Application of Mathematical Symmetrical Group Theory in the Creation Process of Digital Holograms

Agustín Pérez-Ramírez,Joel Antonio Trejo-Sánchez,Julian Guerrero Juk,Rafael Sanchez-Lara,Lelio De La Cruz-May

doi:10.1155/2017/5612743

Abstract

This work presents an algorithm to reduce the multiplicative computational complexity in the creation of digital holograms, where an object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image. The image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k-1)N2 for the case of sparse matrices or binary images, where k is the number of pixels other than zero and N2 is the total of points in the image.

Highlights

The problem of reducing the computational time in the creation of digital holograms is becoming a demanding research topic due to the growing range of applications in different areas [1] such as optical microscopy, interferometry, entertainment, security, medicine, and education
This work only focuses on reducing the multiplicative computational complexity in the creation of digital holograms since the time depends on the hardware of the computer and the programming techniques used
It is worth mentioning that the algorithm does not improve the quality of the holograms of Fresnel using the N-LUT method and the peak signal-to-noise ratio (PSNR), and the structural similarity (SSIM) index is the same

Summary

Introduction

The problem of reducing the computational time in the creation of digital holograms is becoming a demanding research topic due to the growing range of applications in different areas [1] such as optical microscopy, interferometry, entertainment, security, medicine, and education. The creation time of holograms has been reduced by using increasingly faster algorithms and hardware such as FPGA (Field Programmable Gate Array) and parallel programming techniques [2] and precomputed lookup tables [3, 4]. An algorithm using group theory is proposed, where MCC (multiplicative computational complexity) is zero in the generation of digital holograms for 2D images, two colors, and different sizes. Unlike other works, it successfully explores the use of the global mathematical.

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