Questions about log polar transform
3.


  The polar image has the black areas, which are parts that can't be
  mapped to the original image because it would be outside of its
  boundaries. However, these areas have nearly disappeared in the log
  polar image. Why is that?


Look at the second (polar) and third (log-polar) images.  What's different about them? 



In the 3rd image, stuff that was on the left is now in the center.  Stuff that was in the center is now on the right.  The "log" in log-polar means that you're stretching the image in a logarithmic way.  So the black out-of-image scallops are still there, they've just been squished very thin against the right side of the image, while details from the left side of the image have been stretched out to take their place.