64 8. DIFFERENTIAL TOPOLOGY AND SHAPE ANALYSIS
e characterization provided by Banchoﬀ correctly distinguishes critical points in dimen-
sion 2 and 3. For higher-dimensional spaces the Betti numbers of the lower link, that is the set of
connected components of the link of a vertex which join points with a height less than that the
vertex, provide a more complete characterization of discrete critical points, as suggested in .
It has to be observed that, since the deﬁnition of critical points has a local nature, small
perturbations of the shapes can considerably inﬂuence the number of critical points detected.
Moreover, in many applications the shapes to be analyzed are likely to have degenerate critical
points. A simple solution is to locally simulate the “simplicity” of the critical points  locally
perturbing the surface but in general it is necessary to further ﬁlter and structure them in a hi-
erarchical structure, as explained in the descriptions and theories described in the chapters 9, 10
Figure 8.3: Critical points for a mathematical surface and a terrain-like surface. Red points corre-
spond to maxima, blue points to minima, and green points are saddles.
8.2 TOPOLOGICAL ANALYSIS THROUGH (LOWER) LEVEL
Topological information about M is captured by the changes of the level sets and the lower level
sets of M relative to the function f . e level set of f corresponding to the real value t is the set
of points V
D fp 2 M j f .p/ D tg D f
.t/; t is called an isovalue of f . e lower level set is
given by M
D fp 2 M jf .p/ tg D f
We begin by studying how the lower level set M
changes as the parameter t changes
(Figure 8.4(a)). Morse theory states that the topology of M
stays unchanged as the parameter
t goes through regular values of f , while changes occur when t passes through a critical value.
More precisely, the following theorem holds:
eorem 8.2 Let a; b be real numbers such that a < b and the set f
.Œa; b/ contains no critical
points for f . en M
have the same topology.