Graphics Reference
In-Depth Information
The estimation of the rigid transformations at the
i
th iteration, the central part of ICP, is
computed from two sets of points with a one-to-one correspondence. These correspondences
are typically formed by taking the vertices of the first surface as the first set of 3D points
and the second set as their closest points in the other surface. Alternatively, the closest points
along the directions of the surface normals are taken. The latter approach has been shown
to demonstrate a higher registration accuracy. Let
P
and
Q
be 3
n
matrices storing the
x
-,
y
- and
z
-coordinates of the first and second point sets, and let
p
i
and
q
i
represent the
i
th
corresponding pair of points. The transformations (
R
i
and
t
i
) that minimize the registration
error,
e
×
=
i
=
1
, can be found in a number of ways. A closed form solution
can find
R
i
and
t
i
by first translating the point sets so that their average positions lie on
the origin,
p
i
=
q
i
−
R
i
p
i
−
t
i
n
i
=
1
p
i
p
i
and similarly
q
i
=
p
i
−
q
i
−
q
i
where
p
i
and
q
i
are
p
i
=
/
1
n
i
=
1
q
i
. The rotational matrix is then found by orthonormalizing the matrix
and
q
i
=
1
/
UDV
; the singular values are replaced by
ones, and if the determinant of the resulting matrix is
P
Q
using single value decomposition
A
A
=
=
−
1, the matrix is multiplied by
−
1, that
UIV
. The translation
t
i
is given by
is
R
=±
−
Rp
+
q
.
Use of ICP in holistic face recognition:
The registration errors between facial surfaces has
been used in 3D face recognition as a dissimilarity measure (Lu et al., 2006). Since ICP is
an approach designed for the registration of rigid objects, its accuracy in 3D face recognition
on the basis of its registration error deteriorates with variations in facial expressions (see
Figure 2.5). Matching facial surfaces using the histograms of the registration errors appears
to handle expression variations better (Amor et al., 2006). The partial ICP approach to 3D
face recognition (Wang et al., 2006), can dynamically extract the regions of the face that are
not affected by expressions (vary according to the expressions) and compute the dissimilarity
measures of these regions. The partial ICP ranks the correspondences (in each ICP iteration)
on the basis of their distances in ascending order and considers only a fixed percentage of
them, that is, those which rank first. The ICP is also used for textured 3D face recognition. It
should be noted that the extension of ICP to multi modal is straightforward (apart from the
complexities related to illuminations variations) as a fixed mapping between the 3D points
and their texture should be maintained.
Principal Component Analysis (PCA)
PCA is an effective approach to the compression of high dimensional data (where each
observation is represented by a multi dimensional vector). The higher dimensional data can
often reside (or approximately reside) in a lower dimensional orthonormal subspace. In this
case, the original data can be approximated or represented by a lower number of variables by
projection on that subspace, the dimension of which is equal to the number of the reduced
variables. These variables were widely used as feature vectors in 2D face recognition and later
in 3D face recognition.
Description of PCA:
Firstly, a lower subspace of the 3D facial data (usually a collection of
range image) is found. As PCA operates on vectorial data, the range images are vectorized
in the same manner. The pixels of each range image are stacked in a vector, row by row or,
equivalently, column by column. Let the vector
f
i
represent the vectorized
i
th range image.
