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where
A
is the
m
n
Jacobian matrix of the
m
-dimensional constraint vector
e
(
x
)
, and
A
†
its pseudo-inverse.
β
is an arbitrary
n
-dimensional vector that is projected into the null space
of the linearized constraints by multiplying it by the matrix
P
×
A
†
A
, also known as the
=
I
−
A
†
e
(
x
)
is the minimum norm solution of Eq.
4.21
. Given
these notations, the
d
x
of Eq.
4.21
can be written as
projector onto
A
's kernel.
d
x
0
=−
=
d
x
0
+
d
x
P
β,
(4.22)
where
β
acts as the new unknown of the problem. This formulation reflects the fact that,
because there are fewer constraints than variables, the projection is not unique. Since
m<n
,
A
†
can be computed as lim
δ
→
0
A
T
(
AA
T
δ
I
)
−
1
, which involves inverting an
m
m
matrix
andcanbedoneevenif
AA
T
itself is non invertible. When
m
n
, which is the case in
practice, performing the inversion in
m
-dimensional rather than
n
-dimensional space helps
reducing the computational cost.
+
×
2. To minimize the criterion of Eq.
4.19
,
β
is taken to be the vector that yields a value of
x
that
solves the equation
M
λ
x
=
b
λ
in the least-squares sense. In other words,
β
is the least-squares
solution of
+
d
x
0
+
P
β)
=
M
λ
(
x
b
λ
,
(4.23)
or, equivalently,
M
λ
P
β
=
b
λ
−
M
λ
(
x
+
d
x
0
).
(4.24)
Solving this equation yields a value of
β
that is used to increment
x
by
d
x
0
+
P
β
. The resulting
coordinate vector can then be used as the new current state, and
A
and
e
(
x
)
can be recomputed.
The process stops when
d
x
becomes small enough. For reconstructions such as those depicted by
Fig.
4.17
where the deformations around the rest shapes are relatively small, the optimization typically
converges to a local minimum in about 10 iterations, which allows for real-time performance. To
account for larger deformations, the same procedure can be used in a frame-to-frame tracking
context, where the initial solution in each frame is taken as the result of the previous one. This
corresponds to the framework proposed in
Shen
et al.
[
2009
], where the shape regularization term
was dropped. As a result, the method of
Shen
et al.
[
2009
] simply involves finding the displacement
within the null-space of the linearized inextensibility constraints that minimizes the reprojection
errors. This, unfortunately, is only possible when correspondences are well-spread over the whole
surface. By contrast, combining the regularization and constraint terms in the above-mentioned way
gives good results even when there are relatively few correspondences.
For the same reasons as those discussed in the context of the
Salzmann and Fua
[
2011
]
approach and depicted by Fig.
4.15
, even better results can be obtained by replacing the length
equality constraints
e
(
x
)
=
0
. As before,
this means that edge lengths can shrink but not extend beyond a certain value. This only involves
a trivial modification of the algorithm above: At each iteration, only currently active constraints are
0
of Eq.
4.19
by inequality constraints of the form
e
(
x
)
≤
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