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Section 12.4 Notes 389

cal models. Matching algorithms naturally yield tracking algorithms, too (Zhong et
al. 2000) There is a large range of active appearance models. Active shape models
are a variant that encodes geometry, but not intensity, and active contour models,
reviewed in Blake (1999), encode boundaries; a particularly important version be-
came famous as a “snake” (Kass et al. 1988). We have chosen a model to expound
for didactic, rather than historical, reasons. Good places to start in this literature
are Cootes and Taylor (1992); Taylor et al. (1998); Cootes et al. (2001); and Cootes
et al. (1994).

Medical Applications
This is not a topic on which we speak with any authority. Valuable surveys include:
Ayache (1995); Duncan and Ayache (2000); Gerig et al. (1994); Pluim et al. (2003);
Maintz and Viergever (1998); and Shams et al. (2010). The three main topics
appear to be: segmentation, which is used to identify regions of (often 3D) images
that correspond to particular organs; registration, which is used to construct
correspondences between images of different modalities and between images and
patients; and analysis of both morphology—how big is this? has it grown?—and
function. McInerney and Terzopolous (1996) survey the use of deformable models.
There are surveys of registration methods and issues in Lavallee (1996) and in
Maintz and Viergever (1998), and a comparison between registration output and
“ground truth” in West et al. (1997).

PROBLEMS
12.1. Show that one line and one point can be used as a frame-bearing group for
2D rigid transformations (i.e., rotations and translations in the plane). The
easiest way to do this is to show that (a) the translation is determined by
placing the source point over the target point; and (b) the rotation can then
be determined by rotating to place the source line over the target line.
(a) Does every such pair yield a rigid transformation? (Hint: think about the
distance from the point to the line.)
(b) Can the point lie on the line and still yield a unique rigid transformation?
(Hint: does the line have symmetries?)
12.2. Use the methods of the previous exercise to establish that all the frame-bearing
groups of Table 12.1 are, in fact, frame-bearing groups.
12.3. Show that three points can be used as a frame-bearing group for 3D rigid trans-
formations (i.e., rotations and translations in the plane). Start by showing the
translation is determined by placing a source point over a corresponding target
point. Now the rotation follows in two steps: rotate to place a second source
point over the corresponding target point, then rotate about the resulting axis
to place the third source point over the third target point.
(a) Does every such triple yield a rigid transformation? (Hint: think about
the distances between the points.)
12.4. Use the methods of the previous exercise to establish that all the frame bearing-
groups of Table 12.2 are, in fact, frame-bearing groups.
12.5. Check that a weak-perspective camera is equivalent to an orthographic camera,
calibrated up to unknown scale.
12.6. Assume that we are viewing objects in an orthographic camera, calibrated up