Virtual Character Definition and Animation within the MPEG-4 Standard - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 44

an affine (or not) transformation, T i to C i is equivalent in defining a deformation
function

ϕ Ω

on such that:

i i

[ ]

∈

∀ v ∈ Ω , ϕ ( v ) = µ ( v ) ω T ( ξ ) − ξ

, (1)

i i i k i k k

ξ ψ ( v )

k i

where µ is the affectedness measure associated with C i and ω is a weigh

i k

coefficient.
In practice, the transformation T i is applied to the controller and is propagated
within the influence volume V ( S i ) according to the affectedness measure

µ

,

i

ψ ( v )

which plays the role of a weighting function. When is reduced to a single

i

element, Equation (1) is simplified:

[ ]

∀ v ∈ Ω , ϕ ( v ) = µ ( v ) T ( ψ ( v )) − ψ ( v )

. (2)

i i i i i i

This controller-based deformation framework enables the unification of the
different deformation techniques reported in the literature with respect to the
dimension of the controller support. Typically, the most representative technique
of a volume controller-based approach is the lattice-based deformation model.
In this case, the 3D grid is considered as the controller volume. The 1D
controller-based approach covers most of the deformation techniques currently
used, namely: spline-based and skeleton-based. The 0D controller principle is
used in the case of deformation tables (a particular case being described in the
previous section in the case of FBA), cluster-based and morphing-based
approaches.
In practice, choosing an appropriate controller results from a trade-off between:
(1) the complexity of representing the controller directly linked to its dimension;
and (2) the distribution of mesh vertices affected by the controller, specifically
by choosing the most appropriate influence volume.
An optimal balance is obtained by using a 1D controller. The support of the
controller is thus easy to control (the number of parameters is small) and the
corresponding influence volume covers a large class of configurations. The new
specifications of the MPEG-4 standard support this approach for animating an
articulated virtual character in the case of two specific 1D controllers, namely
a segment and a curve defined as a NURBS, referred to as bone and muscle-
based deformation, respectively.
The following sections describe in details each of these 1D controllers.