Graphics Reference
In-Depth Information
an effective tool for dimensionality reduction of facial data, PCA is the de facto technique for
feature extraction in many 3D face recognition approaches that further process these features
to counteract feature variations and, as a result, enhance the recognition accuracy. Examples
of such approaches are some of those based on linear discriminate analysis (LDA), nonlin-
ear manifolds and support vector machines (SVMs). PCA is also a building block in some
approaches to nonrigid 3D face recognition (see Section 2.5).
Linear Discriminant Analysis (LDA)
Description of LDA:
Similar to that of PCA, LDA projects observations of multi-dimensional
data on an orthonormal subspace. However, as opposed to PCA, which aims for a subspace
that captures most of the variance in the data, LDA aims for a better separability among
the various classes of the data (the 3D faces). In order to achieve that, LDA finds a subspace
E
e
k
] that maximizes the Fisher's criterion (the ratio of the variance between the
classes to the variance within the classes) as shown in Equation 2.51.
=
[
e
1
...
trace
E
B
E
E
W
E
J
(
L
)
=
(2.51)
k
e
i
B
e
i
e
i
W
e
i
=
(2.52)
i
=
1
k
e
i
−
W
B
e
i
,
=
(2.53)
i
=
1
where
W
are the between and within covariance matrices that are defined in terms of
a number of
C
class means,
B
and
μ
1
...μ
C
, the overall mean (the average face)
f
, and the vectorized
facial data,
f
ij
, which denotes the
i
th facial vector belonging to the
j
th person (class).
C
f
)(
f
)
.
B
=
(
μ
j
−
μ
j
−
(2.54)
j
=
1
N
j
C
(
f
ij
−
μ
j
)(
f
ij
−
μ
j
)
.
W
=
(2.55)
J
=
i
=
1
1
From Equation 2.53, it can be seen that
J
maximized when the vectors
e
1
...
e
k
are the
−
w
B
. Similarly to PCA, the LDA feature vector of an unseen facial vector
is computed by subtracting the mean face first and then projecting the difference onto the
subspace.
eigenvectors of
Comments on the use of LDA in 3D face recognition:
Because LDA requires the inversion
of the within-class covariance matrix, the total number of the training data samples should be
higher than the dimension of sample vectors, which is usually large (the number of pixels).
Otherwise, the matrix will be singular, referred to as the small sample size (SSS) problem.
