Motion Capture - Computer Vision for Visual Effects - page 268

N and P , then the Jacobian is N

If r ∈ R θ ∈ R × P ; in this section, we assume that

P . 10 This situation occurs in some of the motion editing applications discussed
in Section 7.5 , and is frequently needed for animation. When we have a large number
of motion capture markers, then N

N <

> P and the optimization-based methods in the

next section are more appropriate.
If we know r

( ) θ ( )

, we therefore expect an
infinite number of solutions, since the problem is underconstrained. The general
form of this family of solutions is:

t and are solving Equation ( 7.14 ) for t

d dt = d r ( t ) d θ

0
dt

J † J †

( t ) + ( I P × P − ( t ) J ( t )) (7.17)

dt

where J † ( t ) is the P × N pseudoinverse given by

J † ) ( ) ) − 1 ( t ) = J ( t J ( t ) J ( t (7.18)

θ 0
dt

and d . That is, the second com-

ponent of Equation ( 7.17 ) is the projection of this trajectory onto the non-empty null
space of J

is the derivative of some arbitrary trajectory θ ( t ) 0

( t ) , which expresses the remaining P − N degrees of freedom in the solution.

We can see that choosing θ 0 ( t ) = 0 gives one solution as

dt d r ( t )

d

J † = ( t ) (7.19)

dt

which is just the usual least-squares solution in which θ ( t ) has minimum norm —

that is, it minimizes the cost function

2

d r ( t ) d θ ( t )

C ( θ ( t )) = − J ( t ) (7.20)

dt dt

To reconstruct the actual values of θ ( t ) from the recovered derivatives, we just

compute

t

d dt (τ )

θ ( t ) = θ ( 0 ) + d τ (7.21)

0

where is the known starting position of the joints, which can often be estimated

by initializing the motion capture in a standard pose (e.g., a “T” pose with arms
outstretched).
A related approach that avoids problems with singularities of the Jacobian (that is,
when the trajectory passes through or near a region where the Jacobian is not rank
N ) is to replace the pseudoinverse in Equation ( 7.18 ) with a damping factor

θ ( 0 )

J ∗ ( ) ( ) + λ

2 I N × N

) − 1

t ) = J ( t J ( t ) J ( t (7.22)

which corresponds to minimizing the damped or regularized cost function

2

2 2

d r ( t ) d θ ( t ) d θ ( t )

C reg ( θ ( t )) = − J ( t ) + λ (7.23)

dt dt dt

Yamane and Nakamura [ 559 ] used this approach to compute joint angles for a
kinematic model in which some end effectors were fixed (“pinned”) and another

10 Recall that we also defined a Jacobian in the context of matchmoving (Section 6.5.3.2 ), although
the Jacobian in that case had N P .