NURBSWarps
This section introduces a new warp, the NURBSWarp, which is built upon tensorproduct NonUniform Cubic BSplines.
We show that the NURBSWarp naturally appears when replacing the affine projection by a perspective one in the image formation model of the previous section.
As our experimental results will show, the NURBSWarp performs better in the presence of perspective effects.
All the necessary details about the NURBS model have been presented in section 2.3.3.
Following the same reasoning than for the BSWarp in the previous section, we show that the NURBSWarp corresponds to perspective imaging conditions.
This is illustrated in figure 6.3.
Figure 6.3:
A NURBSWarp can be seen as the result of a threedimensional BSpline surface projected under perspective conditions.

Let
be the perspective projection:

(6.8) 
with
the matrix of intrinsic parameters for the second camera.
is the homogeneous to affine coordinates function, i.e.
where
are the homogeneous coordinates of
(
).
We assume that the image coordinates are chosen such that the origin coincides with the principal point:

(6.9) 
Replacing
by its expression of equation (6.3) in equation (6.8) leads to:

(6.10) 
Defining
,
and
, equation (6.10) is the very definition of a tensorproduct NURBS with control points
and weights (see section 2.3.3).
We denote
this new warp and call it a NURBSWarp:

(6.11) 
Here, the warp parameters, i.e. the control points and the weights, are grouped into a vector
.
Using the NURBSWarp in the setup used for the experiment of figure 6.2 leads to a transfer error consistently smaller than pixels.
The NURBSWarp defined by equation (6.11) can be expressed with homogeneous coordinates.
We note
the NURBSWarp in homogeneous coordinates:

(6.12) 
We observe that in the homogeneous version of equation (6.12), our NURBSWarp does a linear combination of control points in homogeneous coordinates, as opposed to the classical BSWarp of equation (6.2) that does a linear combination of control points in affine coordinates.
This is what makes our NURBSWarp able to model perspective projection, thanks to the division `hidden' in the homogeneous coordinates.
Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010)  Florent Brunet
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