The differential structure of a surface is captured by the local Hessian matrix, which may be approximated in terms of surface normals by

(3) |

where and denote the

The principal curvatures of the surface are the eigenvalues of the Hessian matrix, found by solving
for ,
where **I** is the identity matrix.
Koenderink and van Doorn[19] developed a single-value, angular measure to describe local surface topology in terms of the principal curvatures. This *shape index* is defined as

(4) |

and may be expressed in terms of surface normals thus

Figure 1 shows the range of shape index values, the type of curvature which they represent, and the grey-levels used to display different shape-index values. Dark regions correspond to concavities, such as ruts, troughs and spherical caps, whilst light regions indicate caps, domes and ridges.