Virtual Character Definition and Animation within the MPEG-4 Standard - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 42

by applying a rigid geometric transformation to each segment. In order to
overcome this limitation, the object should be considered as a seamless mesh and
animated by means of deformations. The generic principle is described and
illustrated below.
Let { } M (Ω ) be a seamless mesh, where Ω = v , 1 v ,... v is the set of the mesh

0 n

vertices and let (Ω

i )

be a family of non-empty subsets of Ω (Figure 8a). A local

i

3

deformation function ϕ : Ω → R makes it possible to move a vertex v ∈ Ω into

i i

the new position expressed as v + ϕ ( v ) (Figure 8b and c). Here, ϕ is extended

i i

from Ω to Ω as the null function, i.e., ∀ v ∈ Ω \ Ω , ϕ ( v ) = 0 . Note that the

i i i

family ( Ω

i )

is not necessarily a partition of

Ω . In particular,

Ω can be a strict

i i

subset of Ω (some vertices may remain unchanged) and for two distinct subsets

∩Ω can be non-empty. The deformation
satisfies the superposition principle, i.e., the deformation induced by both

Ω Ω

and Ω , the intersection

i i j

j

ϕ

and

i

ϕ +
(Figure 8d). In order to achieve a compact description and an efficient implemen-
tation of a deformation model, the notion of a deformation controller is
introduced. It is defined as a triplet made of: (1) the support S associated with
a n dimensional ( n D) geometric object (

ϕ ( v ) ϕ ( v )

at a vertex v belonging to Ω ∩ Ω is expressed as the sum

j i j

i j

∈ n ); (2) an influence volume
V ( S ) associated with S ; and (3) the affectedness measure µ , defined on V ( S ) and
characterizing the intrinsic deformation properties of the influence volume.

{0, 1, 2, 3}

Figure 8. Mesh deformation principle.

(a) Mesh partition.

ϕ

(b) Deformation function

i

Ω applied to the subset .

i