Geometric modeling for computer vision

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Abstract
Computer vision faces the challenge of exploiting the CAD models available for many manufactured objects. Recently, a variety of new algorithms have been developed that rely on symbolic object representations for recognizing curved three-dimensional objects from images. This paper presents a modeling system for constructing these representations from parametric and implicit algebraic surface specifications. A combination of homotopy continuation and curve tracing is used to compute set operations. Specifically, an algorithm for tracing curves defined implicitly in IRn+1 by n polynomial equations in n + 1 variables is presented that correctly characterizes the topology of the intersection curves and respects singular points. The curve tracing algorithm is also used to render line drawings for which parametric representations of occluding contours are unavailable. Hidden lines are easily removed by explicitly constructing an image structure graph of the smooth curve branches and singular points; it is then sufficient to trace a ray at a single point on each curve branch. A preliminary parallel implementation, distributed over a network of SPARC stations, appears promising. For objects modeled by this system algorithms for constructing their aspect graph, computing their stable poses, and recognizing them in different types of images have been developed.

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