On Invariants of Sets of Points or Line Segments Under Projection

Abstract

We consider the problem of computing invariant functions of the image of a set of points or line segments in $\Re^3$ under projection. Such functions are in principle useful for machine vision systems, because they allow different images of a given geometric object to be described by an invariant `key'. We show that if a geometric object consists of an arbitrary set of points or line segments in $\Re^3$, and the object can undergo a general rotation, then there are no invariants of its image under projection. For certain constrained rotations, however, there are invariants (e.g., rotation about the viewing direction). Thus, we precisely delimit the conditions for the existence or nonexistence of invariants of arbitrary sets of points or line segments under projection.