3: Fourier Transformations
When thinking of images as a superposition of waves and wave patterns, the Fourier transform of an image can be used to identify periodic patterns that can be masked in their Fourier space and converting the image back to its spatial domain allows this usually unwanted periodic pattern to be removed/reduced, hence enhancing the image. The Fourier transform is generated by the series expansion of an image function in terms of “cosine” image basis functions.
Using an image of vertical lines for demonstration, the Fourier space shows us 3 dots. The middle dot stands for the DC component or the zero frequency component. This consists of points in the original image that are not repeating. The other 2 dots on each side correspond to the frequency of the stripes. There are 25 cycles present in the horizontal cosine images where each stripe appears every 4 pixels or 1 cycle every 2 pixels. This results in the dots being halfway between the center and the edge. The father away from the center a dot appears on the Fourier space means, higher the frequency.
When the periodic pattern is tilted, the hypotenuse is used instead, resulting in the distance between the center and frequency points to be larger than the previous example.
A simple example is the lunar orbiter image. There are (periodic) horizontal lines present that appear on the Fourier space as the vertical line in the center above and below the middle point. A mask is generated over this line and the Fourier transform is converted back to see the lines have been removed.
Using a more complicated example where the periodic patterns are very small dots (which may not be visible in the attached image), a mask specific to the requirements of the Fourier space was generated to remove the pattern.
In case the center DC component is masked, the resulting image ends up worse since now only the repeating patterns remain.