# The ICICS/CS Reading Room

## UBC CS TR-93-12 Summary

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

- 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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