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Image Querying with Complex Wavelets: The 2D Discrete Fourier Transform

Module by: Tom Mowad, Venkat Chandrasekaran. E-mail the authors

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Summary: Introduces the 2D DFT.

The two-dimensional Discrete Fourier Transform is another important transform in image processing. It is taken by applying the one-dimensional transform to each row, and then to each column, as seems to be the common practice for increasing the dimension of transforms in signal and image processing.

The 2D DFT has many properties that are useful in image processing; however, most useful is its shift invariance. The DFT of an image and its shifted version differ only by the multiplication with a complex exponential. Since multiplication by a complex exponential changes the phase but not the magnitude, taking the magnitude of the DFT of an image can give us a different kind of “shift-invariance”; that is, the DFT of two versions of the same thing will, rather than being shifted versions of one another, be identical.

Figure 1: Real part of a 2D DFT basis function
High frequency basis function.
High frequency basis function. (real.jpg)

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