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6.1 2D and 3D feature-based alignment 275

y projective
similarity
translation

Euclidean affine
x

Figure 6.2 Basic set of 2D planar transformations

Once we have extracted features from images, the next stage in many vision algorithms is
to match these features across different images (Section 4.1.3). An important component of
this matching is to verify whether the set of matching features is geometrically consistent,
e.g., whether the feature displacements can be described by a simple 2D or 3D geometric
transformation. The computed motions can then be used in other applications such as image
stitching (Chapter 9) or augmented reality (Section 6.2.3).
In this chapter, we look at the topic of geometric image registration, i.e., the computation
of 2D and 3D transformations that map features in one image to another (Section 6.1). One
special case of this problem is pose estimation, which is determining a camera’s position
relative to a known 3D object or scene (Section 6.2). Another case is the computation of a
camera’s intrinsic calibration, which consists of the internal parameters such as focal length
and radial distortion (Section 6.3). In Chapter 7, we look at the related problems of how
to estimate 3D point structure from 2D matches (triangulation) and how to simultaneously
estimate 3D geometry and camera motion (structure from motion).

6.1 2D and 3D feature-based alignment

Feature-based alignment is the problem of estimating the motion between two or more sets
of matched 2D or 3D points. In this section, we restrict ourselves to global parametric trans-
formations, such as those described in Section 2.1.2 and shown in Table 2.1 and Figure 6.2,
or higher order transformation for curved surfaces (Shashua and Toelg 1997; Can, Stewart,
Roysam et al. 2002). Applications to non-rigid or elastic deformations (Bookstein 1989;
Szeliski and Lavall´ ee 1996; Torresani, Hertzmann, and Bregler 2008) are examined in Sec-
tions 8.3 and 12.6.4.

6.1.1 2D alignment using least squares

1

Given a set of matched feature points {(x i , x )} and a planar parametric transformation of
i
the form
x = f(x; p), (6.1)

1 For examples of non-planar parametric models, such as quadrics, see the work of Shashua and Toelg (1997);
Shashua and Wexler (2001).