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
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,
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 ,
or, equivalently,
M λ P β =
b λ
M λ ( x
+ d x 0 ).
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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