However, there are invariant constraints that are almost as useful as true invariants for the task of eliminating the unknown viewpoint. A current research project involves developing and applying these 3D invariant constraints to recognizing objects. This involves invariants of various types of features that can be found in objects, including point sets, lines, conics and general curves. The invariant constraints can be used to dramatically reduce the search space for the correct viewpoint and thus make recognition feasible. An example is shown in Figure 1.
Objects can undergo other types of geometric changes, in addition to viewpoint changes.
For instance, one can recognize apples even though different apples have slightly different shapes. We have applied invariants of such small deformations to characterize classes of objects such as apples and bananas, and to differentiate these classes from each other. Another example of non-viewpoint changes that we are actively studying is identifying articulated objects in range images. Here we can have objects such as tanks that can change not only their positions but their shapes, by rotating the turret or raising the gun barrel. Invariance to these changes again reduces a very big search space to a more manageable one.