Generalizing invariants for 3-D to 2-D matching

Invariant representations of images have proven useful in performing a variety of vision tasks. However, there are no general invariant functions when one considers a single 2-D image of a 3-D scene. One possible response to the lack of true invariants is to attempt to generalize the notion of an invariant by finding the most economical characterization possible of the set of all 2-D images that a group of 3-D features may produce. A true invariant exists when, for each model, we can represent all its images at a single point in some representational space. When this is not possible, it is still very useful to find the simplest and lowest-dimensional representation of each model's images.We show how this can be done for a variety of different types of model features, and types of projection models. Of particular interest, we show how to represent the set of images that a group of 3-D points produces by two lines (1-D subspaces), one in each of two orthogonal, high-dimensional spaces, where a single image group corresponds to one point in each space. We demonstrate the value of our results by applying them to a variety of vision problems. In particular, we describe a space-efficient indexing system that performs 3-D to 2-D matching by table lookup. We also show how to find a least squares solution to the structure-from-motion problem using point features that have associated orientations, such as corners. We show how to determine when a restricted class of models may give rise to an invariant function, and construct a model-based invariant for pairs of planar algebraic curves that are not coplanar.