Graphics Reference
In-Depth Information
few markers attached to normal clothing could drive an animated character in real
time. The low-dimensional input is used to quickly search a database of high-quality
motion capture subsequences that are seamlessly strung together and played back
in real time.
While we only discussed motion capture for full bodies and faces, biomechanical
engineers often use markered systems to study hands (e.g., [
64
]) for understand-
ing dexterity and grasping. These can be augmented with force-feedback sensors to
study how fingers interact with objects they contact [
257
]. Park and Hodgins [
360
]
used about 350 markers finely spaced over a performer's body to collect accurate
data about the motion of skin, muscle, and flesh (e.g., bulging, stretching, jiggling).
Feature films such as the
Lord of the Rings
trilogy have even incorporatedmotion cap-
ture data from horses; of course, this requires an entirely different kinematic model.
Rosenhahn et al. [
398
] studiedmarkerless motion capture of athletes interacting with
machines (e.g., bicycles and snowboards), whichmakes the kinematic skeletonmore
complicated (i.e., since the legs are now connected by the machine into a closed
chain).
7.10
HOMEWORK PROBLEMS
7.1
a)
Show that if
P
,
C
i
, and
V
i
are fixed, then the inner termof Equation (
7.1
)
for one camera
C
i
+
λ
2
P
−
(
i
V
i
)
(7.39)
V
i
P
. Show that the minimum distance of the
is minimized by
λ
=
i
point to the ray is thus
2
C
i
P
C
i
C
i
−
(
P
P
V
i
P
2
−
+
)
(7.40)
Recall that by definition
V
i
is unit length and
V
i
C
i
=
0.
b) Verify Equation (
7.2
) by differentiating Equation (
7.1
) with respect to
P
and using the results of (a).
7.2 Compute the number of degrees of freedomfor the human kinematicmodel
illustrated in Figure
7.8
(including the root).
7.3 Derive the forward and inverse conversion formulas between a rotation
represented as a unit quaternion
q
3
.
7.4 Compute the position of the wrist with respect to the shoulder's coor-
dinate system using a forward kinematics model for the arm illustrated
in Figure
7.9
. Assume that the upper arm is 28 cm long and the fore-
arm is 25 cm long, and that the quaternions specifying the shoulder and
elbow rotations are
q
s
4
and as an axis-angle vector
r
∈ R
∈ R
=[
0.1089, 0.3969,
−
0.6842,
−
0.6021
]
and
q
e
=
respectively.
7.5 Here, we prove that if a three-dimensional rigid motion is specified by a
rotation matrix
R
(corresponding to a rotation
[
0.3410, 0.2708, 0.4564,
−
0.7760
]
ψ
around a unit vector
v
)
3
needed in Equation (
7.10
)
ρ
∈ R
and a translation vector
t
, then the vector
