Subsections
Parameter Estimation
In this section, we show how the BSWarp and the NURBSWarp parameters can be estimated.
Here, we consider the featurebased approach for estimating the warp parameters.
It has the advantage of being one of the most simple approach to estimate the parameters.
We can thus concentrate on the main topic of this chapter, i.e. the representational power of the warps.
As it was explained in section 5.1, this featurebased approach to image registration amounts to solve the following minimization problem:

(6.13) 
The main difficulty here is that problem (6.13) is not linear when using the NURBSwarps.
The dependency of the BSWarp to its parameters is linear.
As a consequence, the optimization problem (6.13) for a BSWarp simply reduces to an ordinary linear leastsquares minimization problem.
The details for solving such a problem have already been detailed in, for instance, section 5.3.
Contrarily to the BSWarp, the dependency of the NURBSWarp to its parameters is not linear.
Problem (6.13) thus leads one to a nonlinear leastsquares optimization problem.
Note that this remark would still be true for other estimation techniques such as the direct approach to image registration.
It is a wellknown fact that this kind of problem can be efficiently solved using an iterative algorithm such as LevenbergMarquardt.
Such an algorithm needs an initial solution (see section 2.2.2.7).
We propose three approaches to compute an initial set of parameters (which are further detailed in the sequel of this section):
 First approach: Act as if the images were taken under affine imaging conditions.
 Second approach: Act as if the warp relating the two images was an homography.
 Third approach: Use an algebraic approximation to the transfer error function.
Since all of these three approaches are relatively cheap to compute, it is possible to test the three of them and choose as initial parameters those that give the smallest transfer error.
An initial solution for the NURBSWarp estimation problem can be computed by setting all the weights to 1.
By doing so, the equation defining a NURBSwarp reduces to the expression of a simple BSWarp.
An initial set of parameters can thus be computed using ordinary leastsquares minimization.
Since the BSWarp corresponds to affine imaging conditions, this approach is expected to give good results when the effects of perspective are limited.
Even if the homographic warp is not suited to deformable environments, it can be a good approximation (particularly if the surface bending is not important).
We denote
the homographic warp.
It is defined by:

(6.14) 
with
.
Minimizing the transfer error of equation (6.13) with the homographic warp
can be achieved iteratively using, for instance, the LevenbergMarquardt algorithm (and by taking the identity warp or the algebraic solution as an initialization).
A complete review of homographic warp estimation can be found in (94).
The third and last approach we propose to initialize the optimization process consists in minimizing an algebraic approximation to the transfer error:

(6.15) 
with
an algebraic distance between the points^{6.1}
and
.
The operator
removes the last element of a 3vector.
are the scaled homogeneous coordinates of
(i.e.
).
If we make the following variable change in equation (6.12) of the homogeneous NURBSWarp:
and
then it is straightforward to see that the algebraic distance is the squared euclidean norm of an expression linear with respect to the parameters , and .
Replacing the algebraic distance by its expression in the initial optimization problem of equation (6.15) leads to homogeneous linear leastsquares.
This problem can be solved using the singular value decomposition presented in section 2.2.2.10.
Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010)  Florent Brunet
Webpage generated on July 2011
PDF version (11 Mo)