Shape-space models

In practice, it is very desirable to distinguish between the spline-vector Q ∈ SQ that describes the basic shape of an object and the shape-vector which we denote X ∈ S, where S is a shape-space. Whereas SQ is a vector space of B-splines and has dimension NQ = 2N _B , the shape-space S _X is constructed from an underlying vector space of dimension N _X which is typically considerably smaller than NQ. The shape-space is a linear parameterisation of the set of allowed deformations of a base curve. The necessity for the distinction is made clear in figure 4.1. To obtain a spline that does justice to the geometric complexity of the face shape, thirteen control points have been used. However, if all of the resulting 26 degrees of freedom of the spline-vector Q are manipulated arbitrarily, many uninteresting shapes are generated that are not at all reminiscent of faces. Restricting the displacements of control points to a lower- dimensional shape-space is more meaningful if it preserves the face-like quality of the shape. Conversely, using the unconstrained control-vector Q leads to unstable active contours and this was illustrated in figure 2.4 on page 31.