Slide 23 of 42
As a quick check to make sure the histogram volume is actually measuring
something useful about the boundaries, we can project it along one of the
axes to get these scatterplots of f' versus f, or f'' versus f. This will
let us test the hypothesis that if there is a boundary in the original
volume, there will be one of those curves we saw before in the histogram
volume. Here's a cross-section of a simple object- one sphere, uniform
interior, uniform boundary, and sure enough, there are those same curves
being traced out in the scatterplots. Here's a slightly more interesting
object- two concentric spherical shells with a hole in the middle- so
there's really three boundaries here- from black to gray, gray to white,
and white back to black. And, correspondingly, in the scatterplots of the
histogram volume, there are three curves, black to gray, gray to white,
and white back to black. Same here. This illustrates that for every
boundary in the original volume, there'll be one of these curves in the
scatterplot.
