Estudos das séries e transformadas de Fourier e suas aplicações em processamento de imagens

Abstract

Signal and systems concepts arise in many fields, and the ideas and techniques associated with these concepts play an important role in diverse areas of science and technology, such as communications, circuit design, economic analysis, acoustics, biomedical engineering, generation systems. and power distribution and voice processing. This work presents the theory of Fourier series and Fourier transforms that allow to represent and study the behavior of certain functions with respect to periodicity properties. Fourier series allow you to represent many periodic functions as an infinite sum of exponential complex. The representation through series is very advantageous when you want to approximate the values of the function. Discretization of the functions of interest is quite desirable in many applications, especially those based on digital techniques. Discretization gives rise to the Discrete Fourier Transform (DFT) which provides a very good approximation for the coefficients of the Fourier series of periodic functions. We will end this work to also include information about some types of applications of the Fourier transform in image processing, giving a natural continuity to this work, expanding the vision to other fields of applications through a brief extension of the theory for the case of two variables and executing certain applications, with the objective of highlighting relevant characteristics, correcting defects, attenuating noise, approaching the elaboration of specific filters to perform these tasks. Thus, we present how it is possible to apply the Fourier transform in an image and then apply the inverse and obtain, practically, the original image.