There are no strokes in the image. Part of the image resembles a "ring" (sort of like a ring around Saturn), with only part of the ring visible. Part of this ring tapers off to a thin point, therefore making a very fine "line." Going back and analyzing the problem, I see that the use of Minus Front always takes off a "hairline's width" from your image, but in most cases, it's not even noticeable. In this case though, the hairline reduction makes a significant part of the "ring" disappear.
My last resort is to add a hairline stroke to the image, outline the path, and then Minus Front. This should compensate for the hairline reduction. However, I was hoping for another way around this problem.