MATCHING 3-D MODELS TO 2-D IMAGES

We consider the problem of analytically characterizing the set of all 2-D images that a group of 3-D features may produce, and demonstrate t hat this is a useful thing to do. Our results apply for simple point f eatures and point features with associated orientation vectors when we model projection as a 3-D to 2-D affine transformation. We show how t o represent the set of images that a group of 3-D points produces with two lines (1-D subspaces), one in each of two orthogonal, high-dimens ional spaces, where a single image group corresponds to one point in e ach space. The images of groups of oriented point features can be repr esented by a 2-D hyperbolic surface in a single high-dimensional space . The problem of matching an image to models is essentially reduced to the problem of matching a point to simple geometric structures. Moreo ver, we show that these are the simplest and lowest dimensional repres entations possible for these cases. We demonstrate the value of this w ay of approaching matching by applying our results to a variety of vis ion problems. In particular, we use this result to build a space-effic ient indexing system that performs 3-D to 2-D matching by table lookup . This system is analytically built and accessed, accounts for the eff ects of sensing error, and is tested on real images. We also derive ne w results concerning the existence of invariants and non-accidental pr operties in this domain. Finally, we show that oriented points present unexpected difficulties: indexing requires fundamentally more space w ith oriented than with simple points, we must use more images in a mot ion sequence to determine the affine structure of oriented points, and the linear combinations result does not hold for oriented points.