where YL and YU are lower bound vector and upper bound vector; [Y.su.M] and YR are midpoint vector and radius vector, respectively.

where PL and PU are lower bound vector and upper bound vector; PM and PR are midpoint vector and radius vector, respectively.

According to continuum mechanics [18,19], the

bound vector to an unit vector (e.g., [[bar.e].sub.1]) by [??] tensor is defined as [[bar.[tau]].sub.1] = [??] x [[bar.e].sub.1].

Each prior shape is described by a normalized set of bound vectors (i.e., the length of each bound vector is 1), whose initial points are discrete points on the boundary of an object model, and directions are identical to normal vectors of the corresponding initial points with respect to the object boundary.

It is simple in terms of computation and it can express significant structure of the object, using a set of bound vectors. Each bound vector consists of a boundary point, and its normal vector.

The objective function f measures the summation of the averages of cosine similarity between each bound vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the transformed model and bound vectors, whose initial points are in its neighboring circle.

Derived sets of bound vectors, then, are normalized into the unit square, and stored into a database as shape models.

The bound vector to [??] by the tensor [6] is vector [[bar.[GAMMA]].sub.v] defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Figure 1 provides a simple geometrical interpretation: A relevant property of Cauchy quadric (defined by [[sigma].sub.v] = const) [7] is that its normal at every point is collinear with bound vector [[bar.[GAMMA]].sub.v] for any direction [??].

From the above analysis, it yields the choice of the reciprocal local base (P, [[bar.e].sup.j]/{j = 1, 2, 3}), that consists of vectors collinear with the bound vectors to directions [[??].sub.1], [[??].sub.2], [[??].sub.3] and defined as:

In order to decode informatiom we have to use the operation of unbinding--it is the inverse (an exact inverse or a pseudo-inverse) of binding enabling us to extract an information from a complex statement, provided that we have one of the

bound vectors or a very similar vector as a cue.

Thus, the ideal and nadir points are the best possible lower and upper

bound vectors for the nondominated set.