Matching 3-D Models to 2-D Images

• Authors: David W. Jacobs
• Venue: International Journal of Computer Vision
• Year: 1997

We consider the problem of analytically characterizing the set of all2-D images that a group of 3-D features may produce, and demonstratethat this is a useful thing to do. Our results apply for simple pointfeatures and point features with associated orientation vectors whenwe model projection as a 3-D to 2-D affine transformation. We showhow to represent the set of images that a group of 3-D points produceswith two lines (1-D subspaces), one in each of two orthogonal,high-dimensional spaces, where a single image group corresponds to onepoint in each space. The images of groups of oriented point featurescan be represented by a 2-D hyperbolic surface in a singlehigh-dimensional space. The problem of matching an image to models isessentially reduced to the problem of matching a point to simplegeometric structures. Moreover, we show that these are the simplestand lowest dimensional representations possible for these cases.We demonstrate the value of this way of approaching matching byapplying our results to a variety of vision problems. In particular,we use this result to build a space-efficient indexing system thatperforms 3-D to 2-D matching by table lookup. This system isanalytically built and accessed, accounts for the effects of sensingerror, and is tested on real images. We also derive new resultsconcerning the existence of invariants and non-accidental propertiesin this domain. Finally, we show that oriented points presentunexpected difficulties: indexing requires fundamentally more spacewith oriented than with simple points, we must use more images in amotion sequence to determine the affine structure of oriented points,and the linear combinations result does not hold for oriented points.