25
C H A P T E R 4
Differential Geometry and
Shape Analysis
In this chapter, we focus on a specific type of shape: surfaces. Surfaces play a fundamental role in
applications as they are used to model boundaries of 3D objects in computer graphics. We will
draw our attention to surfaces with a geometric perspective: geometry is one of the oldest fields
in mathematics, and it is concerned with questions of shape and size measurements. Geometry is
still used to model objects, as our ancestors once did, but today geometric concepts have evolved
to a higher level of abstraction and complexity.
What type of geometric shape properties may we compute on surfaces? e first distinc-
tion to be made is between extrinsic and intrinsic properties. Extrinsic properties are the properties
related to how the surface is laid out in the Euclidean 3D space, and can be described by using
the Euclidean distance between points; Euclidean distances form the basis for most of the earli-
est shape analysis methods in computer vision and computer graphics. Intrinsic properties are the
properties related to the metric structure of the surface and are invariant to a number of defor-
mations.
From the beginning and through the middle of the 18th century, differential geometry
was studied from the extrinsic point of view: curves and surfaces were considered as lying in a
Euclidean space of higher dimension (for example a surface in an ambient space of three dimen-
sions). Starting with the work of Riemann, the intrinsic point of view was developed, in which
one cannot speak of moving outside” the geometric object because it is considered to be given
in a free-standing way. Intrinsic properties started becoming more and more popular in shape
analysis for their suitability to analyze deformable objects, which are ubiquitous in our reality,
from human organs to living beings. Intrinsic properties can be described nicely using geodesic
distances, which measure the length of the shortest path along the surface between two points.
Geodesic and intrinsic metric were defined in section 3.2 for general spaces; here we will in-
troduce their definition for surfaces, that is smooth 2-manifolds (section 4.1). e fundamental
result concerning intrinsic properties is Gauss’s eorema Egregium, which elegantly connects the
total curvature of a smooth manifold without boundary to the Euler characteristic of the space it
represents. Another relevant result is the demonstration that the Gaussian curvature is an intrinsic
invariant of surfaces and this has been used very often in applications to characterize the shape of
surfaces.
26 4. DIFFERENTIAL GEOMETRY AND SHAPE ANALYSIS
4.1 GEODESIC DISTANCES ON SURFACES
In general, surfaces are defined as 2-dimensional manifolds with or without boundary. is gen-
eral definition can be translated into a parametric formulation to ease the formalization of geodesic
and curvature expressions.
A surface is called regular if in a neighborhood of each of its points it can be expressed as
(regular parametrization)
D .u; v/ D x.u; v/; y.u; v/; z.u; v/
where .u; v/ is a regular (i.e., a sufficient number of times differentiable) vector function satis-
fying
u
v
D
0
B
B
@
@x
@u
@y
@u
@z
@u
1
C
C
A
0
B
B
@
@x
@v
@y
@v
@z
@v
1
C
C
A
¤ 0:
Any regular surface can be considered as a metric space with its own intrinsic metric, with
the surface element defined as
ds
2
D d
2
D E.u; v/du
2
C 2F .u; v/dudv C G.u; v/dv
2
where E.u; v/ D h
u
;
u
i, F .u; v/ D h
u
;
v
i, G.u; v/ D h
v
;
v
i. e length of a curve de-
fined on the surface by the equations u D u.t/, v D v.t /, t 2 Œ0; 1, can be computed as
l./ D
Z
1
0
p
Eu
0
2
C 2F u
0
v
0
C Gv
0
2
dt:
en, the geodesic distance between two points on the surface is defined as the infimum of the
lengths of all curves on the surface connecting the two points (cf. sections 3.1 and 3.2).
e expression
I D ds
2
D E.u; v/du
2
C 2F .u; v/dudv C G.u; v/dv
2
is called the first fundamental form. e second fundamental form is defined as
II D Ldu
2
C 2Mdudv C Ndv
2
with L D
uu
n, M D
uv
n, N D
vv
n, and n the normal vector n D
u
v
j
u
v
j
:
Figure 4.1 shows that the distance between two fingers measured along the hand is always
the same, independently of the posture. e use of geodesic distances proved to be effective in a
number of studies, and paved the road to a number of tools for intrinsic non-rigid shape analysis.

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