# Fast 3D Content Update for Wide-Angle Holographic Near-Eye Display
## 3 Update of the Blank View Region  
Calculation of CGHs can be carried out with different approaches depending on the geometry of the problem [33]. For the case of wide-angle CGHs, it is necessary to process the enormous amount of wavefield information efficiently. Propagation-based techniques [20; 34] are not optimal for CGHs with wide-angle, as they do not usually consider geometry where the object is much larger than the hologram. Recently, it has been shown that PAS-based algorithms enable efficient calculation of wide viewing angle CGHs [23]. The PAS algorithm is based on the principle of dividing the hologram plane into smaller segments of size \(N_{s}\times N_{s}\)[35; 36; 37]. At each segment, PAS approximates the spherical wavefront that comes from the \(p\)-th point source of the object with a piece-wise plane wave [36]. It is this property that enables efficient computations. Thus, the PAS algorithm was employed for filling the blank part of the hologram. Moreover, PAS enables easy calculation of hologram information directly from the frequency domain, which further improves execution time [21; 36; 38]. With the aim of coding the proper perspective into the blank region, the rotation angle \(\alpha\) on the \(y\)-axis is applied to the object. Then, information from the point-cloud is coded into each segment in the frequency domain as follows:  
\[\widetilde{O}_{mn}=\sum_{p=1}^{N}\frac{a_{p}}{r_{pmm}}\delta\Big{(}f_{x}-\hat{ f}_{pxmm},f_{y}-\hat{f}_{pymm}\Big{)}e^{ikr_{pmm}}, \tag{2}\]  
where \(r_{pmm}=\Big{(}(x_{2}-x_{2n^{c}})^{2}+(y_{2}-y_{2n^{c}})^{2}+z_{2}^{2}\Big{)}\) is the distance from the \(p\)-th point source to the center of the corresponding segment \((x_{2m}{}^{c},y_{2n}{}^{c})\) within the blank region, \(n=1,\dots,\)\(N\), and \(m=1,\dots,M\) indicate segment position over horizontal and vertical directions in the blank region, and  
\[\hat{f}_{pxmm}=\left\lfloor\frac{1}{\Delta f_{s}}\left(\frac{\sin\theta_{pxmm }}{\lambda}\right)\right\rfloor\Delta f_{s}, \tag{3}\]  
\[\hat{f}_{pymm}=\left\lfloor\frac{1}{\Delta f_{s}}\left(\frac{\sin\theta_{pymm }}{\lambda}\right)\right\rfloor\Delta f_{s}, \tag{4}\]  
are the discrete local directional frequencies [21] where \(\Delta f_{s}=1/N_{s}\Delta\) is the frequency sampling, \(\lfloor\cdot\rfloor\) is the rounding operation, and the angles \(\theta_{pxmm}\), \(\theta_{pymm}\) are the incident angles of object beams on \(x\) and \(y\) directions according to the center position of the segment, respectively. Notably, Equation (2) allows representing the three-dimensional point source information as a set of two-dimensional Dirac deltas [36; 38]. The position of each delta into \(\tilde{O}_{mn}\) is given by the discrete frequencies shown in Equations (3) and (4). After placing all the Dirac deltas for the point-cloud distribution, the inverse FT is applied to obtain the corresponding segment. The accuracy of the PAS algorithm depends on the size of the selected segment. When the size of the segments is smaller than the value based on sampling theory, noise reconstruction is introduced because the discretization error is increased according to the reduction of the number of pixels in the segment. On the other hand, when the segment size is larger to reduce the amount of discretization error, the reconstruction noise will be increased because of the condition from sampling [36]. Thus, the proper size of a segment is a kind of compromise for PAS [38]. The accuracy problem in PAS is improved by using accurate PAS (A-PAS) [36]. A-PAS enlarges the segment in the frequency domain by an integer factor \(q\), which increases the frequency sampling and thereby reduces the discretization error. After applying the inverse FT to the enlarged segment, a submatrix of size \(N_{s}\) is taken and the remaining part is disregarded. The employment of the PAS algorithm for efficient calculation of wide-angle CGHs with occlusion culling has been shown in [23], where it was shown that PAS enables efficient CGH calculations and occlusion culling of unseen areas.