# An Adaptive and Secure Holographic Image Watermarking Scheme
## 3 Encrypted Binary Computer-Generated Hologram Based on Chaos
### Encrypted Binary CGH  
The encrypted CGH is generated as follows:  
(1) In order to decrease the dynamic range of the hologram, a random phase \(\psi(x_{0}\),\(y_{0})\) which is uniformly in the interval [0,1] is multiplied to the image \(f(x_{0}\),\(y_{0})\) first.  
\[f_{1}(x_{0},y_{0})=f(x_{0},y_{0})\exp\{j2\pi\psi(x_{0},y_{0})\}, \tag{9}\]  
(2) Apply the Fourier transform to \(f_{1}(x_{0}\),\(y_{0})\) to get the object wave \(OW(x\),\(y)\).  
\[OW(x,y)=FT[f_{1}(x_{0},y_{0})]=A(x,y)\exp\{j\varphi(x,y)\}, \tag{10}\]  
where FT() is the Fourier transform operator. The amplitude and phase of \(OW(x\),\(y)\) are \(A(x\),\(y)\) and \(\varphi\) (\(x\),\(y\)), respectively.  
(3) Assume that the parallel reference wave is mathematically represented by function \(RW(x\),\(y)=A_{r}\)exp\([j2\pi\varphi\varphi_{r}(x\),\(y)]\). Here, \(\rho\) is the carrier frequency. The amplitude and phase of \(RW(x\),\(y)\) are \(A_{r}(x\),\(y)\) and \(\varphi_{r}(x\),\(y)\), respectively. Sequentially, permute \(OW(x\),\(y)\) and \(RW(x\),\(y)\) to obtain the scrambled \(O_{s}(x\),\(y)\) and \(R_{s}(x\),\(y)\) by use of the proposed chaos-based permutation method shown in Section 3.1 with the parameters \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\).  
(4) The shuffled hologram transmittance \(h(x\),\(y)\) can be achieved according to the following formula.  
\[\begin{array}{rl}h(x,y)=&\big{|}O_{s}(x,y)+R_{s}(x,y)\big{|}^{2}\\ =&\big{|}A_{s}(x,y)\big{|}_{2}+A_{sr}^{2}+2A_{sr}A(x,y)_{s}\cos[2\pi\rho \varphi_{sr}(x,y)-\varphi_{s}(x,y)]\\ =&C+2A_{sr}A_{s}(x,y)\cos[2\pi\rho\varphi_{sr}(x,y)-\varphi_{s}(x,y)],\end{array} \tag{11}\]  
In Equation (11), let \(|A_{s}(x\),\(y)|^{2}+A_{sr}{}^{2}\) be a constant \(C\). \(\varphi_{s}(x\),\(y)\) and \(A_{s}(x\),\(y)\) are the phase and amplitude of \(O_{s}(x\),\(y)\), \(\varphi_{sr}\) (\(x\),\(y\)) and \(A_{sr}(x\),\(y)\) are the phase and amplitude of \(R_{s}(x\),\(y)\)), respectively.  
(5) In the light of [PERSON]'s coding method, let \(|A_{s}(x,y)|_{\max}=1\) and \(A_{sr}=1\), then \(h(x\),\(y)\) becomes  
\[h(x,y)=0.5[1+A_{s}(x,y)\cos[2\pi\rho\varphi_{sr}(x,y)-\varphi_{s}(x,y)]]. \tag{12}\]  
(6) Finally, fabricate the encrypted binary CGH \(EBH\) (\(x\),\(y\)) by quantizing \(h(x\),\(y)\), which is achieved by use of Equation (12), in 1-bit using OTSU algorithm [25].  
The security of this encryption system is enhanced greatly because of the huge key space which is formed by the private keys including \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\).  
The reconstruction process of an encrypted binary CGH is described as follows:  
(1) With the parameters \(SX(1)\), \(SY(1)\), \(SZ\), \(\alpha\), \(\beta\), \(\gamma\), \(t1\), \(t2\) and \(t3\), the encrypted CGH \(EBH\)(\(x\),\(y\)) is scrambled by the proposed inverse permutation process mentioned in Section 3.1 to obtain \(DEBH\)(\(x\),\(y\)).  
(2) With the conjugate reference wave, the binarized reconstruction \(RH\) of the hologram can be achieved via utilizing inverse Fourier transform and OTSU algorithm. To improve the quality of reconstruction, the high-pass filter approach is employed to attenuate the DC item in the reconstructed image.