Graphics Reference
In-Depth Information
2.5 Deformation Modeling of 3D Face Surface
In the previous section, a common strategy used by the rigid approaches for the handling
of expression variations for 3D face recognition is to avoid deformable and/or deformed
regions. This strategy has clearly shown to improve the recognition performance. Yet, the
fact remains that it does not take advantage of any discriminative information available in
the non utilized regions. Another successful strategy for handling expression variations is to
apply deformations to the facial surface before matching, this set of approaches fall under
what is termed nonrigid 3D face recognition. The applied deformations should counter-act
(or neutralize) those attributed to facial expressions but should be able to retain any relevant
discriminative information. In fact, the success of nonrigid approaches to 3D face recognition
largely depends on the extent of achieving this criterion.
Modeling Expressions as Isomorphism
A popular class of 3D face recognition systems, which model facial expression deformations,
is based on the assumption that the geodesic distances on the facial surface are invariant to
facial expression. This is equivalent to assuming that the facial skin does not stretch or shrink
under expression variations. This assumption is intuitive to an extent, but it is violated for
expressions that involve topological changes of the facial surface, notably mouth opening.
This issue, however, can be handled by the detection and closing of the open mouth in a
preprocessing stage. One of the early approaches in this category is that by Bronstein et al.
(2005). They have experimentally shown that geodesic distances on facial surfaces have much
less absolute errors due to expression variations than Euclidian distances.
The approach by Bronstein et al. (2005) first deforms the facial surface to a canoni-
cal form (a surface of a standard shape in 3D space) subject to the preservation of the
geodesic distances. During the matching phase, the canonical forms are matched against
each other using ICP or other feature-based matching. The deformation is carried out itera-
tively and through the optimization of an objective function. The coordinates of the surface
vertices (or sample points on the surface) are varied (displaced in the 3D space) such that
the objective function that incurs costs proportional to the discrepancies in geodesic dis-
tances (with respect to the original surface) is minimized. Hence, the computation of the
geodesic distances is required at every iteration of deformation. The geodesic distances can
be computed from every point on the facial surface to every other point on that same sur-
face. However, this is computationally expensive. The computational load of the approach is
reduced by only computing distances from certain points (the tip of the nose and the nose
apex in their case). These geodesic distances are efficiently computed by the fast marching
method (FMM), which emulates wave propagation on the 3D facial surface from these two
selected points.
Alternative methods to the deformation of the facial surfaces have also been developed. They
extract features on the basis of the geodesic distances of the surface rather than using them as
constraints for deformation. In the work by Smeets et al. (2009), the geodesic distances among
the 3D points of the facial surface are first computed and stored in a 2D matrix. The largest
singular values of that matrix are then taken as features. These features are independent of the
order of the surface points used during the computation of that matrix. It is worth mentioning
that the isometric modeling of expression deformations only captures the intrinsic properties
