Graphics Reference
In-Depth Information
that are able to identify surface protrusions, and are often well aligned with perceptual features.
In Figure
5.1
some segmentations of a human model induced by the nodal domains of different
eigenfunctions are shown, chosen among the first 15 in the spectrum. At different scales, the
segmentations capture the symmetry of the shape, the arms, legs, hands and feet of the model.
Additional examples are given in Figure
5.2
.
Pose transfer
e pose transfer approach proposed in Lévy [
125
] is based on the Fourier de-
composition of the manifold embedding coordinates. ere the “layout” (pose) of the shape
X
is
transferred to the shape
Y
while preserving the geometric details of
Y
. In a formal way, given two
shapes
X
and
Y
embedded in
R
3
with the corresponding harmonic bases
X
i
and
Y
i
, and the cor-
responding Fourier decompositions of the embeddings
XD
P
P
i1
a
i
X
i
and
YD
i1
b
i
Y
i
,
P
P
N
iD1
a
i
Y
i
C
a new shape
Z
is composed according to the rule
ZD
i>N
b
i
Y
i
with the first
N
low frequency coefficients taken from
X
, and higher frequencies taken from
Y
. Figure
5.3
represents how the pose of the shape
X
is transferred to
Y
generating a new shape
Y
t
.
Figure 5.3:
e shape
Y
t
is obtained from
Y
with the transfer of the pose of
X
.
5.2 HEAT EQUATION
e heat equation describes the distribution of heat (or variation in temperature) in a given region
over time. In case the shape
S
is a compact two-dimensional Riemannian manifold, the diffusion
process on
S
is described by the partial differential equation:
@
@t
C
f.t;x/D0;
(5.3)
where
denotes the positive-semidefinite Laplace-Beltrami operator associated with the Rie-
mannian metric of
S
. e heat equation governs the distribution of heat from a source point on
the surface. e initial condition of the equation is some initial heat distribution
f.0;x/
at time
tD0
; if
S
has a boundary, appropriate boundary conditions must be added.
Search WWH ::
Custom Search