet transform on the host image and embed watermark by using the following formula

i i iC C W Cα= + (5) where iC is the framelet coefficient,. {1, 1}iW ∈ − is the pixel value of the binary watermark image, and α is the parameter controlling watermark embedding strength.

After embedding the watermark in the lowest subband, we use three-level inverse framelet transform to obtain the watermarked image.

B. Watermark extracting Generally, the procedure of watermark extracting is

an inverse process of watermark embedding. The framework is as follows

Figure 3 Watermark extracting procedure

Three-level framelet decomposition is performed on watermarked image. For each coefficient of the lowest

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subband we extract the watermark bit by using the following formula

' '( ) /i i iW C C α= − (6) Where '

iC is the framelet coefficient of the watermarked image, and iC is the corresponding framelet coefficient of the original image. Using formula (6) we can extract each watermark bit, and then reproduce the watermark image after Arnold transform.

IV. EXPERIMENTAL RESULTS Our proposed algorithm is implemented in Matlab6.5.

The framelet used in our algorithm has one low pass filter and two high pass filters. The filter coefficients are given below:

-0.03243267,-0.20234149,0.27938597,-0.10133703,2

0.27938597,-0.20234149,-0.03243267

0.03294782,0.20555539,-0.32060556,0,0.32060556,1 2

-0.20555539,-0.03294782

-0.00580348,-0.03620686,0.2 2

H

G

G

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞

= ⎜ ⎟⎝ ⎠

=25881349,0.56632033,

0.25881349,-0.03620686,-0.00580348⎛ ⎞⎜ ⎟⎝ ⎠

We use this filter group to apply three-level framelet pyramid decomposition on host image.

The watermark embedding strength α is a key parameter. If α is too large, the watermark has good robustness, but the watermarked image quality will be too poor. Otherwise, if α is too small, the watermark’s robustness is poor, and it can’t resist image attack. Considering the purpose of watermark is usually for copyright protection, so the precondition is that the watermark embedding should not make noticeable visible noise to host image. In our paper, the parameter α value is 0.03. Under this condition, the watermarked image quality PSNR(compared with host image) is above 35dB. Past research has shown that no difference can be noticed by human eyes between the watermarked image and the host image when the watermarked image quality reaches to 35dB.

Four kinds of attacks were chosen in our experiment to evaluate the performance of our proposed method. We chose PSNR(Peak Signal Noise Ratio) to evaluate the watermarked image’s quality which verify the imperceptibility of the watermark, and chose NC(Normalized Correlation) to describe the correlation between the original watermark and the extracted watermark which can verify the robustness of the watermark. Here, NC is calculated by the following formula:

'1 ( , ) /i iNC XOR X X N= − (7)

where XOR is exclusive or operation, iX and '

iX represent the original and extracted watermark respectively, and [1, ]i N∈ , N is the total pixel number of watermark image.

We chose several 256×256×8 bit gray images as host images. Considering paper length, we just give the result on standard image Lena, however similar results are also obtained on other images.

In Fig. 4, we give the host image Lena and the binary watermark image(32×32).

From the experimental results we can see that our watermarked image still keep good visual quality compared with the original host image. This proves the imperceptibility of our watermark is good. Moreover, under different kinds of watermark attacks the extracted watermarks are clear enough for recognition. This proves that the robustness of our watermark is also very good.

Figure 4 Host image and watermark image

Test 1: free attack

(a) (b)

Figure 5 (a)watermarked image PSNR=37.58dB (b)extracted watermark NC=1

The watermarked image’s PSNR is 37.58dB. By contrasting with the host image, we can see the watermark embedding cause little influence on host image quality. The value of NC is 1, which shows that the watermark can be perfectly extracted under no attack.

Test 2: Gaussian noise attack, and the attack parameter is 0.005

Figure 6 (a)Gaussian noise attack PSNR=22.89

(b)extracted watermark NC=0.894

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Test 3: JPEG compression attack, compression parameter is 80%

Figure 7 (a)attacked by JPEG compression PSNR=29.78

(b)extracted watermark NC=0.992

Test 4: Image cropping attack

Figure 8 (a)cropping image, PSNR=10.77 (b)extracted watermark NC=0.948

Test 5: Gaussian low pass filter, the filter parameter is [3,0.5]

(a) (b)

Figure 9 (a)Gaussian low pass filter, PSNR=33.76 (b)extracted watermark NC=0.997

In order to evaluate our algorithm’s performance compared with wavelet, we also embedded the same watermark in the lowest frequency subband of Lena, which is decomposed by three-level Daubechies7/9 wavelet filter. We adopted the same watermark embedding formula (5), and set the embedding strength

parameter as 0.03. After inverse wavelet transform, we get the watermarked image, whose PSNR is 36.45dB. However, as mentioned above, the PSNR of watermarked image by using framelet is 37.58dB. This means that framelet watermarking can achieves better imperceptibility.

V. CONCLUSION A digital image watermark method based on framelet

is proposed in this paper. Experimental results show that our method achieves good imperceptibility. At the same time, it has good robustness, and can resists attacks such as Gaussian noise, JPEG compression and image cropping etc. However, in this paper we just embedded the watermark into lowest subband of framelet. The future work is to consider combining the framelet coefficient characteristic with human visual system to determine the watermark embedding position and strength, which can improve the watermark imperceptibility further.

Acknowledgment

This research is sponsored by PhD Youth Teacher Funds of North China Electric Power University, and here shows special thanks for its support.

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