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images. However, for 3D point clouds and meshes, the fitting of the analytical surface to the
local 3D points requires a parameterization of the surface. A simple parameterization approach
is to project the local 3D points on the tangential plane. Their positions on the tangential plane
are then expressed in any orthonormal coordinates, u and v .
c x 3
2
c x 5
2
u 2
v 2
x
=
c x 0 +
c x 1 u
+
c x 2 v
+
+
c x 4 uv
+
.
(2.47)
c y 3
2
c y 5
2
u 2
v 2
y
=
c y 0 +
c y 1 u
+
c y 2 v
+
+
c y 4 uv
+
.
(2.48)
c z 3
2 u 2
c z 5
2
v 2
z
=
c z 0 +
c z 1 u
+
c z 2 v
+
+
c z 4 uv
+
.
(2.49)
2.3 3D Face Segmentation
The segmentation of 3D surfaces consists in the partitioning of the surface into disjoint
sub-surfaces of the same representation. The segmented subsurfaces may correspond to well-
defined surface parts, which are referred to as part-segmentation. Examples of this type include
the segmentation of the facial surface from a 3D scan and/or the subparts of the face such
as the nose, the forehead, and the eyes. Alternatively, the result of the segmentation may
not correspond to natural and visually distinct surface parts, yet the segmentation is still
meaningful. In this case, it is referred to as patch-segmentation. The latter segmentation is
objectively carried out on the basis of metric criteria and properties of the surface without
aiming to obtain segments neatly corresponding to distinctive parts.
There are many applications of 3D surface segmentation, including parameterization (which
allows texture mapping and curvature extraction), modeling, and morphing. However, the
primary purpose of surface segmentation in 3D face recognition is the extraction (from the
segmented sub-surfaces) of surface features for matching. Many general approaches for the seg-
mentation of 3D surfaces have been proposed in the literature. To some extent, these approaches
fall in a specific class of clustering algorithms but with a different variety of distance measures
and constraints pertaining to 3D surfaces. A recent survey of (generic) 3D surface segmentation
algorithms is provided by Shamir (2008) and another one for CAD applications is provided
by Agathos et al. (2007). In the next subsections, we limit our focus to 3D segmentation
approaches that are specific to facial surfaces.
2.3.1 Curvature-based 3D Face Segmentation
An number of 3D face recognition systems use surface curvatures for segmentation. As
curvatures are independent of the transformation/pose of the face and can be computed at
every 3D point of the surface, they are handy as measures for surface segmentation. They are
equally applicable for both patch-segmentation, part-segmentation, and the detection of facial
landmarks. For patch-segmentation, either principal curvatures,
κ 2 , or the mean H and
Gaussian K curvatures are computed at every point of the surface and are used to assign each
points to a surface segment. Typically, each of the segments has consistent geometric properties
such as convexity (peak), concavity (pit), ellipticity (shaped like an oval cup), hyperbolicity
(saddle shaped), cylindricity (both concave and convex corresponding to valleys and ridges
κ 1 and
 
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