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so that groups of markers work together to define the position and rotation of joints.
For example, the four markers on the front and back waist form a quadrilateral that
defines the motion of the pelvis. We'll discuss the relationship between the mark-
ers and the underlying skeleton in more detail in Section
7.4
. It's also common to
add additional redundant markers to help the capture system cope with occlusions,
or to easily distinguish one performer from another. Menache [
323
] gives a detailed
discussion of marker placement and its biomechanical motivation.
7.2
MARKER ACQUISITION AND CLEANUP
The first problemis todetermine the three-dimensional locations of themarkers from
their projections in the cameras' images. Thismay seemdifficult since all themarkers
in a typical motion capture setup look exactly the same. However, since the camera
array has been precisely calibrated, we can compute the epipolar geometry between
any pair of cameras (see Section
6.4.1
), as well as higher-order image relationships
like the trifocal tensor. This means that correct correspondences are generally easy to
obtain, since there are only tens of markers visible in each image and it's unlikely that
incorrect matches will be consistent with the epipolar geometry between multiple
pairs of images.
Therefore, the problem of 3D marker estimation is one of
triangulation
, as dis-
cussed inSection
6.4.2
.Moreprecisely, let's assume that amarker is observedat image
coordinates
in the image from camera
i
, and that we have
M
total images of
the marker.
M
is usually less than the total number of cameras in the system due to
the substantial self-occlusion of the human body (for example, the sternum marker
will not be visible to a camera looking at the back of a performer). Of course,
M
must
be at least two to obtain a triangulation; we deal with the case of missing markers
later in this section.
A good initial guess for the marker location is the point in 3D that minimizes the
sum of squared distances to each of the
M
rays from the camera centers through the
observed image coordinates, as illustrated in Figure
7.6
.
The ray for camera
i
can be expressed as
C
i
+
λ
(
x
i
,
y
i
)
V
i
, where
V
i
is the 3D unit vector
pointing from the camera center
C
i
to the 3D point given by
(
x
i
,
y
i
)
on the image
plane, and
C
i
is computed as
C
i
=
C
i
V
i
)
C
i
−
(
the distance
M
C
i
+
λ
2
min
λ
i
X
−
(
i
V
i
)
(7.1)
i
=
1
is given by
I
3
×
3
−
1
1
M
M
M
1
M
V
i
V
i
C
i
=
−
X
(7.2)
i
=
1
i
=
1
This solution can be refined by using it as an initial point for the nonlinear min-
imization of the sum of squared reprojection errors of
X
onto the image planes, as
described by Andersson and Betsis [
14
].
is not the camera center, but a different point on the ray so that
C
i
5
C
i
V
i
=
0.
