Morse and Morse-Smale Complexes - Mathematical Tools for Shape Analysis and Description - page 78

10.2 CONCEPTS IN ACTION
Generalization of Morse-Smale complexes Topology-based methods used for visualization and
analysis of scientific data are becoming increasingly popular. eir main advantage lies in the
capability to provide a concise and effective synthesis of important structures in scientific data
sets, where interactive exploration of data is a means to understand phenomena at hand. Consider
for instance, simulations of turbulent mixing between fluids, or in molecular shape analysis: the
interactive exploration of data in this field requires the handling of huge, complex and possibly
noisy data. erefore, flexible hierarchical, or multi-resolution, representations are welcome in
order to allow for an adaptive data simplification and for detecting important structures [ 32 - 34 ].
In the case of Morse-Smale complexes, it has been shown how multi-resolution represen-
tations can be obtained by successively coupling and eliminating pairs of critical points. For a 2D
scalar field, the so-called cancellation consists of collapsing a maximum-saddle pair into a max-
imum, or a minimum-saddle pair into a minimum. A cancellation simulates the smoothing of
the scalar field by modifying the gradient flows around two critical points. Figure 10.2 shows an
example of cancellation on a surface network. e various techniques proposed in the literature
differ in the criterion used to filter the critical points, and in the order they are eliminated. For
example, in [ 32 , 70 ], a saddle s is chosen together with its adjacent maximum at lower elevation,
or its adjacent minimum at higher elevation; the order in which the pairs of points are cancelled
is determined by their persistence, according to the principle that persistence is equivalent to
significance (cf. chapter 11 ).
e case of 3D Morse-Smale complexes has been investigated in [ 96 ] and more recently
in [ 49 , 95 , 97 ]. ese methods extend 2D techniques to functions defined over 3-manifolds. e
extension is not trivial, since in 3D there are three possible types of legal cancellations: mini-
mum and 1-saddle, 1-saddle and 2-saddle, and 2-saddle and maximum. While the simplification
involving any extremum (minimum or maximum) are similar to their 2D counterparts, the saddle-
saddle cancellation does not have an analogue in lower dimensions. To ensure that minima and
maxima originally separate by two saddles do not connect, additional cells have to be introduced
in the Morse-Smale complex, which are removed by subsequent saddle-extremum cancellations
as proposed in [ 96 ].
As an application of the multi-resolution generalization of Morse-Smale complexes, Bre-
mer et al. [ 34 ] propose the rendering of a 1201-by-1201 single-byte integer value terrain data set
of the Grand Canyon, and the identification of oil extraction sites in an underground oil reservoir,
as local minima in the simplified Morse-Smale complex of oil pressure data.
In [ 33 ] two other applications are shown. One is molecular biology, where the focus is
the segmentation of a molecular surface into cavities and protrusions. Given the skin surface of
a protein complex and the atomic density function over this surface, the ascending manifolds
of minima of this function segment the surface into protrusions; simplifying the Morse-Smale
complex captures protrusions at coarser and coarser level, as shown in Figure 10.3 .