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components of sets
D.p/\A.q/
, for all critical points
p
and
q
of function
f
[
69
,
70
]. Each
cell of the Morse-Smale complex is the union of the integral lines sharing the same origin
p
of
index
i
and the same destination
q
of index
j
. e dimension of the cell is given by the difference
of the indices. Examples of descending and ascending manifolds on a surface and the resulting
Morse-Smale complex are shown in Figure
10.1
.
Figure 10.1:
(a) Descending manifolds of maxima and saddles; (b) Ascending manifolds of minima
and saddles; (c) e Morse-Smale complex given by the overlay of these ascending and descending
manifolds.
e Morse-Smale complex is characterized by cells with a regular connectivity. In the 2D
case, each saddle point
p
has four incident
1
-cells, two joining
p
to maxima, and two joining
p
to
minima. Such
1
-cells alternate in a cyclic order around
p
. Also, the
2
-cells are quadrangles whose
vertices are critical points of
f
of index
1;0;1;2
(i.e., saddle, minimum, saddle, maximum) in
this order. In the 3D case, all
2
-cells are quadrangles whose vertices are a minimum,
1
-saddle,
2
-saddle,
1
-saddle in this order (
quadrangles of type 1
), or a
1
-saddle, a
2
-saddle, a maximum, a
1
-saddle in this order (
quadrangles of type 2
). A
1
-cell connecting a
1
-saddle and a
2
-saddle is on
the boundary of four quadrangles that alternate between quadrangles of type
1
and type
2
. e
3
-cells are called
crystals
and are bounded by quadrangles [
69
,
70
].
e
1
-skeleton of a Morse-Smale complex is a 1-complex formed by integral lines joining
critical points. Similar structures among critical points have been widely studied in the literature
under the name of
critical nets
. A graph representation of the critical net in a two-dimensional
Morse-Smale complex is the so-called
surface network
[
163
,
174
], widely used in spatial data
processing for morphological terrain modeling and analysis (see [
165
] for an interesting collection
of contributions on this specific topic).
Morse and Morse-Smale complexes have been extensively studied, mainly for the under-
standing and visualization of scalar fields, but also for more general applications in shape analysis,
by using as function the curvature [
133
,
155
], or the Connolly function [
41
]. A considerable num-
ber of algorithms has been developed for extracting critical points and lines, with a specific focus
on terrain modeling and analysis. In general, region-based methods aim at extracting a Morse
complex, while boundary-based approaches typically focus on the computation of a Morse-Smale
complex. In case of simplicial models the most popular algorithms for the extraction of the Morse
and Morse-Smale complexes are [
31
,
41
,
54
,
55
,
69
,
70
,
97
,
131
,
151
,
158
].
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