# A Content Hiding Method for Digital Hologram Using Multiple Fresnel Diffraction
## 3 Proposed Digital Hologram Encryption
### Content Hiding Method  
Figure 5 shows the hologram encryption algorithm and an example result from each processing step, respectively. First, we add the bias value to the input hologram and perform the Fresnel transformation twice as mentioned above. The bias is experimentally obtained. Then, the concentrated area due to biasing is calculated and extracted from the result of the second Fresnel transform. The extracted data are encrypted using a kind of encryption algorithm. Then the encrypted result is replaced in its original location and the result is inverse-Fresnel-transformed twice.  
In order to extract the area for encryption, the region concentrated on by the biasing is extracted from the result from the second Fresnel transform. Figure 6 shows the SEs in the Fresnel transforms to determine the concentrated area due to biasing in the second Fresnel transform. In the Fresnel transform, the SE of the diffraction plane can be expressed by Equation (4) [15], where \(\delta x\) is the sampling distance of the input plane.  
Figure 4: Fresnel transform results for holograms. Top four are real holograms and bottom ones are imaginary holograms. (**a**) Result of 1 st Fresnel transform; (**b**) result of second Fresnel transform; Fresnel transform result of hologram after biasing. (**c**) Result of 1 st Fresnel transform; (**d**) result of second Fresnel transform.  
Figure 5: Hologram encryption algorithm.  
\[SE\leq\frac{\lambda z}{\delta x} \tag{4}\]  
In Figure 6, \(SE1_{Max}\) and \(SE2_{Max}\) are the sizes of the diffraction planes to be output and can be obtained by the input planes. \(SE1_{DC}\) and \(SE1_{DC}\) are the sizes of the areas encrypted by the \(SE\)s of the areas generated by the biases of the input holograms. Although the area depends on the hologram and bias value, it has a similar size to the input image because of its relatively small spatial frequency. The encryption area \(S_{c}\) (size of ciphering) can be calculated by Equation (5). \(S_{c}\) is obtained by using \(S_{ID}\) (size of input region in diffraction plane) which is the same area as the size of the input image in the second Fresnel transform and scaling parameter \(w_{s}\).  
\[S_{c}=w_{s}S_{ID} \tag{5}\]  
\[IN1:SE1_{Max}=S_{ID}:IN2 \tag{6}\]  
\[S_{ID}=\frac{IN1 IN2}{SE1_{Max}}=\frac{N\delta x_{1}N\delta x_{2}}{\frac{ \lambda_{1}z_{1}}{\lambda z_{1}}}=\frac{N^{2}\delta x_{1}^{2}\delta x_{2}}{ \lambda_{1}z_{1}} \tag{7}\]  
\[R_{c}:N=S_{c}:SE2_{Max} \tag{8}\]  
\[R_{c}=\Big{\lfloor}\frac{NS_{c}}{SE2_{Max}}\Big{\rfloor}=\Bigg{\lfloor} \frac{Nw_{s}\frac{N^{2}\delta x_{1}^{2}\delta x_{2}}{\lambda_{1}z_{1}}}{ \frac{\lambda_{2}z_{2}}{\lambda_{2}}}\Bigg{\rfloor}=w_{s}\Big{\lfloor}\frac{ N^{3}\delta x_{1}^{2}\delta x_{2}^{2}}{\lambda_{1}\lambda_{2}z_{1}z_{2}}\Big{\rfloor} \tag{9}\]  
\(S_{ID}\) can be expressed by Equation (7) using the parameters of the first and second Fresnel transforms as the proportional expression of Equation (6). The discrete domain \(R_{c}\) for encryption in the result of the second Fresnel transform is expressed by Equation (9) using the proportional expression of Equation (8). \(N\) is the resolution of the input hologram plane and the output planes of the Fresnel transform. \(\delta x_{1}\), \(\lambda_{1}\), and \(z_{1}\) are the parameters of the first Fresnel transform; and \(\delta x_{2}\), \(\lambda_{2}\), and \(z_{2}\) are the parameters of the second Fresnel transform. Therefore, the encryption key of our scheme is the combination of the bias values, the parameter values for each Fresnel transform (\(z,\delta x,\lambda\) of Equation (3)), the scale parameter for extracting the encrypted region, and the ciphering key for the cipher.