Discrete Fourier Transform

The Fourier transform method has a long mathematical history and we are not going to discuss it here (it can be found in any digital signal processing or digital image processing theory book). As far as image processing is concerned, we shall focus only on 2D Discrete Fourier Transform (DFT). The basic idea behind the Fourier transform method is that an image can be thought of as a 2D function, f, that can be expressed as a weighted sum of sines and cosines (Fourier basic functions) along two dimensions.

We can transition from a set of grayscale pixel values in the image (spatial/time domain) to a set of Fourier coefficients (frequency domain) using the DFT, and it is discrete since the spatial and the transform variables to be used can only take a set of discrete consecutive integer values (typically the locations of a 2D array representing the image).

In a similar way, the frequency domain 2D array of Fourier coefficients can be converted back into the spatial...