Gross - 10 
Fig. 7 Same geometry, Gouraud shaded and viewed from above (from [2]): 
a) initial triangular patches. 
b) 16 subdivisions. 
c) 64 subdivisions. 
The inherent paradigm behind the deformable surface model is to find the minimum of 
an energy functional defined by internal stretching and bending forces as well as by external 
forces. The computation of this variational functional follows the finite element procedures 
and carries out the 3D shape as a result. This is further stressed in fig.8, where deformations 
of a simple 128 element patch under different external forces are shown. 
Note, that the respective boundary conditions, such as vertex displacement and tangent 
planes, have to be set in advance. 
The above paradigm allows to perform even very complex, physically-based modeling 
tasks. A final example for human faces is given below, where the goal is to answer questions 
like alterations after facial surgery or the simulation of aging in 3D. Figure 9 depicts the 
different processing steps necessary to set up a physically based model of the human face 
and scull. The required input data stems from a high-res CT or MR image and from a range 
scanner. After initial visualization and segmentation steps a geometric model of the face- 
scull anatomy has to be generated. This is accomplished by a NURB modeler, which also 
handles all interactions necessary to manipulate the model. Additional processing is de 
manded to achieve registration of CT and range data or to accomplish adaptive triangula 
tions of the surface. The results are feed into a physically-based model of the face anatomy, 
which is illustrated on the right hand side. In this first approach, the FEM-shape is attached 
via springs with the scull. The accurate visualization and analysis of the final shape is fig 
ured out using a ray-tracer for the finite element shape functions.