Abstract
Fourier presented a memoir to the Institut de France in 1807 where he claimed that any periodic function can be represented as a series of harmonically related sinusoids. This idea had a profound impact in mathematical analysis, physics and engineering, but it took one and a half centuries to understand the convergence of Fourier series and complete the theory of Fourier integrals. Fourier was motivated by the study of heat diffusion, which is governed by a linear differential equation. However, the Fourier transform diagonalizes all linear time-invariant operators, which are the building blocks of signal processing. It is, therefore, not only the starting point of our exploration but the basis of all further developments. Two -dimensional Fourier Transform is a straightforward extension of the one-dimensional Fourier transform. The two-dimensional case is briefly reviewed for image processing applications.
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