# Efficient Hogel-Based Hologram Synthesis Method for Holographic Stereogram Printing
## 1 Introduction  
Holography is a two-step technique that records and reconstructs the intensity and phase information of a wavefront scattered from a three-dimensional (3D) object [1; 2; 3]. Recently, many researchers have been paying attention to the holographic printers as a technique to visualize the holographic contents of a 3D object. The 3D virtual models can be easily implemented by a holographic printer, which can be divided into two main types: wavefront printer and a holographic stereogram (HS) printer. The wavefront printer records an arbitrary object field while retaining full control of both the amplitude and phase distributions of the 3D object as does the traditional analogue hologram [4; 5; 6; 7; 8]. Among the various holographic representation approaches, the holographic stereogram (HS) is the most common incoherent reconstruction technique. In HS, the angular information is stored as a holographic pixel or hogel and a sub-hologram is commonly generated from many perspectives of a 3D scene. A hogel is the smallest holographic element of a light field hologram and an array of these elements forms the entire image. Each hogel image diffracts the light as a plane wave where the intensity can change depending on the viewing angle. The advantage of the HS printer is that there is no complex diffraction calculation compared with the wavefront printer, which requires complex field extraction from a 3D object. Digital content generation is relatively simple for the HS printer. The HS printer records a set of two-dimensional (2D) parallax-related images created from an array of perspective projections of a 3D scene [9; 10; 11; 12; 13; 14]. For the hogel-based approaches, hogel images are usually re-arranged by a conventional pixel re-arrangement algorithm, which was proposed by [PERSON] and [PERSON] [15]. One drawback of this pixel re-arrangement technique is that there exists a trade-off problem between spatial-angular resolution of the reconstructed HS. Therefore, it needs a high memory usage and computation time to acquire a large number of perspectives. Various approaches have been proposed to realize the holographic stereogram printing technology with or without a hogel re-arrangement technique. A hogel image for HS can be created from phase-modulated or phase kinoforms using e-beam lithography [16; 17]. Refreshable HSs can be recorded rapidly and continuously without an X-Y translation stage and replaced it with an uninterrupted translation belt holding holographic lenses [18]. Non-hogel-based calculations, which generate HSs with continuous wavefronts for each object point has been presented recently [19]. Unlike conventional approaches, a non-hogel based method synthesizes holograms without hogel configuration, generating a converging parabolic wave for each object point with a continuous wavefront compared with conventional hogel-based approaches. A new method based on effective perspective image segmentation and mosaicking (EPISM) for single-step exposure was proposed [20]. The hogels are exposed using the synthetic effective images in sequence by mosaicking under the optimized modelling parameters. Recently, authors in Reference [21] have reviewed the development and research status of the synthetic holographic stereogram printing technique. Authors in Reference [22] show that the light-field based CGH techniques have advantages to explore near-eye light field displays for presenting high-definition 3D scenes with continuous depth and support of view-dependent occlusion. By combining light-field rendering and a Fresnel diffraction calculation, the aliasing in both over-sampling and under-sampling can be solved. In this paper, we present an efficient hologram synthesis method based on 3D rendering of computer-generated HSs by a pixel re-arrangement algorithm to limit the trade-off problem between spatial-angular resolution of the reconstruction. This novel method improves the angular resolution of HS without increasing the hogel size, while keeping the spatial resolution without degradation.