Journal of Mathematical Imaging and Vision

, Volume 25, Issue 1, pp 87–105

# Representation of the Fourier Transform by Fourier Series

Article

## Abstract

The analysis of the mathematical structure of the integral Fourier transform shows that the transform can be split and represented by certain sets of frequencies as coefficients of Fourier series of periodic functions in the interval $$[0,2\pi)$$. In this paper we describe such periodic functions for the one- and two-dimensional Fourier transforms. The approximation of the inverse Fourier transform by periodic functions is described. The application of the new representation is considered for the discrete Fourier transform, when the transform is split into a set of short and separable 1-D transforms, and the discrete signal is represented as a set of short signals. Properties of such representation, which is called the paired representation, are considered and the basis paired functions are described. An effective application of new forms of representation of a two-dimensional image by splitting-signals is described for image enhancement. It is shown that by processing only one splitting-signal, one can achieve an enhancement that may exceed results of traditional methods of image enhancement.

## Keywords

Fourier transform Fourier series paired transform wavelet transform signal decomposition

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