The Shape Index

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

$$\mathcal{H}= \left(\begin{array}{cc} -\left(\frac{\partial \mathb... ...)_x & -\left(\frac{\partial \mathbf n}{\partial y}\right)_y \end{array} \right)$$ (3)

where $\left(\cdots\right)_x$ and $\left(\cdots\right)_y$ denote the x and y components of the parenthesized vector respectively.

The principal curvatures of the surface are the eigenvalues of the Hessian matrix, found by solving $\left\vert\mathcal{H}-\kappa\mathbf I\right\vert = 0$ for $\kappa$, 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

$$s = \frac{2}{\pi}\arctan{\frac{\kappa_2 + \kappa_1}{\kappa_2 - \kappa_1}} \qquad\qquad \kappa_1\geq\kappa_2$$ (4)

and may be expressed in terms of surface normals thus

$$s = \frac{2}{\pi}\arctan{\frac{\left(\frac{\partial \mathbf n... ...ght)_y \left(\frac{\partial \mathbf n}{\partial y}\right)_x}}}$$ (5)

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.

  

Figure: The shape index scale ranges from -1 to 1 as shown. The shape index values are encoded as a continuous range of grey-level values between 1 and 255, with grey-level 0 being reserved for background and flat regions (for which the shape index is undefined).