Beginning in the 1970s, computer scientists used mathematical algorithms to search through computer code to find ways to reduce the file size. Since then, there has been an ever-growing need, spurred on by the development of the Internet, to create better compression schemes and reduce the size of any given file as much as possible. (McClain)
File compression is a crucial part of the internet's inner workings. It allows files to be transferred that would otherwise consume too much bandwidth and time. Because of this, it’s vital for companies like Spotify and Netflix, in addition to their users. Compressing, while still preserving quality, makes their operations more efficient. (Stegner)
Fourier expansion is a mathematical model used to represent an arbitrary function in terms of a sum of sinusoidal waves. Throughout this study, large amounts of data is manipulated and stored as list of values known as Fourier Coefficients. While there are fewer of them, their summation provides the same function as the original data, but requires less computational space to do so.
Mathematical modeling has a variety of applications and is becoming increasingly important. Current compression techniques make use of optimization and redundancy techniques that far exceeds this paper's ability to describe compression. Nonetheless, through the use of rotating vectors, Fourier analysis, and optimization, this essay considers and alternative method of storing 2D and 3D data from a mathematical perspective. Simultaneously, the essay also explores the feasibility of the concept through detailed investigations and published results.
The Following Research question is investigated:
Do Fourier Transformations provide a superior approach to storing and rendering 3D objects in comparison to conventional methods?
This study of data storage and Fourier Analysis aims to be as straightforward as possible. In this regard, it is important to have a logical starting point to develop from in order to consider deeper levels of understanding and application throughout the study.
In terms of resources, it is also important to note that the majority of sources of information will be secondary, with the exception of primary analytic investigations in order to provide exemplars and promote a greater understanding. As such, they aim to come from trusted subject-related encyclopedias and peer-reviewed papers in order to be as reliable as possible. In order to accomplish this, searches will be conducted through scholarly databases in addition to external library catalogs such as the Massachusetts Institute of Technology Library and ETH-Bibliothek (ETH Zurich Library).
This paper is created using LaTeX, a typesetting system.