Graphics Reference
In-Depth Information
(a)
(b)
Figure 7.7.
(a) Interpolation of known marker positions (white dots) can help estimate missing
marker positions (gray dots). (b) Extrapolation of partial trajectories can help determine when
broken trajectories should be merged.
Equation (
7.3
) into known and unknown parts:
X
k
X
u
U
k
U
u
µ
k
µ
=
β
+
(7.4)
u
Solving the top half of Equation (
7.4
) for
β
using the known information gives
U
k
U
k
)
−
1
U
k
(
β
=
(
X
k
−
µ
k
)
(7.5)
and plugging this back into the bottom half of Equation (
7.4
) gives the unknown
marker locations as
U
k
U
k
)
−
1
U
k
(
X
u
=
U
u
(
X
k
−
µ
k
)
+
µ
u
(7.6)
Since a global linear model like Equation (
7.3
) will probably do a poor job of
representing the underlying nonlinear relationships in motion capture data, we can
replace Equation (
7.6
) with a weighted combination of linear models, each learned
locally on a small segment of simple behavior. Since missing markers are likely only
weakly correlated with faraway known markers (e.g., an ankle marker will not help
much to estimate a missing wrist marker), better results may be obtained by learning
a PCA model only over the missing marker and the markers nearby on the body.
However, this approach works best when the markers are spaced very densely on the
body (e.g., as described in the experiment by Park and Hodgins [
360
]). Lou and Chai
[
302
] proposed to incorporate time lags of the vectors into the PCA, as well as a robust
estimator to reduce the influence of outliers.
If a marker trajectory contains high-frequency jitter, it can be smoothed, but we
must be careful not to remove the nuanced motions that motivate motion capture
in the first place; these details are inherently high-frequency. Sharp motions like
punches or kicks also contain important high-frequency components whose removal
would make the captured motion useless. We should also be aware that a too-low-
dimensional PCA basis may abstract away subtle performance details that project
onto the higher-order modes.
