Fast calculation of computer-generated hologram using the circular symmetry of zone plates

Improving the imbalance of the light intensity of 3D wire-frame projection with electro-holography by superimposing a phase error

The CG-line method is an algorithm for generating computer-generated holograms (CGHs), a digitally recording medium for three-dimensional images in electro-holography. Since the CG-line method is specialized for projecting three-dimensional wireframe objects, it can calculate CGH with a very low computational load. However, the reconstructed image of the conventional CG-line method suffers from unintended light imbalance depending on the object shape, which disturbs the understandability of the projecting image. Therefore, we propose a method for reducing light imbalance by imposing phase error that controls light according to the line shape. Consequently, we reduced light imbalance by maintaining the high computational speed.

Fast numerical propagation in high-NA imaging using the resampling angular spectrum method

Numerical propagation calculation is a fundamental research topic in optical engineering. The standard angular spectrum method (ASM) is accurate but time- and memory-consuming, especially for high-NA systems. In this work, we propose a fast and simple numerical propagation method, the resampling ASM (RS-ASM). Numerical propagation can be accelerated by combining a resampling technique with interpolation methods in the angular spectrum domain of a constrained object at the focal plane. RS-ASM has three main advantages: simple implementation, faster calculation than the standard ASM, and SNR enhancement. Here we validate RS-ASM using theory, simulation and experiment. Using the "bilinear" ASM with a proper resampling factor can result in a speed-up factor of up to 20x (for a transformation from the angular spectrum to the E field) and 4x (for a transformation from E field to the angular spectrum), together with a SNR improvement of approximately 2x. For an application example of Gerchberg-Saxton phase reconstruction, the "bilinear" RS-ASM can converge 2.6x faster than the standard ASM.

Review of computer-generated hologram algorithms for color dynamic holographic three-dimensional display

Holographic three-dimensional display is an important display technique because it can provide all depth information of a real or virtual scene without any special eyewear. In recent years, with the development of computer and optoelectronic technology, computer-generated holograms have attracted extensive attention and developed as the most promising method to realize holographic display. However, some bottlenecks still restrict the development of computer-generated holograms, such as heavy computation burden, low image quality, and the complicated system of color holographic display. To overcome these problems, numerous algorithms have been investigated with the aim of color dynamic holographic three-dimensional display. In this review, we will explain the essence of various computer-generated hologram algorithms and provide some insights for future research.

Variable-intensity line 3D images drawn using kinoform-type electroholography superimposed with phase error

Three-dimensional (3D) display using electroholography is a promising technology for next-generation television systems; however, its applicability is limited by the heavy computational load for obtaining computer-generated holograms (CGHs). The CG-line method is an algorithm that calculates CGHs to display 3D line-drawn objects at a very high computational speed but with limited expressiveness; for instance, the intensity along the line must be constant. Herein, we propose an extension for drawing gradated 3D lines using the CG-line method by superimposing phase noise. Consequently, we succeeded in drawing gradated 3D lines while maintaining the high computational speed of the original CG-line method.

Pincushion point-spread function for computer-generated holography

Point-spread functions (PSFs) are non-stationary signals whose spatial frequency increases with the radius. These signals are only meaningful over a small spatial region when being propagated over short distances and sampled with regular sampling pitch. Otherwise, aliasing at steep incidence angles leads to the computation of spurious frequencies. This is generally addressed by evaluating the PSF in a bounded disk-shaped region, which has the added benefit that it reduces the required number of coefficient updates. This significantly accelerates numerical diffraction calculations in, e.g., wavefront recording planes for high-resolution holograms. However, the use of a disk-shaped PSF is too conservative since it only utilizes about 78.5% of the total bandwidth of the hologram. We therefore derive a novel, to the best of our knowledge, optimally shaped PSF fully utilizing the bandwidth formed by two bounding hyperbola. A number of numerical experiments with the newly proposed pincushion PSF were performed, reporting over three-fold reductions of the signal error and significant improvements to the visual quality of computer-generated holograms at high viewing angles.

Faster generation of holographic video of 3-D scenes with a Fourier spectrum-based NLUT method

In this article, a new type of Fourier spectrum-based novel look-up table (FS-NLUT) method is proposed for the faster generation of holographic video of three-dimensional (3-D) scenes. This proposed FS-NLUT method consists of principal frequency spectrums (PFSs) which are much smaller in size than the principal fringe patterns (PFPs) found in the conventional NLUT-based methods. This difference in size allows for the number of basic algebraic operations in the hologram generation process to be reduced significantly. In addition, the fully one-dimensional (1-D) calculation framework of the proposed method also allows for a significant reduction of overall hologram calculation time. In the experiments, the total number of basic algebraic operations needed for the proposed FS-NLUT method were found to be reduced by 81.23% when compared with that of the conventional 1-D NLUT method. In addition, the hologram calculation times of the proposed method, when implemented in the CPU and the GPU, were also found to be 60% and 66% faster than that of the conventional 1-D NLUT method, respectively. It was also confirmed that the proposed method implemented with two GPUs can generate a holographic video of a test 3-D scene in real-time (>24f/s).

GPU-accelerated calculation of computer-generated holograms for line-drawn objects

The heavy computational burden of computer-generated holograms (CGHs) has been a significant issue for three-dimensional (3D) display systems using electro-holography. Recently, fast CGH calculation methods of line-drawn objects for electro-holography were proposed, which are targeted for holography-based augmented reality/virtual reality devices because of their ability to project object contours in space with a small computational load. However, these methods still face shortcomings, namely, they cannot draw arbitrary curves with graphics processing unit (GPU) acceleration, which is an obstacle for replaying highly expressive and complex 3D images. In this paper, we propose an effective algorithm for calculating arbitrary line-drawn objects at layers of different depths suitable for implementation of GPU. By combining the integral calculation of wave propagation with an algebraic solution, we successfully calculated CGHs of 1, 920 × 1, 080 pixels within 1.1 ms on an NVIDIA Geforce RTX 2080Ti GPU.

An interactive holographic projection system that uses a hand-drawn interface with a consumer CPU

Holography is a promising technology for photo-realistic three-dimensional (3D) displays because of its ability to replay the light reflected from an object using a spatial light modulator (SLM). However, the enormous computational requirements for calculating computer-generated holograms (CGHs)—which are displayed on an SLM as a diffraction pattern—are a significant problem for practical uses (e.g., for interactive 3D displays for remote navigation systems). Here, we demonstrate an interactive 3D display system using electro-holography that can operate with a consumer’s CPU. The proposed system integrates an efficient and fast CGH computation algorithm for line-drawn 3D objects with inter-frame differencing, so that the trajectory of a line-drawn object that is handwritten on a drawing tablet can be played back interactively using only the CPU. In this system, we used an SLM with 1,920 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}× 1,080 pixels and a pixel pitch of 8 μm × 8 μm, a drawing tablet as an interface, and an Intel Core i9–9900K 3.60 GHz CPU. Numerical and optical experiments using a dataset of handwritten inputs show that the proposed system is capable of reproducing handwritten 3D images in real time with sufficient interactivity and image quality.

Fast calculation of computer-generated hologram of line-drawn objects without FFT

Although holographic display technology is one of the most promising three-dimensional (3D) display technologies for virtual and augmented reality, the enormous computational effort required to produce computer-generated holograms (CGHs) to digitally record and display 3D images presents a significant roadblock to the implementation of this technology. One of the most effective methods to implement fast CGH calculations is a diffraction calculation (e.g., angular spectrum diffraction) based on the fast-Fourier transform (FFT). Unfortunately, the computational complexity increases with increasing CGH resolution, which is what determines the size of a 3D image. Therefore, enormous calculations are still required to display a reasonably sized 3D image, even for a simple 3D image. To address this issue, we propose herein a fast CGH algorithm for 3D objects comprised of line-drawn objects at layers of different depths. An aperture formed from a continuous line at a single depth can be regarded as a series of aligned point sources of light, and the wavefront converges for a sufficiently long line. Thus, a CGH of a line-drawn object can be calculated by synthesizing converged wavefronts along the line. Numerical experiments indicate that, compared with the FFT-based method, the proposed method offers a factor-56 gain in speed for calculating 16-k-resolution CGHs from 3D objects composed of twelve line-drawn objects at different depths.

Full-scale one-dimensional NLUT method for accelerated generation of holographic videos with the least memory capacity

A full-scale one-dimensional novel-look-up-table (1-D NLUT) method enabling faster generation of holographic videos with the minimum memory capacity is proposed. Only a pair of half-sized 1-D baseline and depth-compensating principal-fringe-patterns (PFPs) is pre-calculated and stored based on the concentric-symmetry property of the PFP, and from which a set of half-sized 1-D PFPs for all depth planes are generated based on its thin-lens property, which enables minimization of the required memory size down to a few KB regardless of the number of depth planes. Moreover, all those hologram calculations are fully one-dimensionally performed with a set of half-sized 1-D PFPs based on its shift invariance property, which also allows minimization of its overall hologram calculation time. From experiments with test videos, the proposed method has been found to have the shortest hologram calculation time even with the least memory in comparison with several modified versions of the conventional NLUT and LUT methods, which confirms its feasibility.

Compression scheme of electro-holography based on the vector quantization of point light sources

Video holography has attracted attention after its invention in 1947; however, the enormous amount of data involved in recording and transmitting three-dimensional (3D) images remains a serious issue in electro-holography. Majority of the studies that have investigated holography transmission target the system that transmits the 3D images by compressing the holograms created on the distributor side using various compression techniques such as the conventional video compression techniques. However, the importance of the information in frequency space and characteristics, such as the correlation between adjacent pixels and frames, is different in natural images and holograms; therefore, these approaches are not always effective. In this study, we propose an effective electro-holography compression scheme based on the vector quantization of point light sources (PLSs). Instead of directly compressing a hologram, our method compresses and transmits PLSs from the distributor side and generates a hologram on the receiver side. To reduce the computational load that is required for creating a computer-generated hologram (CGH) on the receiver side, a fast CGH calculation technique has been developed for the vector-quantized PLS data based on the lookup tables (LUTs). This reduces the data rate by 76% when compared to that observed in case of uncompressed CGH transmission with 2K resolution and results in a calculation speed that is 1.34 times faster than that obtained using the conventional LUT method.

Highly efficient calculation method for computer-generated holographic stereogram using a lookup table

A highly efficient calculation method to synthesize a computer-generated holographic stereogram using a lookup table (LUT) is proposed. The complete phase distribution (CPD) of a spherical wave that is larger than the hologram is precalculated and stored as a LUT. The wavefront converging to each viewpoint can be directly obtained by adding a specific part of the precalculated CPD to the parallax image, instead of computing the wavefronts pixel by pixel in the conventional method. The computation amount and calculation time are effectively reduced to one-third of the conventional method. The working memory size is reduced effectively using the symmetry of the phase distributions of different viewpoints. A simulation and an optical experiment are performed to verify the proposed method.

Fast computer generated hologram calculation with a mini look-up table incorporated with radial symmetric interpolation

The amount of heavy computation in Computer Generated Hologram (CGH) can be significantly reduced by pre-computing look-up tables. However, the huge memory usage of look-up tables is a major challenge. To address this problem, the Look-up tables can be efficiently compressed by methods such as radial symmetric interpolation. In this paper, we notice that there is still data redundancy in the look-up table of radial symmetric interpolation method and the table size can be further compressed to 5%-10% or even less of original, by our proposed mini look-up table approach based on Principal Component Analysis (PCA). The compressed look-up table in our scheme only occupies a memory size of around 200-300 KB or even less. Moreover, the proposed scheme will introduce almost no extra cost of computation speed slowdown and reconstructed image quality degradation, compared to conventional method.

Sampling and error analysis of radial symmetric interpolation for fast hologram generation

In this paper, we present a fast hologram pattern generation method by radial symmetric interpolation, which exploits concentric redundancy of a point hologram pattern to reduce computational complexity in hologram pattern calculation, and analyze the quality degradation sources in the proposed method. Compared to the analytic method in which phase and amplitude information is directly calculated from a wave equation, in our method a Fresnel zone plate is periodically sampled along a diagonal line and the wave information of a point hologram is calculated by linear interpolation. During these sampling and interpolation processes, the wave information can be modified from the original signal and the reconstruction quality can be degraded compared to the analytic pattern calculation method. The effects of sampling and linear interpolation are investigated in spatial and frequency domains.

Simple and fast cosine approximation method for computer-generated hologram calculation

The cosine function is a heavy computational operation in computer-generated hologram (CGH) calculation; therefore, it is implemented by substitution methods such as a look-up table. However, the computational load and required memory space of such methods are still large. In this study, we propose a simple and fast cosine function approximation method for CGH calculation. As a result, we succeeded in creating CGH with sufficient quality and made the calculation time 1.6 times as fast at maximum compared to using the look-up table of the cosine function on CPU implementation.

Aerial projection of three-dimensional motion pictures by electro-holography and parabolic mirrors

We demonstrate an aerial projection system for reconstructing 3D motion pictures based on holography. The system consists of an optical source, a spatial light modulator corresponding to a display and two parabolic mirrors. The spatial light modulator displays holograms calculated by computer and can reconstruct holographic motion pictures near the surface of the modulator. The two parabolic mirrors can project floating 3D images of the motion pictures formed by the spatial light modulator without mechanical scanning or rotating. In this demonstration, we used a phase-modulation-type spatial light modulator. The number of pixels and the pixel pitch of the modulator were 1,080 × 1,920 and 8.0 μm × 8.0 μm, respectively. The diameter, the height and the focal length of each parabolic mirror were 288 mm, 55 mm and 100 mm, respectively. We succeeded in aerially projecting 3D motion pictures of size ~2.5 mm³ by this system constructed by the modulator and mirrors. In addition, by applying a fast computational algorithm for holograms, we achieved hologram calculations at ~12 ms per hologram with 4 CPU cores.

Fast calculation of computer-generated hologram using run-length encoding based recurrence relation

Computer-Generated Holograms (CGHs) can be generated by superimposing zoneplates. A zoneplate is a grating that can concentrate an incident light into a point. Since a zoneplate has a circular symmetry, we reported an algorithm that rapidly generates a zoneplate by drawing concentric circles using computer graphic techniques. However, random memory access was required in the algorithm and resulted in degradation of the computational efficiency. In this study, we propose a fast CGH generation algorithm without random memory access using run-length encoding (RLE) based recurrence relation. As a result, we succeeded in improving the calculation time by 88%, compared with that of the previous work.

Spatial bandwidth analysis of fast backward Fresnel diffraction for precise computer-generated hologram design

Designing near-field computer-generated holograms (CGHs) for a spatial light modulator (SLM) requires backward diffraction calculations. However, direct implementation of the discrete computational model of the Fresnel diffraction integral often produces inaccurate reconstruction. Finite sizes of the SLM and the target image, as well as aliasing, are major sources of error. Here we present a new design prescription for precise near-field CGHs based on comprehensive analysis of the spatial bandwidth. We demonstrate that, by controlling two free variables related to the target image, the designed hologram is free from aliasing and can have minimum error. To achieve this, we analyze the geometry of the target image, hologram, and Fourier transform plane of the target image to derive conditions for minimizing reconstruction error due to truncation of spatial frequencies lying outside of the hologram. The design prescription is verified by examples showing reconstruction error versus controlled parameters. Finally, it is applied to precise three-dimensional image reconstruction.