# Fast 3D Analytical Affine Transformation for Polygon-Based Computer-Generated Holograms
## 1 Introduction  
With the rapid development of display and computer technology, 3D display technology has made great progress [1; 2]. The current 3D display technologies generally include binocular parallax display [3], volume 3D display [4], light field display [5] and holographic 3D display. Holographic 3D display is a technology to reconstruct 3D wave-front by using light wave diffraction information based on the principle of wave optics. Because the wave-front information of the 3D scene is reconstructed, it can provide all the depth cues required by the human eye [6; 7; 8]. The wave-front recording process of holography can be completed and simulated by a computer to generate the so-called computer-generated holograms (CGHs). The use of CGHs avoids the setup of complicated optical paths in optical holography [9; 10], as long as the mathematical description of the 3D scene is obtained and transformed into the wave-front distribution in the hologram plane by algorithms. Therefore, the algorithm of calculating CGHs is key, because it directly determines the computational efficiency of the hologram and the quality of the reconstructed image [11; 12]. Indeed, the mathematical description of a 3D scene can also be expressed in many forms. According to the geometric information of its surface, it can be discretely expressed as a collection of point sources, giving the so-called point-based method [13]. The wavefront information of the whole 3D object is obtained by calculating and then adding the light field distribution of the spherical wave emitted by each point light source in the hologram plane. However, the required number of point sources is usually as high as millions, leading to computational bottlenecks. By using a look-up tables (LUTs) [14; 15; 16] and graphic processing units (GPUs) [17; 18], we can accelerate the calculations greatly. Another popular CGH calculation method is the use of polygons (usually represented by triangle meshes) to approximate the surface of a 3D scene, giving to the polygon-based method [19; 20; 21; 22; 23]. Compared with the point-based method, the number of polygons can be greatly reduced, and the mature theory of computer graphics can be used to make the reconstructed scene more realistic. There are other methods to decompose 3D objects, such as the layer-based [33; 34] or line-segment based method [35]. In polygon-based methods, the propagation of light field can be calculated for each polygon by using the angular spectrum theory if the polygon is parallel to the hologram plane [6]. The essence of polygon-based methods is the diffraction calculation between non-parallel planes. Polygon-based methods can be divided into two categories; one is the traditional method based on sampling [8]. This method needs to sample each tilted polygon that is not parallel to the hologram in the spatial and frequency domains. Due to the polygon rotation, sampling in the spatial and frequency domains creates uneven sampling intervals between the two domains. Time-consuming interpolation is needed to alleviate sampling distortion. The other method is the analytical method [8]. Unlike the traditional method, sampling in the spatial domain to obtain the spectrum of a tilted polygon is not needed. Instead, the spectrum of a tilted polygon can be expressed in terms of the spectrum of a unit triangle (also called primitive triangle), which is known analytically, through the 2D affine transform [23]. Therefore, only sampling is needed in the frequency domain for each polygon, bypassing the need for interpolation in the traditional method. Compared with the traditional method, the analytical method effectively reduces the amount of calculations. However, shaping and texture mapping are not easily included in the analytical methods as compared with the traditional methods [8]. [PERSON] et al. [25; 26] have developed an analytical method utilizing 3D affine transformation.