Fourier analysis and its applications in image processing

Abstract

Fourier Analysis is a theory of pivotal relevance in many fields, as it allows any periodic function in a finite interval to be represented as a sum of sines and cosines. The Fourier Transform extends this concept to non-periodic functions by decomposing them into their frequency components. This paper aims to present the fundamentals of Fourier Analysis, covering the key properties and results of the Fourier Series and the Fourier Transform. Subsequently, the discrete version of the Fourier Transform, known as the Discrete Fourier Transform (DFT), will be discussed. Additionally, we will examine the correct methods for sampling continuous signals, addressing issues of sampling and aliasing. The paper will then introduce the groundbreaking work of Cooley and Tukey (1965) on the Fast Fourier Transform (FFT), an algorithm that reduces the computational cost of the DFT from $O\left(N^2\right)$ to $O(N \log N)$. Finally, the application of Fourier Analysis in the field of image processing will be explored, demonstrating its practical significance and versatility.