# The ICICS/CS Reading Room

## UBC CS TR-92-27 Summary

- No on-line copy of this technical report is available.

- 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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