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of the face surface to the Euclidean distance between two canonical surface points. Canonical
surfaces were obtained from face surfaces by warping according to a topology preserving
transformation. Finally, face models were represented with the geometric moments up to
the fifth order computed for the 3D face canonical forms. However, although the effect of
expressions was attenuated, a similar attenuation also occurred for discriminating features
such as the eye sockets and the nose. This approach was improved in Bronstein et al. (2006b)
where the authors handled the challenge of missing parts. They embedded the probe facial
surface into that of the gallery. Faces belonging to the same subject are nearly isometric and
thus result in low embedding error, whereas different subjects are expected to have different
intrinsic geometry, and thus produce higher embedding error. The open mouth corrupts the
isometric model. This problem was handled later by the authors in Bronstein et al. (2007) by
using a geodesic mask that excluded the mouth region. The authors first detected and removed
the lips, then the computation of geodesic distance geodesics were calculated on the surface
in the presence of a hole corresponding to the removed part. This was done while avoiding
passing in mouth area.
The assumption of the isometric model has motivated several authors to use geodesic
distance on facial surface. In Samir et al. (2009a), the geodesic distance to the tip of the nose
was used as a surface distance function. Differential geometry was used to compare 3D level
curves of the surface distance function. This approach was an improvement upon an earlier
work by the authors Samir et al. (2006), where they had used the level curves of the height
function to define facial curves. The use of geodesic distance in (Samir et al., 2009a) allows
this approach to handle facial expressions. However, the open mouth corrupts the shape of
some level curves, and this parametrization did not address this problem; hence, experiments
were restricted to a small subset of FRGC v2.0 database.
A similar geodesic polar parametrization of the face surface was proposed in Mpiperis
et al. (2007), but rather than studying the shape of curves, they studied local geometric
attributes under this polar parametrization. To handle data with open mouths, they modified
their geodesic polar parametrization by disconnecting the lips. Therefore, their approach
required lips detection, as was the case in Bronstein et al. (2007).
In Berretti et al. (2010b), the authors used the geodesic distance on the face to extract iso-
geodesic facial stripes. Equal-width iso-geodesic facial stripes were used as nodes of graph and
edges between nodes were labeled with descriptors, referred to as 3D weighted walkthroughs
(3DWWs), that captured the mutual relative spatial displacement between all the pairs of
points of the corresponding stripes. Face partitioning into iso-geodesic stripes and 3DWWs
together provided an approximate representation of local morphology of faces that exhibits
smooth variations for changes induced by facial expressions.
A common limitation of the previously described approaches is that they assume that the
facial shape deforms isometrically, that is, the surface distances between points are preserved,
which is not valid in the case of large expressions. Actually, the movement of mimic muscles
can stretch and/or shrink the face surface and not only bending it.
Deformable Template-Based Approaches
In recent years there has been focus on deforming surfaces, one into another, under a chosen
criterion. Grenander's
deformable template
theory (Grenander, 1993) has been successfully
