Invariants and Their Applications
Invariants are intrinsic to the idea of recognition in both 2-D and 3-D. The basic idea is to identify some parameter or parameters that do not vary between different instances of the same object. Unfortunately, perspective projection makes the issue far harder in the general 3-D case. This chapter explores the problem and demonstrates a number of useful techniques.
• how a ratio of distances between features along the same straight line can act as a convenient invariant under weak perspective projection.
• how a ratio of ratios (or “cross ratio”) can act as a convenient invariant under full perspective projection.
• how the cross ratio type of invariant can rather cunningly be generalized to cover many wider possibilities.
• how the cross ratio type of invariant seems largely unable to provide invariance outside any given plane.
Although this chapter considers only one aspect of 3-D vision, it is extremely useful both in helping to cue into complex images (see particularly the egomotion example of Fig. 19.4 and the facial analysis example of Section 24.12) and in taking shortcuts around the tedious analysis of 3-D geometry (see, for example, Sections 20.8 and 20.9).