Image Processing Reference

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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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