Before the rush

Before the rush
by evan-pak

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I am reading a book about image processing. I have found mention of a method for processing images.

It is said that a special effect could be reached when we multiply two results of the Fast Fourier Transform. But in the book, it does not mention the function of the multiplication.

Then my question is what the function of the multiplication of two results of the Fast Fourier Transform? If this question is too abstract to answer, can someone give me some practical usages of FFT in image processing?

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Fast Fourier Transforms are simply Fourier transforms computed... fast :-) That is, in your question it is not relevant that they are fast, you could also compute them very slowly. – Francesco Mar 27 '12 at 5:43
Chuck have you read the link about Gaussian blur? It is an example of convolving a gaussian kernel with your image, to achieve the desired blur. Convolution is typically done via Fourier Transforms. – Francesco Mar 27 '12 at 5:53
Francesco, thanks. I will read this right away. – Chuck Wang Mar 27 '12 at 5:57
I have added a more detailed (and very nice) link. – Francesco Mar 27 '12 at 6:03

You should add more details with respect to the specific special effect.

In general, given two function f(x) and g(x) and their Fourier transforms F(k), G(k), the product of the two transform F(k)G(k) when brought back to the coordinate space (that is, when anti-transformed) is equal to the convolution of the original f and g.

Suppose that you are convoluting a function f with a gaussian kernel (to achieve a gaussian blur). Instead of doing the convolution integral you could find easier to multiply the Fourier transforms of the f and of the kernel, and then transform back.

Here you find a nice description of blurring and unblurring which showcases the idea without burdening the explanation with excessive mathematical notation .

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Sorry that I did not provide enough information. – Chuck Wang Mar 27 '12 at 5:47
Interesting link -- that's a great reference. – D. Lambert Mar 27 '12 at 14:13

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