# Enhancing the Quality of Sampled Phase-Only Hologram (SPOH) Based on Time-Division Comb Filtering
## 2 Sampled Phase-Only Hologram (SPOH)
### Generating a Sampled Phase-Only Hologram  
The above mentioned problem can be overcome by down-sampling the intensity of the object with a grid-cross lattice prior to generation of the hologram. A grid cross lattice is comprising of regular spaced sampling lines along the horizontal, vertical, and diagonal directions. Down-sampling the intensity image effectuates an optical diffuser, resulting in a hologram with roughly homogeneous magnitude distribution that can be discarded without causing too much degradation to the reconstructed image. The grid-cross lattice is a two-dimensional binary image \(D(m,n)\) with '0' and '1' representingnon-sampling and sampling points. Let \(\tau\) and \(mod(A,B)\) denote the sampling interval, and the remainder of \(A/B\). The following parameters are defined as  
The down-sampling grid-cross lattice is given by  
\[C_{1}(m)=mod(m,\tau), \tag{3a}\] \[C_{2}(n)=mod(n,\tau)\] (3b) \[C_{3}(m,n)=C_{1}(m)-C_{2}(n)\] (3c) \[C_{4}(m,n)=C_{1}(m)-[\tau-C_{2}(n)]\] (3d) \[D(m,n)= \begin{cases}1&\text{if }C_{1}(m)\times C_{2}(n)\times C_{3}(m,n) \times C_{4}(m,n)=0\\ 0&\text{otherwise}\end{cases}  \tag{4}\]  
An example of the grid-cross lattice for \(\tau=8\), and \(m,n=[0,8]\) is shown in Figure 1.  
The down-sampled image \(I_{D}(m,n)\) is  
\[I_{D}(m,n)=I(m,n)\times D(m,n) \tag{5}\]  
\(I_{D}(m,n)\) comprises pixels of the original image that are sampled at uniformly spaced interval along the horizontal, vertical, and diagonal directions. Next a Fresnel hologram \(H_{D}(m,n)\) of the down-sampled image \(I_{D}(m,n)\) is generated, and the phase component of \(H_{D}(m,n)\) is extracted to be the sampled phase-only hologram (SPOH) \(H_{P}(m,n)\), as given by  
\[H_{P}(m,n)=arg\big{\{}\mathcal{F}^{-1}\big{[}\overline{I}_{D}(\omega_{m}, \omega_{n})\times\overline{f}(\omega_{m},\omega_{n};z_{0})\big{]}\big{\}} \tag{6}\]  
where \(\mathcal{F}^{-1}[. ]\) is the inverse Fourier transform operation, \(arg\{\psi\}\) represents the phase component of the complex variable \(\psi\), and \(\overline{I}_{D}(\omega_{m},\omega_{n})\) is the frequency spectrum of \(I_{D}(m,n)\). From the phase-only hologram \(H_{P}(m,n)\) an image \(I^{\prime}_{D}(m,n)\), can be reconstructed from the SPOH. Denoting the conjugate of \(\overline{f}(\omega_{m},\omega_{n};z_{0})\) as \(\boldsymbol{f}^{\prime}(\omega_{m},\omega_{n};z_{0})\), the intensity of the reconstructed image is given by  
\[I^{\prime}_{D}(m,n)=\Big{|}\mathcal{F}^{-1}\big{[}\overline{H}_{P}(\omega_{m},\omega_{n})\times\boldsymbol{f}^{\prime}(\omega_{m},\omega_{n};z_{0})\big{]} \Big{|}, \tag{7}\]  
where \(\overline{H}_{P}(\omega_{m},\omega_{n})\) is the Foruier transform of \(H_{P}(m,n)\), and \(|A|\) is the magnitude of the complex-valued function \(A\). In general the reconstructed image \(I^{\prime}_{D}(m,n)\) is similar to the down-sampled image \(I_{D}(m,n)\)., which is also masked with the pattern of the down-sampling lattice. As an example, the plane image in Figure 2a is converted to a phase-only hologram with the optical settings in Table 1. Equation (1) is first applied to generate a Fresnel hologram directly from the original image, and the phase component  
Figure 1: An example of the grid-cross down-sampling lattice for \(\tau=8\), and \(m,n=[0,8]\). The cells in white and black, represent sampling and non-sampling points, respectively. is retained as the phase-only hologram. The reconstructed image, as shown in Figure 2b, is mostly invisible with most of the contents removed but with slight edge information. Figure 2c-e are the reconstructed images of the SPOH with down-sampling factor \(\tau=8\), \(12\), and \(16\). The dynamic range of the intensity is normalized to \([0,1]\), with \(0\) and \(1\) representing the smallest and the largest intensity in the image. A greyscale bar is associated with each image to reflect the relative intensity. It can be seen that the shaded region is preserved in all the \(3\) cases.