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Definition of a Fourier Transform
The conventional mathematical representation of an image is a function of two spatial variables, f(x,y). The function at a particular location, (x,y), is the intensity at that point. The term transform means an alternative mathematical representation of the image.
A Fourier Transform uses a series of complex exponentials (sinusoids) with different frequencies to represent an image.
The Fourier Transform has applications in image processing, feature recognition, signal processing etc.
If f(x,y) are spatial variables in the continuous domain the two dimensional Fourier Transform is defined as:
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(1) |
where f(x,y) is the light intensity at point (x,y). (u,v) are the horizontal and vertical spatial frequencies respectively. As e-j2p ux =cos2p ux–jsin2p ux, F(u,v) is composed of an infinite series of sine and cosine terms. Inversely, the Fourier Transform can be transformed back to the spatial domain by the inverse Fourier transform:
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(2) |
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