A Logarithmic Quantization Index Modulation Data Hiding Using the Wavelet Transform

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A Logarithmic Quantization Index Modulation Data Hiding Using the Wavelet Transform

Jinhua Liu, Peng Ye

Sichuan Jiuzhou Electric Group Co.ˈLtd Sichuanˈ621000ˈP.R.C [email protected]

Abstract—Conventional quantization-based data hiding algorithms used uniform quantization. This scheme may be easily estimated by averaging on a set of embedded signals.

Furthermore, by uniform quantization, the perceptual characteristics of the original signal are not considered and the watermark energy is distributed uniformly within the original signal, which introduces visual distortions in some parts of it.

Therefore, we introduce a logarithmic quantization-based data hiding method based on the visual model by using the wavelet transform that takes advantage of the properties of Watson’s visual model and logarithmic quantization index modulation (LQIM). Its improved robustness is due to embedding in the high energy blocks of original image and by applying the logarithmic scheme. In the detection scheme, we model the wavelet coefficients of image by the generalized Gaussian distribution (GGD). Under this assumption, the bit error probability of proposed method is analytically calculated.

Performance of the proposed method is analyzed and verified by simulation. Results of experiments demonstrate the imperceptibility of the proposed method and its robustness.

Keywords-Data hiding;logarithmic quantization index modulation; generalized Gaussian distribution; wavelet transform.

I. INTRODUCTION

Data hiding is a process in which some information is embedded within a digital media so that the inserted data becomes part of the media. This technique has been widely studied in recent years for a variety of applications, such as copyright protection, data authentication, etc ^[1]. It is known that current data hiding algorithms can be divided into two classifications: one is the spatial domain methods and the other is the transform domain methods which embed the watermark by modifying the coefficients of a properly chosen transform domain such as discrete cosine transform ^[2]and wavelets ^[3]. The transform domain-based data hiding has been proved to be a better choice for the robust image watermarking scheme ^[4].As a result, the proposed data hiding method is introduced in the transform domain.

Among many quantization-based watermarking algorithms presented so far, the class of Quantization Index Modulation (QIM) methods have been proposed in [5] has grabbed the attention of researchers due to its good rate distortion-

robustness tradeoffs. However, the disadvantage of QIM has been its extreme sensitivity to valumetric scaling. Even small changes in the amplitude of a signal, can result in dramatic increases in bit-error rate (BER).Several papers have addressed this issue. Jiao Li and Cox proposed an adaptive QIM (AQIM) watermarking ^[6]by using modified Watson’s perceptual model. Gonzalez et al developed a rational dither modulation (RDM)^[7] based on using a gain-invariant adaptive quantization step-size at both encoder and decoder. The uniform quantization is optimum when the host signal follows uniform distribution. However, by uniform quantization, the impact on the signal with high amplitude and the signal with low amplitude is the same. Furthermore, the same noise may not affect the signal with high amplitude, but it may seriously affect the signal with small amplitude.

Inspired by [8] and [9], we propose an improved logarithmic quantization by which stronger watermark can be inserted that introduces less distortion to the host signal. First, the host image is segmented into non-overlapping blocks, and the higher ranking blocks are selected in estimated energy measure for watermarking purposes. Second, the discrete wavelet transform is used to transform each block, and the selected wavelet coefficients are transformed using the logarithm function. Then, distortion-compensated quantization is performed by using the optimal quantization step sizes on transformed signal. Finally, the inverse of logarithm function is applied to obtain the watermarked signal. The rest of the paper is organized as follows. In Section 2, a brief review of scalar logarithmic QIM (SLQIM) scheme is presented. The proposed logarithmic quantization-based data hiding algorithm using DWT is introduced in Section 3. Section 4 gives experimental results about the imperceptibility and robustness of the proposed method against common attacks. Finally, Section 5 concludes this paper.

II. BRIEF INTRODUCTION TO THE SLQIM

The scalar logarithmic quantization index modulation (SLQIM) scheme was introduced in [8]. As mentioned in [8], the well-known -Law compression function is studied.

Inspired by the -Law rule, they developed a logarithmic quantization data hiding scheme. To perform the SLQIM, the original image must be transformed as:

2013 Third International Conference on Instrumentation, Measurement, Computer, Communication and Control

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ln 1

, 0, 0

ln 1

s

x

c X X

(1)

where

denotes a parameter defining the compression level andX_srepresents the parameter that scales the original image.

The bestX_s value is the value which spreads most of the original image samples into the range [0, 1]. The values that are larger than one can be converted to the logarithmic domain and be used for embedding. Then, the transformed signal is used for data insertion. In their scheme, the transformed signal was quantized uniformly. Then the watermarked signal can be obtained through expanding the quantized signal as follows:

sgn( ) ^s 1 ^z 1

w

x x X (2) where sgn( ) is the sign function,zis the quantized signal in the transformed domain, andx_wis the watermarked signal. In order to extract the watermark signal, the Euclidean distance decoder is applied. In the logarithmic scheme, zero and one are embedded in the received signal r using the SLQIM method resulting inr₀andr₁, respectively. Thus the watermark signal can be extracted as:

ˆ arg min _i 2, {0,1}

m rr i (3) where mˆrepresents the extracted watermark signal.

According to Eq. (1), the optimum value for parameteris derived by minimizing the quantization distortion. In order to obtain the optimum value for, the watermark power should be found and it can be minimized with respect to.According to [8], the optimum_optof SLQIM can be obtained as:

2 2

2 2 2

(0, )

arg min 1 2 [ ] ln (1 )

[ ] [ ]

s s

opt

X X

E x E x E x

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III. PROPOSED DATA HIDING

A. Data Embedding

The proposed method performs the data embedding in the following steps.

Step1: Apply Pseudo-random Noise (PN) generator to create watermark signal. Letbi { 1,1} be the watermark signal.

Step2: Segment the original image into non- overlapping L L blocks and choose the k higher ranking blocks in estimated entropy measure for data hiding purposes.

Step3: The three-level DWT is applied to each selected

image block. Thus the

3 ₁ ₁ ₁ ₂ ₂ ₂, 3 3 3

LL , LH , HL , HH , LH , HL , HH LH , HL , HH subband images are obtained respectively. Then we compute the averagel₁-norm (or energy) of subband images at each decomposition level. Finally, the highest energy subband coefficients are chosen as the watermark embedding space.

LetI {I_j( , )}m n be the selected wavelet coefficients in the

highest energy subband, and letx[ ,x x₁ ₂,...,x_N]be the 1-D vector by zigzag scanning forI ,whereN m n.

,

1 1

1 ( , )

s s

M N

j j

m n

s s

E I m n

M N

(5) where M_sN_s denotes the size of the subband image I_j,j1, 2,3,1, 2,3, 4 ⁱⁿ ^each block. 1, 2,3, 4 represents the low-frequency direction, horizontal direction, vertical direction and diagonal direction respectively. Where j denotes the j -^th level of DWT decomposition.

Step4: Transform the set of wavelet coefficients[ ,x x₁ ₂,...,x_N] by using logarithmic function (1).

Letc[ ,c c₁ ₂,...,c_N]be the set of transformed coefficients.

After compression using (1), DC-QIM (Distortion Compensated- Quantization Index Modulation) method is applied to the transformed signalc_ito embed the watermark signal as:

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

i i

i b i i b i

z Q c ! c Q c i N (6)

where ( ) ( )

i

i i

b i i

c b

Q c round " " "b

" .b_i { 1,1}is thei th watermark bit, and ( )

bi i

Q c is the adaptive quantizer,which applied to the transformed signerc_i. "is the quantization step size, ! represents the quantization scalar factor.

Step5: The watermark signal is embedded into the selected wavelet coefficients. Thus, the watermarked signaly_ican be obtained as follows:

sgn( ) ^s 1 ^zⁱ 1

i i

y z X

(7) Step6: The inverse discrete wavelet transform to the watermarked image is utilized, then combining with the non- watermarked blocks to obtain the watermarked image.

B. Watermark Detection

The proposed method performs the watermark decoding in the following steps.

Step 1: Segment the received image into non- overlappingL L blocks and choose thekhigh entropy image block for watermark detection purposes.

Step 2: The three-levels DWT is applied to each block.

According to equation (5), the received signal is the wavelet coefficients with high energy in each block. Thus the received signal sample should be first transformed into logarithmic domain using equation (1) which results incˆ_i.

Step 3: For each block, decoding can be performed as the following function

ˆ arg min ˆ ( ) ,ˆ { 1,1}

i i bi i i

b c Q c b (8) wherecˆ_i is obtained using equation (1), ˆ

bi is the extracted watermark data.

Step 4: The watermark signal can be obtained by combining the extracted watermark of each block.

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IV. EXPERIMENTAL RESULTS

In order to test the proposed data hiding scheme, two well- known 512×512 images Lena and Barbara are used for performance evaluation from two aspects: imperceptibility and robustness.

A. Imperceptibility Test

In this section, our experiments on various block sizes and watermark message lengths indicated that the case of 16×16 block size and 256 higher entropy blocks are the best compromise. Then, the low-frequency coefficients of the third level are used for quantization, resulting in embedding 1024 bits in a 512×512 original image. Fig.1 shows the experimental results of the imperceptibility test for the well- know images. As can be seen, the invisibility of proposed method is satisfied. The Peak-Signal-to-Noise-Ratio (PSNR) between the original and the watermarked images are 50.3221 dB and 51.1502 dB, respectively.

More importantly, the proposed method provides image- dependent watermark which has strong components in the complex part of the image, such as in the complex texture region, which is hard to see. Therefore, we can embed strong watermark into the complex texture region of image. Thus, the perceptual quality of the watermarked image can be kept at acceptable level. Hence; the proposed method has good invisibility without any attacks.

⁵⁰ ¹⁰⁰ ¹⁵⁰ ²⁰⁰ ²⁵⁰ ³⁰⁰ ³⁵⁰ ⁴⁰⁰ ⁴⁵⁰ ⁵⁰⁰

50

100

150

200

250

300

350

400

450

500

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Fig.1. Originalˈwatermarked and difference images using the proposed method for Lena and Barbara. For each image, the left one is the original image, the middle one is the watermarked image, and the right one is absolute difference between the watermarked and the original image.

B. Robustness Test

Apart from the imperceptibility test, for the purpose of saving paper’s space, in this regard, we have computed the probability of error of proposed method under AWGN and JPEG compression attacks. It is well known that the wavelet coefficients of image are highly non-Gaussian; therefore the generalized Gaussian distribution (GGD) has been used for modeling the transform domain, Fig. 2 shows that the histograms of wavelet coefficients and the histogram of wavelet coefficients together with a plot of the fitted GGD.

From Fig.2, the fits are quite good. Therefore the marginal distribution of wavelet coefficients can be well modeled by the two parameters of GGD model.

-60 -40 -20 0 20 40 60

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

mu = 0.03,alpha = 2.1997, beta = 0.8581

Histogram

^-60 ^-40 ^-20 ⁰ ²⁰ ⁴⁰ ⁶⁰

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

mu = -0.115,alpha = 1.7904,beta =0.671

Histogram

(a) (b)

Fig.2 Wavelet subband coefficient histogram fitted with a generalized Gaussian distribution for Lena image (a) horizontal part (b) vertical part.

In order to derive the bit error probability, the Additive White Gaussian Noise (AWGN) channel is considered.

Error in detection occurs when noise causes the received signal to fall into a wrong region. The bit error probability is calculated by using the method from [8]. Fig.3 shows the probability of error under AWGN and JPEG compression attacks for Lena image respectively. From Fig.3, it can be seen that the proposed method has slightly better performance than [8] and [9] from the results as shown in Fig.3.The main reasons are summarized as follows:

(1)The watermark signal is embedded into the high energy of host image. By using this scheme, the robustness of watermarking system can be improved.

(2)A logarithmic quantization approach is used to embed the watermark signal by modulating a set of wavelet coefficients. Using logarithmic scheme, the effect of quantization distortion on the signal with small amplitude can be reduced.

(3)The wavelet coefficients are modeled by the generalized Gaussian distribution. Using this model, it can well capture the features of wavelet coefficients and the performance of watermark can be improved through GGD model.

Furthermore, thanks to the good multi-resolution of discrete wavelet transform and the similar with human vision perceptual model.

Noise variance

Probabilityoferror

DM-logarithmic[8]

LQIM[9]

Proposed method

Quality factor

Probabilityoferror

DM-logarithmic[8]

LQIM[9]

Proposed method

(a)JPEG compression (b) AWGN attack

Fig.3. Probability of error under JPEG compression and AWGN attack.

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V. CONCLUSIONS

In this paper, an innovative logarithmic domain-based quantization index modulation watermarking by applying the discrete wavelet transform for still gray level images in the logarithmic domain from the method of [8] is proposed, which exploits the characteristics of the compression function

^-law

standard for quantization. Due to the use of logarithmic function, smaller step sizes are devoted to smaller amplitudes and larger step sizes are associated with larger amplitudes. As well, due to the use of entropy and the optimal quantization scalar factor, the probability of error was obtained and the experimental results are verified in comparison with previous quantization-based watermarking. Simulations show better robustness of the proposed method in comparison with previous quantization-based algorithms, in similar perceptual qualities of watermarked images. Further work will focus on developing robust watermarking system against geometric distortion attacks using logarithmic quantization scheme.

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