# Fast 3D Content Update for Wide-Angle Holographic Near-Eye Display
## 2 Principles of Hologram Updating  
The aim of a holographic display is to reconstruct an object which can be seen from any viewpoint. The viewer should be free to move their eyes in any direction (\(x,y,z\)-shift of a viewpoint) and observe corresponding perspectives of the object (rotation of the viewpoint). The viewer's location in respect of the object determines the geometry for hologram computation. Therefore, when this geometry is altered, a different 3D image should be encoded. Figure 1 illustrates the change in mutual object-hologram geometry as the eye position changes. At the first location, a viewer sees the front view of an object, while at the next, the viewer observes other parts of the object from a different angle. The straightforward way to accommodate this change is to recalculate the entire CGH for each geometry. However, the computation load can be very high. To reduce the computation time, it is necessary to use the view approximation method [28; 30]. Unfortunately, known methods are based on paraxial approximation, which cannot be used for wide-angle displays. Figure 1: Illustration of viewpoint change. The proposed hologram update method enables faster content generation for a wide-angle holographic display system where the hologram plane is related to the eye position. Figure 2a shows the studied geometry, for which the hologram plane (\(x_{1}\), \(y_{1}\)) is located at a distance \(z\) away from the image plane (\(x_{3}\), \(y_{3}\)). The square hologram O has a size of \(B_{Ox}\times B_{Ox}\), where \(B_{Ox}=N_{x}\Delta\), \(N_{x}\) is the pixel size and \(\Delta\) the pixel pitch. The pixel dimension determines the viewing angle as \(FoV_{x}=2\sin^{-1}(\lambda/2\Delta)\) and the corresponding size of the image, \(B_{x3}=2\pi\)tan(\(\frac{1}{2}FoV_{x}\)). In our research, a hologram of resolution 4K \(\times\) 4K pixels and pixel pitch \(\Delta=0.62\) um is investigated. For the given pixel pitch and wavelength \(\lambda=640\) nm, the \(FoV_{x}=62^{\circ}\), while the physical size of the hologram O is \(2.54\times 2.54\) mm\({}^{2}\). The starting distance between the hologram and image plane is set to \(z=1000\) mm, for which the image space is of size 1180 \(\times\) 1180 mm\({}^{2}\). Figure 2b illustrates data transfer of the hologram update method for two consecutive viewpoints. The investigated case considers viewpoint location change in the \(z\)-\(x\) plane for simplicity. Two holograms, the initial O\({}_{1}\)(\(x_{1}\), \(z_{1}\)) and updated O\({}_{2}\)(\(x_{2}\), \(z_{2}\)), of size \(B_{Ox}\), correspond to the first and second viewpoint positions, respectively. Hologram O\({}_{2}\) is shifted and rotated with respect to the initial hologram O\({}_{1}\). The spatial shift from the center of O\({}_{1}\) to the center of O\({}_{2}\) is determined by the vector, \(v_{12}\). For this geometry, the normal vectors of O\({}_{1}\) and O\({}_{2}\) are directed to the center of the observed object. Therefore, the rotation angle of hologram O\({}_{2}\) is equal to the angle \(\alpha\) between optical axes \(z_{1}\) and \(z_{2}\). For a small change of viewpoint, the holograms O\({}_{2}\) and O\({}_{1}\), looking from the object position, share the common area. The common hologram area for the new viewpoint will be called the overlap region (blue part of O\({}_{1}\) and O\({}_{2}\)), while the non-overlapping part will be called the blank region (green part of O\({}_{2}\)). The division of hologram O\({}_{2}\) into the overlap and blank regions is the basis for the proposed hologram update algorithm, where the overlap information of hologram O\({}_{1}\) can be reused to fill the overlap region of hologram O\({}_{2}\) and only the blank region of hologram O\({}_{2}\) is filled with new data.