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Section 12.1 Registering Rigid Objects 371

Transformation Frame-bearing groups
Three points, or
Rigid (Euclidean) one line and one point off the line, or
two intersecting lines
Three points, or
Rigid and scale one line and one point off the line, or
two intersecting lines and a point off their plane.
Affine Four points, no two co-planar

TABLE 12.2: Some frame-bearing groups for estimating transformations from 3D to 3D.
Assume we have one such group in the source, another in the target, and a correspondence
between the items in the group; then, we can estimate the transformation uniquely.

FIGURE 12.1: On the left, frames from a video taken by an aircraft overflying an airport.
These frames are rectified to one another to form a mosaic on the right, which reveals (a)
the overall structure of what was seen and (b) the flight path of the aircraft. This figure
was originally published as Figure 1 of “Video Indexing Based on Mosaic Representations,”
by M. Irani and P. Anandan, Proc. IEEE, v86 n5, 1998, c IEEE, 1998.

One application is building larger images. There are several other important appli-
cations. For example, imagine we have image frames taken by, say, an orthographic
camera attached to an aircraft; then, if we register the frames to one another, we
see not only the pictures taken by the aircraft in a form that exposes all that it
saw, but also a representation of the flight path, and so of what it could have seen
(Figure 12.1). As another example, imagine we have a fixed camera, that collects
video. By registering the frames with one another, we can make estimates of (a)
the moving objects and (b) the background, and expose this information to viewers
in a novel way (Figure 12.2). As yet another example, we could build either a
cylindrical panorama, a set of pixel samples that mimic the image produced by a
cylindrical camera, or even a spherical panorama, a set of pixel samples that mimic
the image produced by a spherical camera. One feature of these panoramas is that
it is easy to query them for a set of pixels that looks like a perspective image. In
particular, it is easy to use these panoramas to imitate what one would see if a
perspective camera were to rotate about its focal point.
Building mosaics is a useful application of registration. In the simplest case,
we wish to register two images to one another. We do so by finding tokens, deciding
which ones should match, and then choosing the transformation that minimizes