| Main Site areas:
Where can you go from here? |
The discrete Fourier Transform
As computational methods deal with discrete quantities a discrete Fourier transform, (DFT), is used instead of continuous. In the discrete domain the computational efficient Fast Fourier Transform, (FFT), is used. The DFT is only usually defined for a discrete function f(x,y) that is nonzero only over a finite region, 0£ x£ M-1 and 0£ y£ N-1, where M by N is the resolution of the spatial image. The two dimensional M by N DFT and inverse DFT are defined as:
The relationship between the sampling increments in the spatial, (D x,D y), and the frequency domain, (D u,D v), is defined as:
|
|
|
(3) |
The FFT of an image is a two dimensional array of complex numbers that represent the spatial image. The low frequencies represent the slowly varying content of the image whilst the higher frequencies represent the smaller details and pixel noise.
The FFT of an image can be displayed either using its real and imaginary components or using its magnitude and phase. A plot of the magnitude is known as the Fourier spectrum whilst the square of the magnitude is known as the Power spectrum or Spectral density. The phase angle is given by:
|
|
(4) |
| OELWeb Features: |
| Undergraduate course notes can be found
here.
Download FRAN, our fringe analysis software that's free for non-commercial use. Ever heard an opera singer shake the house down? See what they are doing to themselves. Do you know how an internal combustion engine works? Find out here. |