Images and Fourier Analysis

We want to teach students to understand the concept of Fourier spectrum and its usage in image processing using Octave. Though students know formulae, they have no physical understanding of Fourier domain and how to use coefficient to gain something useful from an image or signal. We have simple programs to clarify some fundamental ideas about Fourier domain.

I created programs for their use:

1. Simple routines to create images with sinusoidal waves. They should understand the concept of 'spatial waves' and 'spatial frequencies'.
   

2. Then students are asked to look at a simple circular step function in an image and its Fourier Transform (Sinc function plotted using SQRT). They are expected to vary the amplitude and size of the step function in the image and understand the correspondence with the Sinc function in the Fourier spectrum.

3. Students are then given two Gaussians of different widths, with some separation. One can now use Fourier filters to smooth the image. As one throws away 'higher spatial frequencies', one would see some details lost. How much of smoothing leads to which details being lost?
4. Finally, we use a textbook image to grasp the idea that different 'structures' in an image are nothing but grey-level variations of certain spatial sizes. These sizes correspond to 'spatial wavelengths'.