Virtual Character Definition and Animation within the MPEG-4 Standard - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 48

In practice, the computations are performed in the local coordinate system
attached to the bone. In order to perform the scaling with respect to a pre-defined
direction, the matrix SR b performs a pre-orientation of the bone. Thus, in this
system, the T transformation is expressed as:

T = TR b ⋅ C b ⋅ R b ⋅ SR b ⋅ S b ⋅ ( SR b ) -1 ⋅ ( C b ) -1 (5)

where matrix C b allows it to pass from the bone local coordinate system to the
world coordinate system.
Once the bone geometric transformation is computed, it is possible to compute
the new position of the skin vertices in the bone influence volume according to
Equation (2).
The SMS framework does not limit the number of bones used for animating a
virtual character. Consequently, local deformation effects can be obtained by
creating and animating additional bones. If such an approach can be used with
a certain success for simulating limited local deformation, the efficiency may
strongly decrease when considering realistic animations. To address this kind of
animation, the second controller, ensuring muscle-like deformation was intro-
duced.
The muscle influence volume is constructed as a tubular surface generated by
a circle of radius r moving along the NURBS curve (Figure 10b). The
affectedness function is then defined as follows:

( () )

0 δ v , ψ v > r

i i

( () )

r − δ v , ψ v

( () )

µ ( v ) = ,

i i

(6) f δ v , ψ v ≤ r

i i

r

where δ denotes the Euclidean distance, f ( ⋅ ) is to be chosen among the following

π

3 2 1 / 2

x , x , x , sin( x ), x x / 3

functions: and , and ψ is the function assigning to

2

v its correspondent point on the muscle curve.
A muscle is designed as a curve, together with an influence volume on the virtual
character's skin. In order to build a flexible and compact representation of the
muscle shape, a NURBS-based modeling is used. Animating the muscle consists
here of updating the NURBS parameters. The main advantages of NURBS-
based modeling are the accurate representation of complex shapes. The control