In this chapter, we consider a pinhole camera viewing a plane in the world. In these circumstances, the camera equations simplify to reflect the fact that there is a one-to-one mapping between points on this plane and points in the image.
Mappings between the plane and the image can be described using a family of 2D geometric transformations. In this chapter, we characterize these transformations and show how to estimate their parameters from data. We revisit the three geometric problems from Chapter 14 for the special case of a planar scene.
To motivate the ideas of this chapter, consider an augmented reality application in which we wish to superimpose 3D content onto a planar marker (Figure 15.1). To do this, we must establish the rotation and translation of the plane relative to the camera. We will do this in two stages. First, we will estimate the 2D transformation between points on the marker and points in the image. Second, we will extract the rotation and translation from the transformation parameters.
Two-dimensional transformation models
In this section, we consider a family of 2D transformations, starting with the simplest and working toward the most general. We will motivate each by considering viewing a planar scene under different viewing conditions.
Euclidean transformation model
Consider a calibrated camera viewing a fronto-parallel plane at known distance, D (i.e., a plane whose normal corresponds to the ω-axis of the camera).