# NURBS-Warps

This section introduces a new warp, the NURBS-Warp, which is built upon tensor-product Non-Uniform Cubic B-Splines. We show that the NURBS-Warp 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 NURBS-Warp performs better in the presence of perspective effects. All the necessary details about the NURBS model have been presented in section 2.3.3.

#### Perspective interpretation.

Following the same reasoning than for the BS-Warp in the previous section, we show that the NURBS-Warp corresponds to perspective imaging conditions. This is illustrated in figure 6.3.

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 tensor-product NURBS with control points  and weights  (see section 2.3.3). We denote  this new warp and call it a NURBS-Warp:

 (6.11)

Here, the warp parameters, i.e. the control points and the weights, are grouped into a vector  .

Using the NURBS-Warp in the setup used for the experiment of figure 6.2 leads to a transfer error consistently smaller than  pixels.

#### Homogeneous NURBS-Warp.

The NURBS-Warp defined by equation (6.11) can be expressed with homogeneous coordinates. We note  the NURBS-Warp in homogeneous coordinates:

 (6.12)

We observe that in the homogeneous version of equation (6.12), our NURBS-Warp does a linear combination of control points in homogeneous coordinates, as opposed to the classical BS-Warp of equation (6.2) that does a linear combination of control points in affine coordinates. This is what makes our NURBS-Warp 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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