Fourier Methods

Lets consider a 1D Fourier transform example:

Consider a complicated sound such as the noise of a car horn. We can describe this sound in two related ways:

• sample the amplitude of the sound many times a second, which gives an approximation to the sound as a function of time.
  
• analyse the sound in terms of the pitches of the notes, or frequencies, which make the sound up, recording the amplitude of each frequency.

Similarly brightness along a line can be recorded as a set of values measured at equally spaced distances apart, or equivalently, at a set of spatial frequency values.

Each of these frequency values is referred to as a frequency component.

An image is a two-dimensional array of pixel measurements on a uniform grid.

This information be described in terms of a two-dimensional grid of spatial frequencies.

A given frequency component now specifies what contribution is made by data which is changing with specified x and y direction spatial frequencies.

 

• What do frequencies mean in an image?
  
• Fourier Theory
  
• Convolution
  
• Fourier Transforms and Convolutions
  
• The Fast Fourier Transform Algorithm