Slide 27 of 42
Okay, the first stage- the position function. This maps from data
value to "position relative to the boundary". That is, the signed
distance, from wherever you are, along the gradient direction, to the
middle of the boundary, as marked by the max in f' or the zero-
crossing in f''. Here's that same cut through the sample object
again, and we're graphing the data value as a function of position.
Now notice that at the middle of a the boundary, we hit value v0.
Therefore, at value v0, the position is 0, because we're at the
middle of the boundary. And for values higher than v0- inside the
object- the position is positive. For values lower than v0- outside
the object- the position is negative. The important thing to remember
about the position p(v) is that the position it gives is not some
(x,y,z) position inside the volume, but is really the distance between
the isosurface at v and the middle of the boundary. The useful thing
about the position function is that it can be calculated,
automatically, from the histogram volume. How exactly that happens is
something you can read about in the paper, but I will note that it
relies on both the measurement of f' and of f''- there's no way to
recover the position from having just one of those derivatives, so it
was definately important to measure them both.
