Graphics Reference
In-Depth Information
individual nuances of joint motion (especially around wrists and feet, or in cases
section lead into the more general algorithms for 3D data acquisition discussed in
Chapter
8
.
7.7.1
The Dynamical System Model
Most markerless techniques use a
dynamical system
model, in which we want to
estimate the underlying
state
of a system based on a sequence of
observations
. For
motion capture, we define the state
as a random variable specifying the underly-
ing pose of the human (for example, a parameterization of the root position and joint
angles of a kinematic model). The state can't be directly observed, but instead must
be inferred based on a series of observations up to the current time,
θ
(
t
)
{
(
)
...
(
)
}
.In
motion capture, these observations are features extracted from images from a set of
synchronized cameras surrounding a performer.
The relationships between successive states and between the states and obser-
vations are described by probabilistic models, respectively defined by the
state
transition probability
r
1
,
,
r
t
p
(
θ
(
t
)
|
θ
(
1
)
,
...
,
θ
(
t
−
1
))
(7.33)
and the
observation likelihood
p
(
r
(
1
)
,
...
,
r
(
t
)
|
θ
(
1
)
,
...
,
θ
(
t
))
(7.34)
These are usually simplified using the
Markov property
and the assumption that
the current observation only depends on the current state to
p
(
θ
(
t
)
|
θ
(
1
)
,
...
,
θ
(
t
−
1
))
=
p
(
θ
(
t
)
|
θ
(
t
−
1
))
t
(7.35)
p
(
r
(
1
)
,
...
,
r
(
t
)
|
θ
(
1
)
,
...
,
θ
(
t
))
=
p
(
r
(
i
)
|
θ
(
i
))
i
=
1
We therefore take a Bayesian approach, searching for the maximum (or multiple
modes) of a
posterior
probability distribution
p
(
θ
(
t
)
|
r
(
1
)
,
...
,
r
(
t
))
∝
p
(
r
(
t
)
|
θ
(
t
))
p
(
θ
(
t
)
|
r
(
1
)
,
...
,
r
(
t
−
1
))
∝
p
(
r
(
t
)
|
θ
(
t
))
p
(
θ
(
t
)
|
θ
(
t
−
1
))
p
(
θ
(
t
−
1
)
|
(7.36)
p
(
θ
(
t
−
1
)
|
r
(
1
)
,
...
,
r
(
t
−
1
))
d
θ
(
t
−
1
)
Therefore, we can recursively update the posterior density based on its previous
estimate and our models for the state transition and observation likelihoods. Mark-
erless motion capture approaches differ in how the observation
r
is extracted from
the current image and related to the state, how the various probability densities are
represented, and how the posterior is used to obtain the current state estimate.
(
t
)
15
To be fair, many algorithms in this section aren't designed for highly accurate motion capture but
for robust human detection, pose estimation, and tracking in video sequences, where the results
are sufficient.
