Image Processing Reference
In-Depth Information
defines a mapping from the low-dimensional representation to the high-dimensional one. This
mapping can be written as
=
w
i
f
i
(
c
)
+
,
x
(2.5)
i
where
w
i
are the weights of the possibly nonlinear functions
f
i
of the low-dimensional representation
of the manifold
c
. By placing a simple Gaussian prior on the weights
w
i
, they can be marginalized
out. This yields a multivariate Gaussian conditional density for the data, which can be written as
exp
2
tr
K
−
1
XX
T
,
1
(
2
π)
ND
1
|
C
,)
=
−
p(
X
(2.6)
D
|
K
|
where
X
and
C
are the matrices containing the
ND
-dimensional training examples and their latent
representations respectively.
K
is a positive-definite covariance matrix whose elements are obtained
by evaluating a kernel function
k
, such that
K
i,j
=
k(
c
i
,
c
j
)
. This kernel function is entirely defined
by its hyper-parameters
, which are optimized at training together with the latent variables
C
,so
as to maximize
p(
X
|
C
, )p(
C
)p()
. At inference, the predictive distribution
p(
x
∗
|
c
∗
,
C
,
X
)
of a
new deformation
x
given its latent representation
c
is a Gaussian with mean and variance
∗
∗
X
T
K
−
1
k
μ(
c
)
=
(2.7)
∗
∗
k
T
K
−
1
k
∗
,
σ(
c
∗
)
=
k(
c
∗
,
c
∗
)
−
(2.8)
∗
N
×
1
is the vector containing the covariance function evaluated between the training
and the test data. This has the advantage of modeling the uncertainty on the output space to account
for the high or low density of training examples in different regions of the space. As a consequence,
it allows to build a prior for the shape and its latent representation.
Several extensions of the original GPLVM have been proposed. For instance, to extend the
GPLVM to motion data, the Gaussian Process Dynamical Model (GPDM)
Wang
et al.
[
2005
]
was introduced. The GPDM allows to model nonlinear relationships between the latent variables
corresponding to consecutive frames in a sequence. In a different context, to overcome the burden of
evaluating the kernel function between each training latent variable, and thus of having a computation
time cubic in
N
, sparse representations were proposed
Lawrence
[
2007
]. In this sparse GPLVM, the
kernel is defined in terms of a much smaller number of inducing variables. This makes the GPLVM
practical for problems involving many degrees of freedom, therefore requiring large training sets, as
is the case of deformable surfaces.
where
k
∗
∈
2.2.2 LEARNEDMODELS FORNON-RIGIDMODELING
In Computer Vision, the linear learning techniques quickly became very popular. The original Ac-
tive Shape Models
Cootes and Taylor
[
1992
] were extended to full 2D Active Appearance Models
(AAM)
Cootes
et al.
[
1998
],
Matthews and Baker
[
2004
] to track 2D face deformations. In this
case, the model is separated into shape and texture components, both modeled as linear combi-
nations of basis vectors. Adaptations of this were also proposed to group appearance and shape in
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