Groups for Grouping: A Strategy for the Exploitation of Geometrical Constraints

Special shape configurations, such as the two halfs of a mirror symmetric pattern, are related by specialized projective transformations when observed with a perspective camera. These projectivities belong to subgroups of the plane projective transformations. Thus, simpler than general projective invariants exist to describe and recognize such shapes. Moreover, they can be used to probe for the specific nature of the deformations and the assumptions they entail. This paper tries to engage in a systematic treatment of the different subgroups of the projectivities, by starting from the image structures they keep fixed. In particular, a classification of subgroups according to the number of fixed points and lines is given and invariants are derived for each of them. Their usefulness is illustrated by a practical example.