segments that are later filtered out. This could cause false segments (or ghosts) in a complicated scene. Nevertheless, the overall hierarchical organization is probably fairly stable, as there are only a limited number of ways to compute the geometric tokens using two-variable functions. Thus the number of levels in the hierarchy is likely to remain constant.
A major weakness of the current simulation is that only feedforward connections are used. This means that the system is essentially open loop and measurement errors propagate upwards through the levels. Feedback connections that enforce consistency between the networks should greatly improve the accuracy of the measurements.
The accuracy problem was exacerbated by the fact that the distributed encoding scheme was not used. Thus, the grain size of the parameter network was kept judiciously small. We expect that when the distributed scheme is implemented, this restriction could be lifted.
Symmetry played an unanticipated important role. For the simple shape used in the experiment, the projection of its parts into the orientation-cue subspace results in a nearly symmetric representation in the subspace. Consequently, many incorrect rotations of the object appear almost as good as the correct one. Although this caused no problems for our example, one can imagine that more complicated cases could produce false results due to false symmetries in the orientation-cue subspace. This issue will have to be resolved by more experimentation.
Finally, although the network is specialized to polyhedral constraints, there is no reason why it cannot be generalized to curved surfaces as well. The idea is essentially that of matching coordinate frames that can be associated easily and reliably with the objects and scene. If a way to do this is found for curved objects, then the transformational matching can be applied. We are currently working to extend our results in this direction.
This work was developed over several years during which time several people contributed to the development of the ideas expressed herein. I am especially grateful to Lydia Hrechanyk, Hiromi Tanaka, Shmuel Tomer, John Sullins, and Lucy Lin for various early implementations. Cesar Quiroz implemented the simulator primitives. Thanks also go to Jerry Feldman, Chris Brown, and members of the Rochester vision group who provided helpful critiques. Peggy Meeker handled all aspects of the preparation of this chapter.
This research was supported in part by the National Science Foundation under Grant DCR-8405720 and the National Institutes of Health under Public Health Service Grant 1R01NS22407-01.