Affine Interpretation of the BSWarps
In this chapter, we denote
the BSWarp.
As a reminder of section 2.3.2, a BSWarp is defined as:

(6.2) 
where the functions
are the BSpline basis functions and the
are the control points (grouped in the parameter vector
).
The scalars and are the number of control points along the and directions respectively.
Note that we consider as coincident the knot sequences used to define the BSpline basis functions.
If we consider that the observed surface is modelled by a threedimensional tensorproduct BSpline, the BSWarp corresponds to the transformation between the two images under affine imaging conditions (see figure 6.1 for an illustration).
Figure 6.1:
A BSWarp can be seen as the result of a threedimensional BSpline surface projected under affine conditions.

Let
be the 2D3D map between the first image and the threedimensional surface:

(6.3) 
with
the 3D control points of the surface.
Let
be the affine projection of the surface into the second image:

(6.4) 
with
the matrix which models the affine projection, assuming that the 3D surface is expressed within the coordinate frame of the second camera:

(6.5) 
Given these notations, the warped point
can be written
which, after expansion, gives:

(6.6) 
Equation (6.6) matches the definition of equation (6.2) of a BSWarp.
As we just demonstrated, BSWarps are obtained under affine imaging conditions.
However, this does not prove that they are not suited for perspective imaging conditions.
In this paragraph, we experimentally illustrate that BSWarps are indeed not suited for perspective imaging conditions.
This comes from the fact that the division appearing in a perspective projection is not present in the BSWarp model.
The bad behavior of the BSWarp in the presence of perspective effects is illustrated in figure 6.2.
In this experiment, we simulate a set of point correspondences by transforming a regular grid with an homography parametrized by a scalar that controls the amount of perspective effect.
The
matrix of this homography,
, is given by:

(6.7) 
where indicates equality up to scale.
The larger , the more important the perspective effect.
Figure 6.2 c clearly shows that the transformation can be correctly modelled by a BSWarp only when , i.e. when the perspective effect is barely existant.
Figure 6.2 d shows that the number of control points of a BSWarp must be significantly large in order to correctly model a perspective effect.
Figure 6.2:
Bad behavior of the BSWarp in the presence of perspective effects. (a) Data points on a regular grid (first image). (b) Transformed points simulating a perspective effect with an homography (second image). (c) Influence of the perspective effect on a BSWarp with 16 control points. (d) The perspective effect (
) can be modeled with a BSWarp but it requires a large amount of control points.

Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010)  Florent Brunet
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