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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