Gaussian (K) and Mean (H) curvatures are the most widely used indicators for surface shape classification in range image analysis. The HK segmentation[1,2,4] was introduced by Besl in 1986. He defines Gaussian and Mean curvatures, which are calculated from the two principal curvatures and . The Gaussian curvature equals the product of the principal curvatures. The Mean curvature equals the arithmetic average of the principal curvatures.
Image points can be labelled as belonging to a viewpoint- independent surface shape class type based on the combination of the signs from the Gaussian and Mean curvatures as shown in Table 1.
Table 1: Classification for the HK segmentation based on the sign H and K
Koenderink defined an alternative curvature representation. His approach (SC classification) decouples the shape and the magnitude of the curvedness. The surface in terms of relative curvature remains invariant under changes in scale. He defined a shape index S, which is a number in the range [-1,1]. The index covers all shapes except for the planar shape which has an indeterminate shape index (). The shape index provides a continuous gradation between shapes, such as concave shapes (-1 < S < -1/2), hyperboloid shapes (-1/2 < S < 1/2) and convex shapes (1/2 < S < 1). The image points can be classified as shown in Table 2. We use the positive principal curvatures ( for convex objects. Beside the shape index, Koenderink introduced the positive value C for describing the magnitude of the curvedness at a point. It is a measure of how highly or gently curved a point is. At a point that has no curvedness the value becomes zero. Therefore, this variable can be used to recognise a plane surface.
Table 2: Classification for Koenderink's approach based on the shape index (S)