Introduction

In this chapter, we bring several contributions. We first demonstrate in section 6.2 that the warps based on tensor-product B-splines (hereinafter abbreviated BS-Warp) corresponds to affine imaging condition, in the sense that it models the affine projection of some 3D surface. We then propose our most important contribution in section 6.3: a novel parametric warp we call NURBS-Warp, that extends the classical BS-Warp to perspective projection. This warp has a simple analytical form: it is obtained as the two-way tensor-product of bivalued Non-Uniform Rational B-Splines (NURBS). Finally, we give in section 6.4 algorithms for the feature-based estimation of our NURBS-Warp. More precisely, we consider that a set $\{\mathbf{q}_k \leftrightarrow \mathbf{q}'_k\}_{k=1,\ldots,r}$ of point correspondences between the two images is known, and show how the parameters that minimize the classical transfer error can be found, by solving:

$$\min_{\mathbf{x}} \sum_{k=1}^r d^2 \left ( \mathbf{q}'_k, \mathcal {W}(\mathbf{q}_k ; \mathbf{x}) \right ),$$ (6.1)

where  $\mathcal {W}$ represents the warp and  $d^2(\mathbf{a},\mathbf{b})$ is the squared euclidean distance between the points  $\mathbf{a}$ and  $\mathbf{b}$. We finally report experimental results in section 6.5 and conclude this section.