Image Processing Reference
In-Depth Information
Figure 4.17: Modeling the deformations of a main sail by minimizing an objective function under
constraint. The black circles in the leftmost image are targets that can be automatically detected and were
used to establish correspondences with the reference configuration. The algorithm can estimate the 3D
deformations at a rate of approximately 10 Hz on a standard PC.
Note that, in this formulation, the length constraints are again equality constraints. Consequently,
the balloon forces are no longer necessary and have been dropped. We will relax the constraints into
inequalities below. Solving the problem of Eq. 4.18 is equivalent to solving in the least-squares sense
=
subject to e ( x ) =
M x
b
0 ,
(4.19)
where M =[
; ]
is the matrix obtained by stacking up the lines of M and those of , b =
M
[
, and e ( x ) is the vector of deviations from the desired lengths. Its components are terms
of the form
0
;−
x o ]
.
Assuming the mesh contains N v vertices and N e edges, this constrained optimization problem
involves n =
v k
v j l j,k , one for each edge in
E
3 N v variables and m = N e edge length constraints, with m<n in all practical cases.
It can therefore be solved very effectively using an iterative algorithm inspired by inverse kinematics
approaches to solving underconstrained problems Baerlocher and Boulic [ 2004 ]. At each iteration,
given the current state x , the computation goes through the two following steps:
1. Project the n -dimensional x onto the space of constraints by finding d x such that
e ( x
+ d x ) = 0 .
(4.20)
By doing a first-order Taylor approximation of the previous equation, we can write this pro-
jection as
A e ( x ) + ( I
A A )β ,
A d x
=−
e ( x ) d x
=−
(4.21)
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