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

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 ( − 1) 2 for the case of sparse matrices or binary images, where is the number of pixels other than zero and 2 is the total of points in the image.

Fast Treetops Counting Using Mathematical Image Symmetry, Segmentation, and Fast k-Means Classification Algorithms

“…Therefore, by the Dirichlet principle [39] there will always be pixels with repeated gray level values. In this way, it is ensured that in a grayscale image there is mathematical symmetry [40][41][42][43].…”

“…In order to use the mathematical symmetry in some mathematical calculation or computational process of an image, it is necessary to calculate the maximum mathematical symmetry group of the image [41]. Mathematically, images can be 2-dimensional, 3dimensional, or n-dimensional.…”

“…Pixel classification based on gray level was performed by using the optimized unsupervised k-means Algorithm [45]. This optimization is achieved by the mathematical symmetry found in all gray level images [40][41][42]. The groups used to classify the objects within the image are: treetop and non-treetop.…”

Construct a New System as a Combining Function for the LFSR in the Stream Cipher Systems Using Multiplicative Cyclic Group

“…Perez Ramirez et al presents an algorithm to reduce the Multiplicative Computational Complexity (MCC) 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 [10]. Theorem (1) [11]: (the fundamental theorem of arithmetic) Any positive integer n>1 can be written uniquely in the following prime factorization form:…”

The Application of Group Theory in Communication Operation Pipeline System

“…Group theory has been used widely in the field of physics and chemistry as a kind of abstract mathematical tools [14,15]. Group theory has also been used in engineering application, for example, to reduce the multiplicative computational complexity in the creation of digital holograms [16]. In the medical image processing field, the computation time is reduced considerably by exploiting symmetry presented in breast models using Group theory in order to detect breast cancer [17].…”