## Abstract

A novel single-channel color-image watermarking with digital-optics means based on phase-shifting interferometry (PSI) and a neighboring pixel value subtraction algorithm in the discrete-cosine-transform (DCT) domain is proposed. The converted two-dimensional indexed image matrix from an original color image is encrypted to four interferograms by a PSI and double random-phase encoding technique. Then the interferograms are embedded in one chosen channel of an enlarged color host image in the DCT domain. The hidden color image can be retrieved by DCT, the improved neighboring pixel value subtraction algorithm,
an inverse encryption process, and color image format conversion. The feasibility of this method and its robustness against some types of distortion and attacks from the superposed image with different weighting factors are verified and analyzed by computer simulations. This approach can avoid the cross-talk noise due to direct information superposition, enhance the imperceptibility of hidden data, and improve the efficiency of data transmission.

© 2007 Optical Society of America

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### Equations (25)

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(1)
$$U\left(x,y\right)=\text{FR}\left\{\left(\text{FR}\left\{f\left({x}_{0},{y}_{0}\right)\mathrm{exp}\left[i2\pi p\left({x}_{0},{y}_{0}\right)\right];\text{\hspace{0.17em}}\lambda ,{z}_{1}\right\}\right)\times \mathrm{exp}\left[i2\pi q\left({x}_{t},{y}_{t}\right)\right];\text{\hspace{0.17em}}\lambda ,{z}_{2}\right\},$$
(2)
$$U\left(x,y\right)=\frac{1}{4{A}_{r}}\left\{\left[{I}_{1}\left(x,y\right)-{I}_{3}\left(x,y\right)\right]+i\left[{I}_{2}\left(x,y\right)-{I}_{4}\left(x,y\right)\right]\right\},$$
(3)
$$h\left(2m-1,2n-1\right)=o\left(m,n\right),h\left(2m-1,2n\right)=o\left(m,n\right)\text{,}$$
(4)
$$h\left(2m,2n-1\right)=o\left(m,n\right),h\left(2m,2n\right)=o\left(m,n\right)\text{.}$$
(5)
$$m,n=1,2\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}N\text{.}$$
(6)
$$B\left(\xi ,\eta \right)=\frac{2}{\sqrt{LK}}\text{\hspace{0.17em}}\gamma \left(\xi \right)\gamma \left(\eta \right){\displaystyle \sum _{u=0}^{L-1}{\displaystyle \sum _{v=0}^{K-1}A\left(u,v\right)}}\times \mathrm{cos}\left[\frac{\left(2u+1\right)\xi \pi}{2L}\right]\mathrm{cos}\left[\frac{\left(2v+1\right)\eta \pi}{2K}\right],$$
(7)
$$A\left(u,v\right)=\frac{2}{\sqrt{LK}}\text{\hspace{0.17em}}{\displaystyle \sum _{\xi =0}^{L-1}{\displaystyle \sum _{\eta =0}^{K-1}\gamma \left(\xi \right)\gamma \left(\eta \right)B\left(\xi ,\eta \right)\times \mathrm{cos}\left[\frac{\left(2u+1\right)\xi \pi}{2L}\right]\mathrm{cos}\left[\frac{\left(2v+1\right)\eta \pi}{2K}\right]}},$$
(8)
$$\gamma \left(\xi \right)=\{\begin{array}{ll}\frac{1}{\sqrt{2}}& \text{for \hspace{0.17em}}\xi =0\\ 1& \text{for \hspace{0.17em}}\xi =1,2\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}L-1\end{array}\text{,}$$
(9)
$$\gamma \left(\eta \right)=\{\begin{array}{ll}\frac{1}{\sqrt{2}}& \text{for \hspace{0.17em}}\eta =0\\ 1& \text{for \hspace{0.17em}}\eta =1,2\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}K-1\end{array}\mathrm{.}$$
(10)
$${D}_{g}\left(2m-1,2n-1\right)={D}_{g}\left(2m-1,2n\right)\mathrm{,}$$
(11)
$${D}_{g}\left(2m,2n-1\right)={D}_{g}\left(2m,2n\right)\mathrm{,}$$
(12)
$$m,n=1,2\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}N\mathrm{.}$$
(13)
$$\text{\hspace{0.17em}}{D}_{g}\prime \left(2m-1,2n-1\right)={D}_{g}\left(2m-1,2n-1\right)+w{I}_{1}\left(m,n\right)\text{,}$$
(14)
$${D}_{g}\prime \left(2m-1,2n\right)={D}_{g}\left(2m-1,2n\right)+w{I}_{3}\left(m,n\right)\text{,}$$
(15)
$${D}_{g}\prime \left(2m,2n-1\right)={D}_{g}\left(2m,2n-1\right)+w{I}_{2}\left(m,n\right)\text{,}$$
(16)
$${D}_{g}\prime \left(2m,2n\right)={D}_{g}\left(2m,2n\right)+w{I}_{4}\left(m,n\right)\text{,}$$
(17)
$${I}_{13}\left(m,n\right)={D}_{g}\prime \left(2m-1,2n-1\right)-{D}_{g}\prime \left(2m-1,2n\right)={D}_{g}\left(2m-1,2n-1\right)+w{I}_{1}\left(m,n\right)-{D}_{g}\left(2m-1,2n\right)-w{I}_{3}\left(m,n\right)\text{.}$$
(18)
$${I}_{13}\left(m,n\right)=w\left[{I}_{1}\left(m,n\right)-{I}_{3}\left(m,n\right)\right],$$
(19)
$${I}_{24}\left(m,n\right)={D}_{g}\prime \left(2m,2n-1\right)-{D}_{g}\prime \left(2m,2n\right)=w\left[{I}_{2}\left(m,n\right)-{I}_{4}\left(m,n\right)\right].$$
(20)
$$U\prime \left(x,y\right)=w\left({I}_{13}-i{I}_{24}\right)=w\left\{\left[{I}_{1}\left(x,y\right)-{I}_{3}\left(x,y\right)\right]+i\left[{I}_{2}\left(x,y\right)-{I}_{4}\left(x,y\right)\right]\right\}\text{.}$$
(21)
$$f\prime \left({x}_{0},{y}_{0}\right)=\text{abs}\left\{\text{IFR}\left\{\left(\text{IFR}\left\{U\prime \left(x,y\right);\text{\hspace{0.17em}}\lambda \text{, \hspace{0.17em}}-{z}_{2}\right\}\right)\times \mathrm{exp}\left[-i2\pi q\left(x,y\right)\right];\text{\hspace{0.17em}}\lambda \text{, \hspace{0.17em}}-{z}_{1}\right\}\times \mathrm{exp}\left[-i2\pi p\left({x}_{0},{y}_{0}\right)\right]\right\}$$
(22)
$$\text{CC}=\frac{\text{COV}\left(h,{h}^{\text{r}}\right)}{{\sigma}_{h}{\sigma}_{{h}^{\text{r}}}},$$
(23)
$$\text{COV}\left(h,{h}^{\text{r}}\right)=\text{E}\left\{[h-\text{E}\left(h\right)][{h}^{\text{r}}-\text{E}\left({h}^{\text{r}}\right)]\right\},$$
(24)
$$P\prime \left\{h={h}_{k}\right\}={p}_{k}\prime \text{,}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}k=1,2,3\text{\hspace{0.17em}}\mathrm{.}\text{\hspace{0.17em}}\mathrm{.}\text{\hspace{0.17em}}\mathrm{.}\text{\hspace{0.17em}}\mathrm{,}$$
(25)
$$\text{E}\left(h\right)={\displaystyle \sum _{k=1}^{\infty}{h}_{k}{p}_{k}\prime}.$$