Graphics Reference
In-Depth Information
Fourier transform remaining frequencies to a value of zero. Then, the inverse of the (DFT) of
that square matrix is computed. To achieve invariance to in plane rotations, the real part of the
DFT inverse is rotated in small steps one complete cycle (360
◦
). The kernels at each step are
then interpolated and their average is found. Finally, the DC component is removed from the
average kernel and the resulting kernel is normalized to sum up to a value of one. The kernels
have shown to detect key points in all regions of the face. A comparison with the well-known
Laplacian of the Gaussian (LoG) kernel, whose response is similar to the DoG, which is also
used to generate blob like images from 2D images reveals that the blobs generated by the
adjacent frequencies approach gives much prominent blob like images.
As descriptors of the detected key points, the intersections of the local surface with five
concentric spheres (of different radii) are found, and each is sampled at fixed angular steps.
A pair of descriptors are matched by first estimating the off set roll rotation between their
samples using circular correlation. The estimation of the off set roll angle is then used to
vectorize the depths (
z
-coordinates) of the two sets of circular samples starting from the same
point (
z
1
and
z
2
, respectively, for the first and second descriptors), giving invariance to roll
rotations. Before the computation of a dissimilarity measure, small out of plane rotations are
adjusted for by adding a plane to one of the two depth vectors so it best fits the other depth
vector, as described in Equations 2.60 and 2.61.
[
uv
]
=
(
L
L
)
−
1
L
(
z
1
−
(
z
2
−
z
c
1
)) and
(2.60)
z
2
=
L
[
uv
]
,
z
2
−
z
c
1
+
(2.61)
where
L
is a two-column matrix holding the
x
- and
y
-coordinates of the samples of the second
descriptor,
u
and
v
are the fitting parameters, and
z
c
1
is the depth at the center of the first
region (at the key-point position). The dissimilarity measure is the summation of the absolute
errors between
z
1
and
z
2
, that is,
e
=
i
|
z
2
(
i
)
z
1
(
i
)
−
|
.
Landmark methods:
Local region matching around predefined landmarks on the face has
been also used in 3D face recognition. This application of landmarks is not generally as
promising as the use of key points in handling facial expressions. The landmarks are very
sparse in comparison to the key points and usually such approaches require the accurate and
full detection of all the landmarks, in contrast to approaches based on key points that are
tolerant to some failure in key-point detection and matching.
The well-known elastic bunch graph matching (EBGM) approach to 2D face recognition
(Wiskott et al., 1997), which has been adapted to perform 2D+3D and 3D face recognition
(Husken et al., 2005; Wang et al., 2001), is the best known approach in this category. EBGM
relies on a set of Gabor filters (of different frequencies, orientations called and scales) to
detect the landmarks and also to describe their local regions. The convolution of a range image
with the Gabor filters produces a vector of responses at each point called a jet. Initially, the
jets at manually localized landmarks on facial range images are computed and stored in the
nodes of a graph (face bunch graph, FBG). Each node stores a bunch of jets computed from
the different range images at a corresponding landmark. Once an initial FBG is obtained, it
can be automatically expanded. The landmarks on an unseen range image are localized by
searching for points whose jets best fit the jets in the FBG and their positions agree with the
FBG nodes.
