We now modify the above flow. In order to be careful about signs,
we simply note that the boundary of a disk initialized so that
the inside of the disk corresponds to a negative value for the signed distance
function 
and a positive value for the signed distance function 
on the outside of the disk has a normal 
which
points outwards away from the center of the disk, and a curvature
defined as 
which is always
positive on all the convex level contours.
Thus, a flow under speed function 
corresponds to the
collapsing curvature flow, since the boundary moves in the direction of
its normal with negative speed, and hence moves inwards.
We need to be further careful about signs and amend a previous definition. We shall refer to a speed function F in the context of the level set equation
thus, from now on, F will give the speed of the front in a direction
opposite to its normal direction. Thus, a curve collapsing under its
curvature will correspond to speed 
. This will be our
convention for the remainder of this paper.
Now, motivated by work on level set methods applied to grid generation [25] and shape recognition [11], we consider two flows, namely
is to allow the inward concave
fingers to grow outwards, while suppressing the motion of the outward
convex regions.
Thus, the motion halts as soon as the convex hull is obtained.
Conversely, the effect of flow under 
is
to allow the outward regions to grow inwards while suppressing the motion of
the inward concave regions. However, once the shape becomes fully convex,
the curvature is always positive and hence the flow becomes the same as
regular curvature flow; hence the shape collapses to a point.
Figure 2: Motion of Curve under Min/Max Flow
We can summarize the above by saying that, for the above case,
flow under 
preserves some of the structure of the
curve, while flow under 
completely diffuses away
all of the information.
Before proceeding, we examine a slightly more complex shape, which
is instructive.
In Figure 6, we show what happens to a double
star-shaped region.
We let the color black correspond to the ``inside'' where 
and the
white correspond to the ``outside'' where 
.
First, in Figure 6a,
we show evolution under plain curvature, that is 
.
Eventually, the shape collapses completely.
In Figure 6b, we show the same curve collapsing
under 
; here, the outer front
moves to the convex hull, while the inner front collapses and disappears.
The last shown state is stable.
In Figure 6c, we show the same curve collapsing
under 
; here, the outer part of the front
collapses, while the inner part expands to its convex hull. Eventually,
the two meet, and the front disappears.
Finally, in Figure 6d, we switch the roles of
black and white; thus flow with speed 
corresponds
to the same flow as in Figure 6d; changing the
colors corresponds to changing between the maximum flow and the minimum flow.
Figure 3: Motion of Complex Region under Various Flows