17.7. Other Fourier Transform Properties

17.7.1. General Properties

../_images/FT_properties.png

Source: Aaron Bobick - Udacity Image Process MOOC from Georgia Tech

17.7.2. What Causes Ringing?

When images are processed, it is very common to see synthetic artifacts around the edges of objects in the image. These artifacts look like a small, light colored shadow of the edge of the object. This is called ringing. Many different filtering algorithm can cause small amounts of ringing.

The primary source of ringing comes from ideal low pass filtering. The ideal low pass filter is like multiplying the frequency response by a box (one period of a square wave). When the image is converted to the spatial domain, the transfer function of the box is convolved with the image. We can see from Common Fourier Transform Pairs that the Fourier transform of a box is what is called a sinc function.

sinc(x) = \frac{\sin(x)}{x}

The sinc function has tails to both sides that oscillate. This oscillation is what causes ringing. We can avoid some ringing by not using the ideal low pass filter, but other image processing algorithms can have the side affect of ringing also.

../_images/ringing.png

Source: Aaron Bobick - Udacity Image Process MOOC from Georgia Tech