|Image Processing Toolbox User's Guide|
Discrete Fourier Transform
Working with the Fourier transform on a computer usually involves a form of the transform known as the discrete Fourier transform (DFT). A discrete transform is a transform whose input and output values are discrete samples, making it convenient for computer manipulation. There are two principal reasons for using this form of the transform:
The DFT is usually defined for a discrete function that is nonzero only over the finite region and . The two-dimensional M-by-N DFT and inverse M-by-N DFT relationships are given by
The values are the DFT coefficients of . The zero-frequency coefficient, , is often called the "DC component." DC is an electrical engineering term that stands for direct current. (Note that matrix indices in MATLAB always start at 1 rather than 0; therefore, the matrix elements
F(1,1) correspond to the mathematical quantities and , respectively.)
The MATLAB functions
fftn implement the fast Fourier transform algorithm for computing the one-dimensional DFT, two-dimensional DFT, and N-dimensional DFT, respectively. The functions
ifftn compute the inverse DFT.
Relationship to the Fourier Transform
The DFT coefficients are samples of the Fourier transform .
fthat is similar to the function f(m,n) in the example in Definition of Fourier Transform. Remember that f(m,n) is equal to 1 within the rectangular region and 0 elsewhere. Use a binary image to represent f(m,n).
fwith these commands.
Discrete Fourier Transform Computed Without Padding
fwhen computing its DFT. The zero padding and DFT computation can be performed in a single step with this command.
fto be 256-by-256 before computing the DFT.
Discrete Fourier Transform Computed with Padding
fftshift, which swaps the quadrants of
Fso that the zero-frequency coefficient is in the center.
|Fourier Transform||Applications of the Fourier Transform|
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