Technical Reports

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UBC CS TR-92-27 Summary

Starshaped Sets, The Radial Function and 3-D Attitude Determination, October 1992 Ying Li and Robert J. Woodham, 18 pages

Attitude characterizes the three rotational degrees of freedom between the coordinate system of a known object and that of a viewer. Orientation-based representations record 3-D surface properties as a function of position on the unit sphere. The domain of the representation is the entire sphere. Imaging from a single viewpoint typically determines a hemisphere of the representation. Matching the visible region to the full spherical model for a known object estimates 3-D attitude.

The radial function is used to define a new orientation-based representation of shape. The radial function is well-defined for a class of sets called {\em starshaped} in mathematics. A starshaped set contains at least one interior point from which all boundary points are visible. The radial function records the distance from the origin of the coordinate system to each boundary point. The novel contribution of this paper is to extend previous mathematical results on the matching problem for convex objects to starshaped objects. These results then allow one to transform the attitude determination problem for starshaped sets into an optimization problem for which standard numerical solutions exist. Numerical optimization determines the 3-D rotation that brings a sensed surface into correspondence with a known model.

The required surface data can be obtained, for example, from laser range finding or from shape-from-shading. A proof-of-concept system has been implemented and experiments conducted on real objects using surface data derived from photometric stereo.

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