Morse and Morse-Smale Complexes - Mathematical Tools for Shape Analysis and Description - page 79

S1

S1

P

S

S2 Q

S2 Q

S3

S3
(A) (B)

Figure 10.2: (a) A surface network S N D.C;A/ . e arcs involved in the cancellation are high-
lighted. (b) e surface network S

0
N

0 0

D.C ;A / obtained from S N by eliminating the saddle s and the

minimum p . e new arcs are highlighted.

Figure 10.3: Segmentation of the atomic density function on a molecule in [ 33 ]. Minima are shown
in blue and ascending paths in gold. ree segmentations into 198 (left); 100 (middle) and 50 (right)
protrusions are depicted.

Another application is the physics simulation of the turbulent mixing between two fluids. In
particular, scientists are interested in bubbles formed during the mixing process and their automatic
segmentation. Given an iso-surface between two mixing fluids extracted from one time-step of a
simulation and the z -coordinate as the Morse function, bubbles can be defined as the descending
manifolds of maxima. Nevertheless, there exist many superfluous maxima caused by noise in the
data set. As it happens with molecular surfaces, a uniform simplification of the MS complex helps
remove most of these artifacts and create a much cleaner segmentation.
Finally, Morse and Morse-Smale complexes are strongly related to visualization of vector
field topology [ 101 , 194 ].