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UBC CS TR-93-12 Summary

Orientation-Based Representations of Shape and Attitude Determination, April 1993 Ying Li, 162 pages

The three rotational degrees of freedom between the coordinate system of a sensed object and that of a viewer define the attitude of the object. Orientation-based representations record 3-D surface properties as a function of position on the unit sphere. All orientation-based representations considered share a desirable property: the representation of a rotated object is equal to the rotated representation of the object before rotation. This makes the orientation-based representations well-suited to the task of attitude determination.

The mathematical background for orientation-based representations of shape is presented in a consistent framework. Among the orientation-based representations considered, the support function is one-to-one for convex bodies, the curvature functions are one-to-one for convex bodies up to a translation and the radial function is one-to-one for starshaped sets.

Using combinations of the support function and the curvature functions for convex bodies, the problem of attitude determination is transformed into an optimization problem. Previous mathematical results on the radial function for convex objects are extended to starshaped objects and the problem of attitude determination by the radial function also is transformed into an optimization problem. Solutions to the optimization problems exist and can be effectively computed using standard numerical methods.

A proof-of-concept system has been implemented and experiments conducted both on synthesized data and on real objects using surface data derived from photometric stereo. Experimental results verify the theoretical solutions.

Novel contributions of the thesis include: the representation of smooth convex objects by the support function and curvature functions; the definition of a new orientation-based representation for starshaped sets using the 3-D radial function; and solutions to the 3-D attitude determination problem using the aforementioned representations. In particular, the scope of orientation-based representations has been extended, both in theory and in practice, from convexity to starshapedness.

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