# Haar-Wavelet-Based Just Noticeable Distortion Model for Transparent Watermark

Authors: Lu-Ting Ko; Jwu-E Chen; Hsi-Chin Hsin; Yaw-Shih Shieh; Tze-Yun Sung
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Watermark transparency is required mainly for copyright protection. Based on the characteristics of human visual system, the just noticeable distortion (JND) can be used to verify the transparency requirement. More specifically, any watermarks whose intensities are less than the JND values of an image can be added without degrading the visual quality. It takes extensive experimentations for an appropriate JND model. Motivated by the texture masking effect and the spatial masking effect, which are key factors of JND, Chou and Li (1995) proposed the well-known full-band JND model for the transparent watermark applications. In this paper, we propose a novel JND model based on discrete wavelet transform. Experimental results show that the performance of the proposed JND model is comparable to that of the full-band JND model. However, it has the advantage of saving a lot of computation time; the speed is about 6 times faster than that of the full-band JND model.


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## Body

## 1. Introduction

Watermarking is a process that hides information into a host image for the purpose of copyright protection, integrity checking, or captioning [1–3]. In order to achieve the transparency of watermark, many commonly used techniques are based on the characteristics of human visual system (HVS) [1–13]. Jayant et al. [14, 15] introduced a key concept known as the just noticeable distortion (JND), against which insignificant errors are not perceptible by human eyes. The JND of an image is in general dependent on background luminance, contrast of luminance, and dominant spatial frequency. It takes extensive experimentations to obtain an appropriate JND model.Perceptual redundancies refer to the details of an image that are not perceivable by human eyes and therefore can be discarded without affecting the visual quality. As noted, human visual perception is sensitive to the contrast of luminance rather than their individual values [16–18]. In addition, the visibility of stimuli can be reduced by nonuniformly quantizing the background luminance [18–20]. The above known as the texture masking effect and the spatial masking effect are key factors that affect the JND of an image. Chou and Li proposed an effective model called the full-band JND model for the transparent watermark applications [21].Wavelet transform provides an efficient multiresolution representation with various desirable properties such as subband decompositions with orientation selectivity and joint space-spatial frequency localization. In wavelet domain, the higher detailed information of a signal is projected onto the shorter basis function with higher spatial resolution; the lower detailed information is projected onto the larger basis function with higher spectral resolution. This matches the characteristics of HVS. Many wavelet-transform-based algorithms were proposed for various applications [22–34].In this paper, we propose a wavelet-transform-based JND model for the watermark applications. It has the advantage of saving a lot of computation time. The remainder of the paper proceeds as follows. In Section2, the full-band JND model is reviewed briefly. In Section 3, the discrete-wavelet-transform- (DWT-) based JND model is proposed. The modified DWT-based JND model and its evaluation are presented in Section 4. Conclusion can be found in Section 5.


## 2. Review of the Full-Band Just Noticeable Distortion (JND) Model

The full-band JND model [21] makes use of the properties of the HVS to measure the perceptual redundancies of an image. It produces the JND file for image pixels as follows:(2.1)JND(i,j)=max⁡⁡{f1(bg(i,j),mg(i,j)),f2(bg(i,j))},
where
(2.2)f1(bg(i,j),mg(i,j))=mg(i,j)×[α(bg(i,j))]+β(bg(i,j)),(2.3)f2(bg(i,j))={17×(1-(bg(i,j)127)1/2)+3forbg(i,j)≤127,3128×(bg(i,j)-127)+3forbg(i,j)>127,(2.4)α(bg(i,j))=bg(i,j)×0.0001+0.115,(2.5)β(bg(i,j))=12-bg(i,j)×0.01,(2.6)bg(i,j)=132∑w=15∑z=15p(i-3+w,j-3+z)×B(w,z),(2.7)mg(i,j)=max⁡k=1,2,3,4{|gradk(i,j)|},(2.8)gradk(i,j)=116∑w=15∑z=15p(i-3+w,j-3+z)×Gk(w,z),
where f1(i,j) and f2(i,j) are the texture mask and the spatial mask, respectively, as mentioned in Section 1, bg(i,j) is the average background luminance obtained by using a low-pass filter, B(w,z), is given in Figure 1, mg(i,j) is the maximum gradient obtained by using a set of high-pass filters, Gk(w,z),k=1,2,3,4, are given in Figure 2, functions α(∘) and β(∘) are dependent on the average background luminance, and p(i,j) is the luminance value at pixel position (i,j).Figure 1
Low-pass filter used in (2.6).A set of high-pass filters used in  (2.8).
(a)(b)(c)(d)


## 3. Discrete-Wavelet-Transform-Based JND Model

In this section, we propose a novel JND model based on discrete wavelet transform. It has the advantage of reducing computational complexity significantly.


### 3.1. Discrete Wavelet Transform

Discrete wavelet transform (DWT) provides an efficient multiresolution analysis for signals, Specifically, any finite energy signalf(x) can be written by(3.1)f(x)=∑nSJ(n)ϕJn(x)+∑l≤J∑nDl(n)ψln(x),
where ℓ denotes the resolution index with larger values meaning coarser resolutions, n is the translation index, ψ(x) is a mother wavelet, ϕ(x) is the corresponding scaling function, ψℓn(x)=2-ℓ/2ψ(2-ℓx-n), ϕℓn(x)=2-ℓ/2ϕ(2-ℓx-n), SJ(n) is the scaling coefficient representing the approximation information of f(x) at the coarsest resolution 2J, and Dℓ(n) is the wavelet coefficient representing the detail information of f(x) at resolution 2ℓ. Coefficients Sℓ(n) and Dℓ(n) can be obtained from the scaling coefficient Sℓ-1(n) at the next finer resolution 2ℓ-1 by using 1-level DWT, which is given by(3.2)Sl(n)=∑kSl-1(k)h(2n-k),Dl(n)=∑kSl-1(k)g(2n-k),
where h(n)=〈ϕ,ϕ-1,-n〉, g(n)=〈ψ,ϕ-1,-n〉, and 〈·,·〉 denotes the inner product. It is noted that h(n) and g(n) are the corresponding low-pass filter and high-pass filter, respectively. Moreover, Sℓ-1(n) can be reconstructed from Sℓ(n) and Dℓ(n) by using the inverse DWT, which is given by(3.3)Sl-1(n)=∑kSl(k)h̃(n-2k)+∑kDl(k)g̃(n-2k),
where h̃(n)=h(-n) and g̃(n)=g(-n).For image applications, 2D DWT can be obtained by using the tensor product of 1D DWT. Among wavelets, Haar wavelet [22] is the simplest one, which has been widely used for many applications. The low-pass filter and high-pass filter of Haar wavelet are as follows:(3.4)h(0)=0.5,h(1)=0.5,g(0)=0.5,g(1)=-0.5.
Figures 3 and 4 show the row decomposition and the column decomposition using Haar wavelet, respectively. Notice that the column decomposition may follow the row decomposition, or vice versa, in 2D DWT:(3.5)LL=A+B+C+D4,LH=A+B-C-D4,HL=A-B+C-D4,HH=A-B-C+D4,
where A, B, C, and D are pixel values, and LL, LH, HL, and HH denote the approximation, detail information in the horizontal, vertical, and diagonal orientations, respectively, of the input image. Figure 5 shows 1-level, 2D DWT using Haar wavelet.Figure 3
The row decomposition using Haar wavelet (A, B, C and D, are pixel values).Figure 4
The column decomposition using Haar wavelet (E, F, G, and H are pixel values).Figure 5
1-level 2D DWT using Haar wavelet (A, B, C, and D are pixel values).TheLL subband of an image can be further decomposed into four subbands: LLLL, LLLH, LLHL, and LLHH at the next coarser resolution, which together with LH, HL, and HH forms the 2-level DWT of the input image. Thus, higher level DWT can be obtained by decomposing the approximation subband in the recursive manner.


### 3.2. Proposed DWT-Based JND Model

As mentioned in Section2, the full-band JND model [21] consists of (2.1)–(2.8), which is essentially computation consuming. In order to simplify computational complexity, a novel JND model based on Haar wavelet is proposed as follows.(3.6)JNDDWT(i,j)=max⁡{f1,DWT(i,j),f2,DWT(i,j)},
where f1,DWT(i,j) and f2,DWT(i,j) are the proposed texture mask and spatial mask, respectively, based on DWT, which are given by(3.7)f1,DWT(i,j)=|Di(i,j)⋅α[LLLL′(i,j)]+β[LLLL′(i,j)]|,(3.8)f2,DWT(i,j)={17×(1-[LLLL′(i,j)127]1/2)+3forLLLL′(i,j)≤127,3128×[LLLL′(i,j)-127]+3forLLLL′(i,j)>127,
where(3.9)α[LLLL′(i,j)]=LLLL′(i,j)×0.0001+0.115,(3.10)β[LLLL′(i,j)]=12-LLLL′(i,j)×0.01,(3.11)LLLL′=T2T1[LLLL]T1TT2T,(3.12)T1=[10100101]4×2,(3.13)T2=[10001000010001000010001000010001]8×4.LLLL′ is the modified LLLL, which replaces the low-pass filter B(w,z) of the full-band JND model. Di(i,j) replaces the maximum gradient, mg(i,j), which is given by(3.14)Di(i,j)=max⁡{|LH′(i,j)|,|HL′(i,j)|,|SLR′(i,j)|,|SLL′(i,j)|},
where(3.15)LH′=T2[LH]T2T,(3.16)HL′=T2[HL]T2T,(3.17)SLR′=T2[SLR]T2T,(3.18)SLL′=T2[SLL]T2T,(3.19)SLR=LH+HL2,(3.20)SLL=LH-HL2.LH′, HL′, SLR′, and SLL′ replace G1(w,z), G2(w,z), G3(w,z), and G4(w,z), respectively.


## 3.1. Discrete Wavelet Transform

Discrete wavelet transform (DWT) provides an efficient multiresolution analysis for signals, Specifically, any finite energy signalf(x) can be written by(3.1)f(x)=∑nSJ(n)ϕJn(x)+∑l≤J∑nDl(n)ψln(x),
where ℓ denotes the resolution index with larger values meaning coarser resolutions, n is the translation index, ψ(x) is a mother wavelet, ϕ(x) is the corresponding scaling function, ψℓn(x)=2-ℓ/2ψ(2-ℓx-n), ϕℓn(x)=2-ℓ/2ϕ(2-ℓx-n), SJ(n) is the scaling coefficient representing the approximation information of f(x) at the coarsest resolution 2J, and Dℓ(n) is the wavelet coefficient representing the detail information of f(x) at resolution 2ℓ. Coefficients Sℓ(n) and Dℓ(n) can be obtained from the scaling coefficient Sℓ-1(n) at the next finer resolution 2ℓ-1 by using 1-level DWT, which is given by(3.2)Sl(n)=∑kSl-1(k)h(2n-k),Dl(n)=∑kSl-1(k)g(2n-k),
where h(n)=〈ϕ,ϕ-1,-n〉, g(n)=〈ψ,ϕ-1,-n〉, and 〈·,·〉 denotes the inner product. It is noted that h(n) and g(n) are the corresponding low-pass filter and high-pass filter, respectively. Moreover, Sℓ-1(n) can be reconstructed from Sℓ(n) and Dℓ(n) by using the inverse DWT, which is given by(3.3)Sl-1(n)=∑kSl(k)h̃(n-2k)+∑kDl(k)g̃(n-2k),
where h̃(n)=h(-n) and g̃(n)=g(-n).For image applications, 2D DWT can be obtained by using the tensor product of 1D DWT. Among wavelets, Haar wavelet [22] is the simplest one, which has been widely used for many applications. The low-pass filter and high-pass filter of Haar wavelet are as follows:(3.4)h(0)=0.5,h(1)=0.5,g(0)=0.5,g(1)=-0.5.
Figures 3 and 4 show the row decomposition and the column decomposition using Haar wavelet, respectively. Notice that the column decomposition may follow the row decomposition, or vice versa, in 2D DWT:(3.5)LL=A+B+C+D4,LH=A+B-C-D4,HL=A-B+C-D4,HH=A-B-C+D4,
where A, B, C, and D are pixel values, and LL, LH, HL, and HH denote the approximation, detail information in the horizontal, vertical, and diagonal orientations, respectively, of the input image. Figure 5 shows 1-level, 2D DWT using Haar wavelet.Figure 3
The row decomposition using Haar wavelet (A, B, C and D, are pixel values).Figure 4
The column decomposition using Haar wavelet (E, F, G, and H are pixel values).Figure 5
1-level 2D DWT using Haar wavelet (A, B, C, and D are pixel values).TheLL subband of an image can be further decomposed into four subbands: LLLL, LLLH, LLHL, and LLHH at the next coarser resolution, which together with LH, HL, and HH forms the 2-level DWT of the input image. Thus, higher level DWT can be obtained by decomposing the approximation subband in the recursive manner.


## 3.2. Proposed DWT-Based JND Model

As mentioned in Section2, the full-band JND model [21] consists of (2.1)–(2.8), which is essentially computation consuming. In order to simplify computational complexity, a novel JND model based on Haar wavelet is proposed as follows.(3.6)JNDDWT(i,j)=max⁡{f1,DWT(i,j),f2,DWT(i,j)},
where f1,DWT(i,j) and f2,DWT(i,j) are the proposed texture mask and spatial mask, respectively, based on DWT, which are given by(3.7)f1,DWT(i,j)=|Di(i,j)⋅α[LLLL′(i,j)]+β[LLLL′(i,j)]|,(3.8)f2,DWT(i,j)={17×(1-[LLLL′(i,j)127]1/2)+3forLLLL′(i,j)≤127,3128×[LLLL′(i,j)-127]+3forLLLL′(i,j)>127,
where(3.9)α[LLLL′(i,j)]=LLLL′(i,j)×0.0001+0.115,(3.10)β[LLLL′(i,j)]=12-LLLL′(i,j)×0.01,(3.11)LLLL′=T2T1[LLLL]T1TT2T,(3.12)T1=[10100101]4×2,(3.13)T2=[10001000010001000010001000010001]8×4.LLLL′ is the modified LLLL, which replaces the low-pass filter B(w,z) of the full-band JND model. Di(i,j) replaces the maximum gradient, mg(i,j), which is given by(3.14)Di(i,j)=max⁡{|LH′(i,j)|,|HL′(i,j)|,|SLR′(i,j)|,|SLL′(i,j)|},
where(3.15)LH′=T2[LH]T2T,(3.16)HL′=T2[HL]T2T,(3.17)SLR′=T2[SLR]T2T,(3.18)SLL′=T2[SLL]T2T,(3.19)SLR=LH+HL2,(3.20)SLL=LH-HL2.LH′, HL′, SLR′, and SLL′ replace G1(w,z), G2(w,z), G3(w,z), and G4(w,z), respectively.


## 4. Modification of the DWT-Based JND Model

In this section, we introduce an adjustable parameter to modify the DWT-based JND model such that the computation time can be reduced significantly while the performance is comparable to that of the benchmark full-band JND model. The test images, namely, Lena, Cameraman, Baboon, Board, and Peppers are shown in the first row of Figure12.


### 4.1. Evaluation of JND Models

Figure6 shows the distortion-tolerant model, which can be used to evaluate JND models. It takes the JND value as noise, adds to the original image, and computes the peak-signal-to-noise ratio (PSNR) defined as(4.1)PSNR=20log⁡(255MSE),
where MSE is the mean squared error, which is defined as(4.2)MSE=1m×n∑i=0m-1∑j=0n-1[JND(i,j)]2,
where image size is m×n. As shown in Table 1, the proposed DWT-based JND model is somewhat different from the benchmark full-band JND model in terms of the PSNR values.Table 1
PSNR comparisons of the benchmark full-band JND model, the proposed DWT-based JND model, and the modified DWT-based JND model.
JND modelLenaCameramanBaboonBoardPeppersFull-band JND32.704129.912234.084525.348630.3052DWT-based JND34.330131.755635.815630.800332.1375The modified DWT-based JND33.711730.961434.093125.719231.7148Figure 6
Distortion-tolerant evaluation model for the proposed JND model.


### 4.2. Modified DWT-Based JND Model

Based on (2.1)–(2.3) and (3.6)–(3.8), one can examine the influences of the dominant mask, texture mask, and spatial mask for the full-band JND model and the DWT-based JND model, respectively. Their respective MSE values are shown in Table 2. As one can see, the proposed texture mask for the DWT-based JND model is less significant than that of the full-band JND model. Thus, we propose an adjustable parameter, K, to modify (3.17) and (3.18) as follows:(4.3)SLR′′=K⋅SLR′,SLL′′=K⋅SLL′,
where SLR′′ and SLL′′ replace SLR′ and  SLL′, respectively. Figures 7, 8, 9, 10, and 11 show the MSE values obtained by modifying the DWT-based texture mask with various K in (4.3). In this paper, the adjustable parameter K is set to 4 after extensive simulations. The performance of the modified DWT-based JND model with K=4 is comparable to that of the full-band JND model in terms of the PSNR and MSE values as shown in Tables 1 and 2, respectively.Table 2
The MSE values due to the dominant mask (case 1), the spatial mask (case 2), and the texture mask (case 3) using the full-band JND model, the DWT-based JND model, and the modified DWT-based JND model.
JND modelCaseLenaCameramanBaboonBoardPeppersFull-band JND134.887666.353425.3878189.765360.6121228.520253.029719.621941.722252.5825310.957418.477212.373165.649212.583DWT-based JND123.992443.402717.041954.082239.7492223.926943.040616.94238.470239.64230.9722.10761.460324.48851.1393The modified DWT-based JND127.625452.112225.3378174.24443.8131223.926943.040616.94238.470239.64236.937913.723314.7008153.40527.8595Figure 7
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Lena image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 8
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Cameraman image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 9
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Baboon image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 10
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Board image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 11
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Peppers image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.(a) The original images, namely, Lena, Cameraman, Baboon, Board, and Peppers; (b), (c), and (d) the noisy images obtained by adding the full-band JND values, the DWT-based JND values, and the modified DWT-based JND values, respectively.
(a)(b)(c)(d)Figure12 shows the noisy images obtained by adding the JND values to the original images (Figure 12(a)) using the full-band JND model (Figure 12(b)), the DWT-based JND model (Figure 12(c)), and the modified DWT-based JND model (Figure 12(d)). It is noted that the images in the second and fourth rows are almost indistinguishable from the original images. As a result, the modified DWT-based JND model is visually comparable to the full-band JND model.


### 4.3. Computation Complexity of the Proposed JND Models

In the full-band JND model, the computation ofbg(i,j) requires 9 multiplications per pixel, the computation of mg(i,j) requires 28 multiplications per pixel, and the computation required for (2.2)–(2.5) is 6 multiplications per pixel. Thus, for an n×n image, it requires 43n2 multiplications. In the proposed DWT-based JND model, the computations of LH′, HL′, LLLL′, SLR′, and SLL′ require (1/4)n2, (1/4)n2, (5/16)n2, (1/4)n2, and (1/4)n2 multiplications, respectively, for an n×n image, and the computation required for (3.7)–(3.10) is also 6 multiplications per pixel. Thus, for an n×n image, it requires 7.3125n2 multiplications. In the modified DWT-based JND model, as the computations of SLR′′ and SLL′′ require no multiplication, it only needs 6.8125n2 multiplications for an n×n image. Figure 13 shows the log plot of numbers of multiplications required for the three JND models versus different image sizes. As a result, the speed of the DWT-based JND model is about 6 times faster than that of the full-band JND model, which is the main advantage.Figure 13
Log plot of numbers of multiplications required for the three JND models versus different image sizes.


## 4.1. Evaluation of JND Models

Figure6 shows the distortion-tolerant model, which can be used to evaluate JND models. It takes the JND value as noise, adds to the original image, and computes the peak-signal-to-noise ratio (PSNR) defined as(4.1)PSNR=20log⁡(255MSE),
where MSE is the mean squared error, which is defined as(4.2)MSE=1m×n∑i=0m-1∑j=0n-1[JND(i,j)]2,
where image size is m×n. As shown in Table 1, the proposed DWT-based JND model is somewhat different from the benchmark full-band JND model in terms of the PSNR values.Table 1
PSNR comparisons of the benchmark full-band JND model, the proposed DWT-based JND model, and the modified DWT-based JND model.
JND modelLenaCameramanBaboonBoardPeppersFull-band JND32.704129.912234.084525.348630.3052DWT-based JND34.330131.755635.815630.800332.1375The modified DWT-based JND33.711730.961434.093125.719231.7148Figure 6
Distortion-tolerant evaluation model for the proposed JND model.


## 4.2. Modified DWT-Based JND Model

Based on (2.1)–(2.3) and (3.6)–(3.8), one can examine the influences of the dominant mask, texture mask, and spatial mask for the full-band JND model and the DWT-based JND model, respectively. Their respective MSE values are shown in Table 2. As one can see, the proposed texture mask for the DWT-based JND model is less significant than that of the full-band JND model. Thus, we propose an adjustable parameter, K, to modify (3.17) and (3.18) as follows:(4.3)SLR′′=K⋅SLR′,SLL′′=K⋅SLL′,
where SLR′′ and SLL′′ replace SLR′ and  SLL′, respectively. Figures 7, 8, 9, 10, and 11 show the MSE values obtained by modifying the DWT-based texture mask with various K in (4.3). In this paper, the adjustable parameter K is set to 4 after extensive simulations. The performance of the modified DWT-based JND model with K=4 is comparable to that of the full-band JND model in terms of the PSNR and MSE values as shown in Tables 1 and 2, respectively.Table 2
The MSE values due to the dominant mask (case 1), the spatial mask (case 2), and the texture mask (case 3) using the full-band JND model, the DWT-based JND model, and the modified DWT-based JND model.
JND modelCaseLenaCameramanBaboonBoardPeppersFull-band JND134.887666.353425.3878189.765360.6121228.520253.029719.621941.722252.5825310.957418.477212.373165.649212.583DWT-based JND123.992443.402717.041954.082239.7492223.926943.040616.94238.470239.64230.9722.10761.460324.48851.1393The modified DWT-based JND127.625452.112225.3378174.24443.8131223.926943.040616.94238.470239.64236.937913.723314.7008153.40527.8595Figure 7
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Lena image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 8
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Cameraman image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 9
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Baboon image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 10
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Board image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.Figure 11
MSE values obtained by modifying the DWT-based texture mask using (4.3) with various K for Peppers image; dashed line: MSE value obtained by using the texture mask of the full-band JND model.(a) The original images, namely, Lena, Cameraman, Baboon, Board, and Peppers; (b), (c), and (d) the noisy images obtained by adding the full-band JND values, the DWT-based JND values, and the modified DWT-based JND values, respectively.
(a)(b)(c)(d)Figure12 shows the noisy images obtained by adding the JND values to the original images (Figure 12(a)) using the full-band JND model (Figure 12(b)), the DWT-based JND model (Figure 12(c)), and the modified DWT-based JND model (Figure 12(d)). It is noted that the images in the second and fourth rows are almost indistinguishable from the original images. As a result, the modified DWT-based JND model is visually comparable to the full-band JND model.


## 4.3. Computation Complexity of the Proposed JND Models

In the full-band JND model, the computation ofbg(i,j) requires 9 multiplications per pixel, the computation of mg(i,j) requires 28 multiplications per pixel, and the computation required for (2.2)–(2.5) is 6 multiplications per pixel. Thus, for an n×n image, it requires 43n2 multiplications. In the proposed DWT-based JND model, the computations of LH′, HL′, LLLL′, SLR′, and SLL′ require (1/4)n2, (1/4)n2, (5/16)n2, (1/4)n2, and (1/4)n2 multiplications, respectively, for an n×n image, and the computation required for (3.7)–(3.10) is also 6 multiplications per pixel. Thus, for an n×n image, it requires 7.3125n2 multiplications. In the modified DWT-based JND model, as the computations of SLR′′ and SLL′′ require no multiplication, it only needs 6.8125n2 multiplications for an n×n image. Figure 13 shows the log plot of numbers of multiplications required for the three JND models versus different image sizes. As a result, the speed of the DWT-based JND model is about 6 times faster than that of the full-band JND model, which is the main advantage.Figure 13
Log plot of numbers of multiplications required for the three JND models versus different image sizes.


## 5. Conclusion

In this paper, an efficient DWT-based JND model is presented. It has the advantage of saving a lot of computation time while the performance is comparable to the benchmark full-band JND model. More specifically, the computation complexity of the proposed DWT-based JND model is only one sixth of that of the full-band JND model. As a result, it is suitable for the real-time applications.



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*Source: 635738-2011-09-07.xml*