Fourier Transforms
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(New page: = The Fourier Transform = The basic idea behind Fourier transforms is rather simple, even fascinating: it is possible to form any function ''f(x)'' as a summation of a series of sine and ...)
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[edit] The Fourier Transform
The basic idea behind Fourier transforms is rather simple, even fascinating: it is possible to form any function f(x) as a summation of a series of sine and cosine terms of increasing frequency. In other words, any space or time varying data can be transformed into a different domain called the frequency space. A fellow called Joseph Fourier first came up with the idea in the 19th century, and it was proven to be useful in various applications, including gene sequencing.
Let us talk about this frequency space before going any further into the details. The term frequency comes up a lot in physics, as some variation in time, describing the characteristics of some periodic motion or behavior. The term frequency that we talk about in computer vision usually is to do with variation in brightness or color across the image, i.e. it is a function of spatial coordinates, rather than time. Some books even call it spatial frequency.
We briefly look at the Fourier transform in the purely mathematical point of view, i.e. we will talk about “continuous” or “infinite” things. Fourier transform of a function is a summation of sine and cosine terms of different frequency. The summation can, in theory, consist of an infinite number of sine and cosine terms.
Now, let "f(x)" be a continuous function of a real variable "x". The Fourier transform of "f(x)" is defined by the equation:
