1.Background
In recent years, computational holography has been developing rapidly thanks to the advances in various technologies such as optics, electronics and computers, as well as new algorithms. Since the existing liquid crystal spatial light modulators have higher modulation ability and diffraction efficiency for pure phase holograms, the optimization algorithms for pure phase holograms have always been a research hotspot. At present, various traditional methods can satisfy different computational time-consuming and reconstruction quality requirements, while new methods such as deep learning and Verdinger flow bring new ideas for pure phase hologram optimization, and these works are conducive to the early realization of real-time, wide-field-of-view, and high-quality holographic 3D displays. Different from the traditional holographic imaging technology, in the field of computer-generated holograms, liquid crystal spatial light modulators bring an unprecedented flexible control ability over wavefront information, which provides a great development space and power for the development of computational holography.
Over the past few decades, there has been a proliferation of computer-generated phase-only hologram algorithms, the core of which is the phase-only hologram optimization problem: given a Complex-Amplitude Hologram (Complex-Amplitude Hologram), encode it as a Phase-Only Hologram (Phase-Only Hologram), so that the optical reconstruction of the image obtained from this pure phase hologram should reproduce the original image as much as possible. The image obtained by optical reconstruction with the phase-only hologram should be restored to the original image as much as possible. These methods are mainly divided into three categories: iterative methods, non-iterative methods and other methods. Iterative algorithms usually start from an approximation of the target hologram, and after a series of repetitive operations continue to optimize the approximation of the hologram until the reconstructed image obtained by this approximation meets certain error requirements; non-iterative algorithms do not need to repeat a large number of optimization calculations, and will be given at once according to the specified steps to the approximate solution. Due to the lower computational load, non-iterative algorithms are more in line with the requirements of real-time holographic display, but at the cost of the reconstruction quality of such methods is not as good as iterative algorithms; other methods are very diverse and have their own characteristics.
2.Introduction to algorithms for generating pure phase holograms
Iterative Algorithm: Gerchberg-Saxton Algorithm
Among the iterative algorithms that can generate pure phase holograms, the Iterative Fourier Transform Algorithm (IterativeFourier Transform Algorithm) is a more representative algorithm, which is characterized by the iterative passing of the Fourier Transform in two planes.
Fig. 1 Flowchart of computer-generated holography
The iterative Fourier transform algorithm, or Error Reduction Algorithm (Error Reduction Algorithm) was proposed as an algorithm for digital holography in the early 1970's, and later modified by Gerchberg and Saxton and applied in the field of phase extraction, which has become the most famous and probably the most utilized method in the iterative algorithm. -The Gerchberg-Saxton (GS) algorithm, whose flowchart is shown in Fig. 2.
Fig. 2 Flow chart of the Gerchberg-Saxton algorithm.
In this algorithm, according to the amplitude distribution of the hologram plane and the reconstructed image plane, the phase information of the light field in the hologram plane is obtained by iteratively performing forward and reverse light wave propagation as well as the constraints imposed on the two planes. This method is very suitable for the calculation of pure phase holograms, and the Fresnel or Fourier transforms can be used to calculate the propagation of the light field.
Iterative Algorithm: Error Diffusion Algorithm
The Error Diffusion Method is another type of iterative algorithm that iterates between pixels in the hologram plane. When the amplitude information of a complex amplitude hologram is directly removed, each pixel point generates an error, and the error diffusion algorithm will scan the pixel points one by one and diffuse the error of each pixel point to the four neighboring pixel points that have not yet been scanned according to a certain weight.
Fig. 3 Schematic diagram of the error diffusion algorithm;
(a)Error diffusion from left-to-right scanning; (b) error diffusion with right-to-left scanning.
3.Non-iterative algorithms
Random phase method is a commonly used non-iterative method in the process of hologram pure phasing. Since the pure phase holographic coding is equivalent to a high frequency filtering process, the reconstructed image only includes the boundary and line parts of the original image, so it is necessary to introduce a Random Phase Mask to make the wavefront of the original image dispersed to the whole hologram in order to improve the reconstruction quality, however, the ensuing speckle noise is also more obvious. In order to reduce this speckle noise, recently there is an improved random phase method, which introduces random phase masks with different frequencies for different images to further reduce the information loss and improve the reconstruction quality. In addition, there are many non-iterative methods for speckle noise reduction, such as the Sampled-Phase-only Hologram method with down-sampling mask, the Patterned Phase-Only Hologram method, the Double- Phase Method, and the Random Phase Method using non-randomized phase masks. Phase Method, and the Random Phase-Free Method using a non-random phase mask.
Fig. 4 Example of the role of random phase on the reconstruction results of pure phase holograms
(a)Original image; (b) without random phase mask added; (c) with random phase mask added.
4.Other methods
In addition to iterative and non-iterative algorithms, there is a direct algorithm that can be used to compute a pure phase hologram. Assuming that a pure phase hologram has M×N pixels and each pixel has Q possible values for the phase value, the search space of the pure phase hologram generation problem is M×N×Q, and the goal is to find all the pixel values of the hologram that minimize the error between the reconstructed image and the original image. There are three main categories of direct algorithms: direct search algorithm (Direct Search Algorithm), simulated annealing algorithm (Simulated Annealing Algorithm), and genetic algorithm (Genetic Algorithm).
Fig. 5 Comparison of three direct algorithms
In addition to the algorithms introduced above, a series of algorithms that have been proposed in recent years are also introduced in the paper, such as: a pure phase hologram generation algorithm between the two classifications of iterative algorithms and non-iterative algorithms, which can save a lot of computation time while maintaining a high reconstruction accuracy and is suitable for applications such as real-time display of holographic dynamics, as well as a deep learning method, which has been rapidly developing in recent years and has been used in the hologram compression In the loop, the CITL technique directly captures the optical reconstruction result of the hologram, and uses the result for further optimization of the hologram, and is able to achieve high reconstruction quality; and the phase extraction method based on Wirtinger Flow proposed by Chakravarthy et al. can transform the phase extraction problem into a first-order optimization algorithm (First-Order-Optimization), which can be used to optimize the hologram. The phase extraction method based on Wirtinger Flow proposed by Chakravarthy et al. can transform the phase extraction problem into a quadratic problem that can be optimized by the First-Order Optimization Method. Using this phase extraction method for hologram optimization can achieve very high accuracy in reconstruction quality at a computational cost comparable to that of the GS algorithm.
At present, both traditional iterative and non-iterative pure phase hologram optimization algorithms have achieved good results, but it is necessary to make a trade-off between computational time-consuming and reconstruction quality, and the continuous emergence of new methods such as deep learning and Verdinger flow has brought new ideas for solving this problem, and all these works are conducive to the early realization of real-time, wide-field-of-view, high-quality holographic 3D displays.
References:
Bu Haozhen,Jiao Shuming. Optimization algorithm for pure phase hologram[J]. Liquid Crystal and Display,2021,36(06):810-826.
DOI: 10.37188/CJLCD.2021-0035