Graphics Reference
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and the Euclidean distances shows that the geodesic distance is more stable than Euclidean
one under facial expression variations.
In other words, this assumption means that
f
preserves the geodesic distance
d
between
every pair of points of facial surface, such that
d
S
1
(
x
,
y
)
=
d
S
2
(
f
(
x
)
,
f
(
y
))
,
∀
x
,
y
∈
S
1
.
(3.1)
This assumption is valid in the case of a small expression. The large expression causes some
combinations of shrinking and/or stretching. Under such variations, the geodesic distance is
no more stable. To quantitatively verify this assumption, we placed four markers on a face and
tracked the change of the geodesic and Euclidean distances under large expression variations.
Figure 3.2 shows that a change in expression from neutral to non-neutral generally results
in a shrinking or a stretching of the face shape, and both the Euclidean and geodesic distances
between points on the face are changed.
In one of the illustrated cases, these distances decrease (from 113 to 103 mm for Euclidean
distance and from 115 to 106 mm for the geodesic distance), whereas in the other two cases they
increase. This clearly shows that large expression variations cause stretching and shrinking of
face shape, and under such elastic deformations, neither the Euclidean distance nor the surface
distance is preserved. Hence, the assumption of isometric deformation of the shape of the face
is not always valid. One way to handle such elastic deformations of faces due to changes in
expressions is to use elastic shape analysis.
Neutral face
Neutral face
Expressive face
Neutral face
Shrinking
Expressive face
Expressive face
Distance along line (Euclidian)
Distance along surface (Geodesic)
Figure 3.2
Significant changes in both Euclidean and surface distances under different face expressions.
Copyright
C
1969, IEEE
