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SCIENTIA SINICA Informationis, Volume 48, Issue 1: 79-99(2018) https://doi.org/10.1360/N112017-00109

Stereo image zero-watermarking algorithm based on ternary polar harmonic Fourier moments and chaotic mapping

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  • ReceivedMay 16, 2017
  • AcceptedAug 1, 2017
  • PublishedNov 16, 2017

Abstract

In recent years, stereo images have attracted extensive attention because of their strong immersion, and the corresponding copyright protection of stereo images is becoming increasingly urgent. At present, most of the watermarking algorithms for image copyright protection are aimed at planar images, and there are few watermarking algorithms for stereo images. Moreover, most of the existing stereo image watermarking algorithms are not very good at reflecting and retaining the specific relationship between the left and right views of the stereo images, which will inevitably affect the robustness of the algorithm. In this paper, ternary polar harmonic Fourier moments (TPHFM) for stereo images is proposed and based on this and chaotic mapping, a robust stereo image zero-watermarking algorithm is presented. First, mixed linear-nonlinear coupled map lattice is used to make the original binary logo image chaotic. Next, the TPHFM of the original stereo image is computed and accurate moments for the zero-watermarking algorithm are selected. Then, the binary feature image is constructed using the accurate moments, and scrambled using Sine mapping and Cosine mapping. Finally, a zero-watermark image is generated using the exclusive-or on the scrambled binary feature image and the chaotic binary logo image. Experimental results show that the proposed zero-watermarking algorithm is excellently robust against common image processing attacks and geometric attacks, and is superior to ternary radial harmonic Fourier moments (TRHFM)-based algorithm and other zero-watermarking algorithms.


Funded by

国家自然科学基金(61672124,61370145,61173183)

“十三五国家密码发展基金(MMJJ20170203)


Supplement

Appendix

引理1的证明

首先将式(9)离散化得到直角坐标系下PHFM的计算公式, 本文基于内切圆映射[17]来计算PHFM, 如下: \begin{equation}{P_{nm}}\; = \frac{8}{{{\pi}{N^2}}}\sum\limits_{p = 0}^{N - 1} {\sum\limits_{q = 0}^{N - 1} {f(p,q){T_n}(r)\exp ( - {\rm{j}}m\theta )} }. \tag{35}\end{equation}

由上式很容易得到${{P}_{00}}=\sqrt{2}\rm{C}$, 此处主要证明$(n,m)\ne~(0,0)$的情况.

如图1所示, 对于单位圆内任一像素$A(r,\theta~)$, 有另外三个像素与之对应, 即$B(r,{\uppi} /(2+ {\theta})$, $C(r,{\pi}~+~\theta~)$和$D(r,3{\uppi}/(2+ {\theta})$. 首先以$(A,B)$和$(C,D)$为例将单位圆内的像素配对, 可得 \begin{equation}\begin{array}{c} \begin{aligned} {P_{nm}} &= \frac{8}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {\left[{\rm{C}}{T_n}(r)\exp ( - {\rm{j}}m\theta ) + {\rm{C}}{T_n}(r)\exp \left( - {\rm{j}}m \left(\frac{{\pi}}{2} + \theta \right)\right)\right]} \\ &= \frac{{8{\rm{C}}}}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {\left[{T_n}(r)\exp ( - {\rm{j}}m\theta ) + {T_n}(r)\exp \left( - {\rm{j}}m \left(\frac{{\pi}}{2} + \theta \right)\right)\right]} \\ &= \frac{{8{\rm{C}}}}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {{T_n}(r)\exp ( - {\rm{j}}m\theta ) \left[1 + \exp \left( - {\rm{j}}m\frac{{\pi}}{2}\right)\right]}. \end{aligned} \end{array} \tag{36}\end{equation}

则当$m$是4的倍数时, $1+\exp~(-{\rm{j}}m\frac{\pi}{2})\ne~0$, 则${{P}_{nm}}\ne~0$, 即${{P}_{nm}}\ne~0$的集合为${{S}_{1}}=\{m,m=4i,i\in~\mathbb{Z}\}$.

然后以$(A,C)$和$(B,D)$为例将单位圆内的像素配对, 可得 \begin{equation}\begin{array}{c} \begin{aligned} {P_{nm}} &= \frac{8}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {[{\rm{C}}{T_n}(r)\exp ( - {\rm{j}}m\theta ) + {\rm{C}}{T_n}(r)\exp ( - {\rm{j}}m({\pi} + \theta ))]} \\ &= \frac{{8{\rm{C}}}}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {[{T_n}(r)\exp ( - {\rm{j}}m\theta ) + {T_n}(r)\exp ( - {\rm{j}}m({\pi} + \theta ))]} \\ &= \frac{{8{\rm{C}}}}{{{\pi}{N^2}}}\sum\limits_{\rm{\{All\;pairs\}} } {{T_n}(r)\exp ( - {\rm{j}}m\theta )[1 + \exp ( - {\rm{j}}m{\pi})]}. \end{aligned} \end{array} \tag{37}\end{equation}

则当$m$为偶数时, $1+\exp~(-{\rm{j}}m{\pi})\ne~0$, 则${{P}_{nm}}\ne~0$, 即${{P}_{nm}}\ne~0$的集合为${{S}_{2}}=\{m,m~\rm{is}~\rm{even}\}$. 取${{S}_{1}}$和${{S}_{2}}$的交集可得满足${{P}_{nm}}\ne~0$的集合为$S=\{m,m=4i,i\in~\mathbb{Z}\}$.

定理1的证明

由式(13)和(14)知, $P_{nm}^{\rm~R}$的计算是由立体图像的左右视角的PHFM ${{P}_{nm}}({{f}_{\rm~L}})$和${{P}_{nm}}({{f}_{\rm~R}})$计算得到.

由引理1可知, 左右视角图像的PHFM满足 \begin{equation*}{P_{nm}}({f_{\rm L}}) = {P_{nm}}({f_{\rm R}}) = \left\{ {\begin{array}{*{20}{l}} {\sqrt 2 {\rm{C}}, {\rm{while}}\;n = m = 0}, \\ {{\rm{nonzero}}, {\rm{while}}\;(n,m) \ne (0,0), m = 4i, i \in {\mathbb{Z}}}, \\ {0, {\rm{otherwise}}}. \end{array}} \right.\end{equation*}

当$n=m=0$时, ${{B}_{nm}}={{C}_{nm}}=\sqrt{2}{\rm{C}},~{{A}_{nm}}=0$, 则$\left|~P_{nm}^{\rm~R}~\right|=\sqrt{A_{nm}^{2}+B_{nm}^{2}+C_{nm}^{2}}=2{\rm{C}}$, 故而可得 \begin{equation}\left| {P_{nm}^{\rm R}} \right| = \left\{ {\begin{array}{*{20}{l}} {2{\rm{C}}, {\rm{while}}\; n = m = 0}, \\ {{\rm{nonzero}}, {\rm{while}}\;(n,m) \ne (0,0), m = 4i, i \in {\mathbb{Z}}}, \\ {0, {\rm{otherwise}}}. \end{array}} \right. \tag{38}\end{equation}


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  • Figure 3

    Comparison of reconstructed images (max moment order ${{n}_{\max~}}=5,10,15,20,25,30$). (a) PHFM of Lena;protectłinebreak (b) RHFM of Lena; (c) PHFM of binary Lena; (d) RHFM of binary Lena

  • Figure 4

    Left and right views of stereo image Art. (a) Left view; (b) right view

  • Figure 5

    Reconstruction images of the stereo image Art using TPHFM (max moment order ${{n}_{\max~}}=5,10,15,20,25,30$). (a) Left view; (b) right view

  • Figure 6

    Flow chart of zero-watermark construction procedure

  • Figure 7

    Flow chart of zero-watermark verification procedure

  • Figure 8

    Stereo images Dolls, Art, Books, Computer. (a) Left views; (b) right views

  • Figure 9

    (Color online) The relationship between the watermark capacity and the max order of TPHFM

  • Figure 10

    (Color online) PSNR to JPEG compression and robustness to JPEG compression. (a) PSNRL; (b) PSNRR; (c) asymmetric attack; (d) symmetric attack

  • Figure 11

    (Color online) PSNR to image rotation and robustness to image rotation. (a) PSNRL; (b) PSNRR;protectłinebreak (c) asymmetric attack; (d) symmetric attack

  • Figure 12

    (Color online) PSNR to image scaling and robustness to image scaling. (a) PSNRL; (b) PSNRR; (c) asymmetric attack; (d) symmetric attack

  • Table 1   Geometric invariance of TPHFM magnitudes
    Attack $\left|~P_{00}^{\rm~R}~\right|$ $\left|~P_{01}^{\rm~R}~\right|$ $\left|~P_{02}^{\rm~R}~\right|$ $\left|~P_{10}^{\rm~R}~\right|$ $\left|~P_{11}^{\rm~R}~\right|$ $\left|~P_{12}^{\rm~R}~\right|$ $\left|~P_{20}^{\rm~R}~\right|$ $\left|~P_{21}^{\rm~R}~\right|$ $\left|~P_{22}^{\rm~R}~\right|$ MRE(%)
    Original image 146.8593 32.6547 17.2196 12.3628 11.6475 6.7317 20.8998 11.8932 4.2694
    Rotation ${{5}^{\circ~}}$ Asymmetrical attack 146.9158 32.7592 17.2662 12.2572 11.6219 6.8192 20.7818 11.9659 3.8423 1.5758
    Symmetrical attack 146.8861 32.5861 17.1113 12.2522 11.6298 6.7887 20.7635 11.8532 3.7080 1.8768
    Rotation ${{45}^{\circ~}}$ Asymmetrical attack 146.7320 31.8293 16.8947 12.4646 10.6857 6.6121 20.8210 12.1691 3.5169 3.9646
    Symmetrical attack 146.7628 32.3142 17.3792 12.4748 9.8521 6.2164 20.9562 11.2425 4.1142 3.9320
    Scaling 0.5 Asymmetrical attack 146.9727 32.8202 17.2606 11.8346 11.4131 6.9127 21.4086 12.0291 4.4020 1.8310
    Symmetrical attack 146.8362 32.8453 17.2104 12.0053 11.4653 6.8465 21.9141 12.0517 4.3768 1.7239
    Scaling 1.5 Asymmetrical attack 146.9728 32.6034 17.1996 12.5180 11.7028 6.6659 20.7984 11.8422 4.2525 0.4856
    Symmetrical attack 147.1410 32.5901 17.2034 12.4688 11.6817 6.6871 20.7374 11.8309 4.2655 0.4099
  • Table 2   TPHFM magnitudes for a $128\times128$ stereo image with constant value 128
    $m$ = 0 $m$ = 1 $m$ = 2 $m$ = 3 $m$ = 4 $m$ = 5 $m$ = 6 $m$ = 7 $m$ = 8 $m$ = 9
    $n$ = 0 255.5213 0.0000 0.0000 0.0000 0.3637 0.0000 0.0000 0.0000 0.5388 0.0000
    $n$ = 1 0.0043 0.0000 0.0000 0.0000 0.0108 0.0000 0.0000 0.0000 0.0091 0.0000
    $n$ = 2 0.6772 0.0000 0.0000 0.0000 0.5144 0.0000 0.0000 0.0000 0.7622 0.0000
    $n$ = 3 0.0087 0.0000 0.0000 0.0000 0.0217 0.0000 0.0000 0.0000 0.0183 0.0000
    $n$ = 4 0.6781 0.0000 0.0000 0.0000 0.5149 0.0000 0.0000 0.0000 0.7631 0.0000
    $n$ = 5 0.0132 0.0000 0.0000 0.0000 0.0327 0.0000 0.0000 0.0000 0.0276 0.0000
    $n$ = 6 0.6795 0.0000 0.0000 0.0000 0.5157 0.0000 0.0000 0.0000 0.7647 0.0000
    $n$ = 7 0.0177 0.0000 0.0000 0.0000 0.0438 0.0000 0.0000 0.0000 0.0370 0.0000
    $n$ = 8 0.6815 0.0000 0.0000 0.0000 0.5169 0.0000 0.0000 0.0000 0.7668 0.0000
    $n$ = 9 0.0224 0.0000 0.0000 0.0000 0.0551 0.0000 0.0000 0.0000 0.0466 0.0000
  • Table 3   Robustness to other attacks
    Attack Art Books Computer Dolls
    Median filtering ($3\times3$) PSNRL (dB) 30.4035 27.3473 27.5332 28.9317
    PSNRR (dB) 30.8843 26.4759 27.6796 29.3416
    Asymmetrical attack 0.0039 0.0078 0.0088 0.0088
    Symmetrical attack 0.0078 0.0195 0.0225 0.0107
    Median filtering ($5\times5$) PSNRL (dB) 25.8941 23.6715 23.7791 24.8816
    PSNRR (dB) 25.9238 22.7944 23.5154 25.1525
    Asymmetrical attack 0.0156 0.0127 0.0215 0.0205
    Symmetrical attack 0.0215 0.0400 0.0371 0.0293
    Gaussian filtering ($3\times3$) PSNRL (dB) 36.9577 34.4840 35.0346 35.5844
    PSNRR (dB) 37.3435 33.8248 34.8005 35.9680
    Asymmetrical attack 0 0.0020 0.0029 0.0029
    Symmetrical attack 0.0020 0.0029 0.0049 0.0049
    Gaussian filtering ($5\times5$) PSNRL (dB) 36.9273 34.4577 35.0088 35.5543
    PSNRR (dB) 37.3123 33.7982 34.7732 35.9377
    Asymmetrical attack 0 0.0020 0.0029 0.0029
    Symmetrical attack 0.0020 0.0029 0.0049 0.0049
    Gaussian noise (0.005) PSNRL (dB) 23.2990 23.1482 23.0811 23.2960
    PSNRR (dB) 23.3135 23.0542 23.0746 23.3485
    Asymmetrical attack 0.0156 0.0166 0.0186 0.0195
    Symmetrical attack 0.0215 0.0264 0.0381 0.0273
    Gaussian noise (0.01) PSNRL (dB) 20.4923 20.1951 20.0748 20.4157
    PSNRR (dB) 20.4772 20.1254 20.1505 20.4921
    Asymmetrical attack 0.0264 0.0215 0.0293 0.0264
    Symmetrical attack 0.0332 0.0361 0.0430 0.0498
    Salt and peppers noise (0.02) PSNRL (dB) 21.6491 21.9930 22.3048 21.7698
    PSNRR (dB) 21.4336 22.1273 22.2937 21.7848
    Asymmetrical attack 0.0215 0.0234 0.0244 0.0225
    Symmetrical attack 0.0254 0.0342 0.0342 0.0371
    Salt and peppers noise (0.03) PSNRL (dB) 19.8839 20.4135 20.5746 19.9893
    PSNRR (dB) 19.8996 20.4051 20.6388 19.9289
    Asymmetrical attack 0.0215 0.0254 0.0273 0.0361
    Symmetrical attack 0.0332 0.0391 0.0410 0.0537
    Upper left corner cropping (1/16) PSNRL (dB) 25.8137 27.2417 26.1823 25.2798
    PSNRR (dB) 25.3407 27.0781 26.4085 26.1656
    Asymmetrical attack 0 0 0 0
    Symmetrical attack 0 0 0 0
    Upper left corner cropping (1/8) PSNRL (dB) 19.9004 21.3112 21.8225 20.9267
    PSNRR (dB) 19.4598 21.2795 21.4054 22.1922
    Asymmetrical attack 0 0 0 0
    Symmetrical attack 0 0 0 0
  • Table 4   Comparison of the robustness of the proposed algorithm and other algorithms
    Attack Proposed TRHFM-based Ref. [5] Ref. [6] Ref. [11] Ref. [13]
    JPEG compression (10) 0.0270 0.0293 0.0164 0.0164 0.0952 0.0332
    JPEG compression (30) 0.0101 0.0111 0.0109 0.0084 0.0742 0.0059
    JPEG compression (50) 0.0062 0.0092 0.0075 0.0056 0.0688 0.0062
    JPEG compression (70) 0.0020 0.0062 0.0068 0.0048 0.0573 0.0026
    JPEG compression (90) 0 0.0036 0.0034 0.0034 0.0535 0.0013
    Rotation with cropping ${{5}^{\circ~}}$ 0.0511 0.0612 0.1825 0.3320 0.5016 0.0059
    Rotation with cropping ${{45}^{\circ~}}$ 0.0544 0.0687 0.1706 0.3155 0.4956 0.0052
    Scaling 0.25 and resizing to original size 0.0423 0.0560 0.0092 0.0117 0.0301 0.0527
    Scaling 4 and resizing to original size 0.0026 0.0042 0.0056 0.0088 0.0315 0.0049
    Median filtering ($3\times3$) 0.0127 0.0215 0.0164 0.0178 0.0334 0.0147
    Gaussian filtering ($3\times3$) 0.0033 0.0042 0.0075 0.0092 0.0176 0.0052
    Gaussian noise (0.01) 0.0374 0.0449 0.0285 0.0285 0.0793 0.0430
    Salt and peppers noise (0.03) 0.0378 0.0488 0.0465 0.0698 0.0818 0.0400
    Upper left corner cropping (1/16) 0 0 0.1098 0.1743 0.0415 0
    Upper left corner cropping (1/8) 0 0 0.1258 0.2256 0.0378 0

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